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late Old English prud, prute "excellent, splendid; arrogant, haughty," probably from Old French prud, oblique case of adjective prouz "brave, valiant" (11c., Modern French preux; cf. prud'homme "brave man"), from Late Latin prode "advantageous, profitable" (cf. Italian prode "valiant"), a back-formation from Latin prodesse "be useful," from pro- "before, for, instead of" (see pro-) + esse "to be" (see essence). Also cf. pride (n.), prowess.
Meaning "elated by some act, fact, or thing" is from mid-13c. To do (someone) proud attested by 1819. Related: Proudness. "The -d- in prodesse is probably due to the influence of forms like red-eo-, 'I go back,' red-imo- 'I buy back,' etc." [OED]. The Old English form with -te probably is from or influenced by pride.
The sense of "have a high opinion of oneself," not found in Old French, might reflect the Anglo-Saxons' opinion of the Norman knights who called themselves "proud." Old Norse pruðr, probably from the same French source, had only the sense "brave, gallant, magnificent, stately" (cf. Icelandic pruður, Middle Swedish prudh, Middle Danish prud). Likewise a group of "pride" words in the Romance languages -- e.g. French orgueil, Italian orgoglio, Spanish orgullo -- are borrowings from Germanic, where they had positive senses (cf. Old High German urgol "distinguished").
Most Indo-European languages use the same word for "proud" in its good and bad senses, but in many the bad sense seems to be the earlier one. The usual way to form the word is by some compound of terms for "over" or "high" and words for "heart," "mood," "thought," or "appearance;" e.g. Greek hyperephanos, literally "over-appearing;" Gothic hauhþuhts, literally "high-conscience." Old English had ofermodig "over-moody" ("mood" in Anglo-Saxon was a much more potent word than presently) and heahheort "high-heart." Words for "proud" in other Indo-European languages sometimes reflect a physical sense of being swollen or puffed up; cf. Welsh balch, probably from a root meaning "to swell," and Modern Greek kamari, from ancient Greek kamarou "furnish with a vault or arched cover," with a sense evolution via "make an arch," to "puff out the chest," to "be puffed up" (cf. English slang chesty).