c. 1200, preove "evidence and argumentation to establish the fact of (something) beyond reasonable doubt," from Anglo-French prove, preove, Old French proeve, prueve "proof, test, experience" (13c., Modern French preuve), from Late Latin proba "a proof," a back-formation from Latin probare "to prove" (see prove). "The devocalization of v to f ensued upon the loss of final e; cf. the relation of v and f in believe, belief, relieve, relief, behove, behoof, etc." [OED].
The meaning "act of proving" is early 14c. The meaning "act of testing or making trial of anything" is from late 14c., from influence of prove. Meaning "standard of strength of distilled liquor" is from 1705, on the notion of "having been tested as to degree of strength." The use in photography is from 1855. The typographical sense of "trial impression to test type" is from c. 1600. The numismatic sense of "coin struck to test a die" is from 1762; now mostly in reference to coins struck from highly polished dies, mainly for collectors.
The adjectival sense "impenetrable, able to resist" (as in proof against) is recorded from 1590s, from the noun in expressions such as proof of (mid-15c.), hence extended senses involving "of tested power against" in compounds such as fireproof (1630s), waterproof (1725), fool-proof (1902), etc. Shakespeare has shame-proof. Expression the proof is in the pudding (1915) is a curious perversion of earlier proof of the pudding shall be in the eating (1708), with proof in the sense "quality of proving good or turning out well" (17c.).