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Theorem crngocom 30228
 Description: The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.)
Hypotheses
Ref Expression
crngocom.1
crngocom.2
crngocom.3
Assertion
Ref Expression
crngocom CRingOps

Proof of Theorem crngocom
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crngocom.1 . . . . 5
2 crngocom.2 . . . . 5
3 crngocom.3 . . . . 5
41, 2, 3iscrngo2 30225 . . . 4 CRingOps
54simprbi 464 . . 3 CRingOps
6 oveq1 6292 . . . . 5
7 oveq2 6293 . . . . 5
86, 7eqeq12d 2489 . . . 4
9 oveq2 6293 . . . . 5
10 oveq1 6292 . . . . 5
119, 10eqeq12d 2489 . . . 4
128, 11rspc2v 3223 . . 3
135, 12mpan9 469 . 2 CRingOps
14133impb 1192 1 CRingOps
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 973   wceq 1379   wcel 1767  wral 2814   crn 5000  cfv 5588  (class class class)co 6285  c1st 6783  c2nd 6784  crngo 25150  CRingOpsccring 30222 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-1st 6785  df-2nd 6786  df-rngo 25151  df-com2 25186  df-crngo 30223 This theorem is referenced by:  crngm23  30229  crngohomfo  30233  isidlc  30242  dmncan2  30304
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