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Theorem rngoass 25180
 Description: Associative law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringi.1
ringi.2
ringi.3
Assertion
Ref Expression
rngoass

Proof of Theorem rngoass
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringi.1 . . . . . 6
2 ringi.2 . . . . . 6
3 ringi.3 . . . . . 6
41, 2, 3rngoi 25173 . . . . 5
54simprd 463 . . . 4
65simpld 459 . . 3
7 simp1 996 . . . . . 6
87ralimi 2860 . . . . 5
98ralimi 2860 . . . 4
109ralimi 2860 . . 3
116, 10syl 16 . 2
12 oveq1 6301 . . . . 5
1312oveq1d 6309 . . . 4
14 oveq1 6301 . . . 4
1513, 14eqeq12d 2489 . . 3
16 oveq2 6302 . . . . 5
1716oveq1d 6309 . . . 4
18 oveq1 6301 . . . . 5
1918oveq2d 6310 . . . 4
2017, 19eqeq12d 2489 . . 3
21 oveq2 6302 . . . 4
22 oveq2 6302 . . . . 5
2322oveq2d 6310 . . . 4
2421, 23eqeq12d 2489 . . 3
2515, 20, 24rspc3v 3231 . 2
2611, 25mpan9 469 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 973   wceq 1379   wcel 1767  wral 2817  wrex 2818   cxp 5002   crn 5005  wf 5589  cfv 5593  (class class class)co 6294  c1st 6792  c2nd 6793  cablo 25074  crngo 25168 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 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586 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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-ov 6297  df-1st 6794  df-2nd 6795  df-rngo 25169 This theorem is referenced by:  rngomndo  25214  zerdivemp1  25227  rngoneglmul  30249  rngonegrmul  30250  zerdivemp1x  30253  isdrngo2  30256  crngm23  30294  crngm4  30295  prnc  30359
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