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Theorem isrngo 25506
 Description: The predicate "is a (unital) ring." Definition of ring with unit in [Schechter] p. 187. (Contributed by Jeff Hankins, 21-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
isring.1
Assertion
Ref Expression
isrngo
Distinct variable groups:   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem isrngo
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4457 . . . 4
2 relrngo 25505 . . . . 5
32brrelexi 5049 . . . 4
41, 3sylbir 213 . . 3
54a1i 11 . 2
6 elex 3118 . . . 4
87a1i 11 . 2
9 df-rngo 25504 . . . . 5
109eleq2i 2535 . . . 4
11 simpl 457 . . . . . . . 8
1211eleq1d 2526 . . . . . . 7
13 simpr 461 . . . . . . . 8
1411rneqd 5240 . . . . . . . . . 10
15 isring.1 . . . . . . . . . 10
1614, 15syl6eqr 2516 . . . . . . . . 9
1716sqxpeqd 5034 . . . . . . . 8
1813, 17, 16feq123d 5727 . . . . . . 7
1912, 18anbi12d 710 . . . . . 6
2013oveqd 6313 . . . . . . . . . . . . 13
21 eqidd 2458 . . . . . . . . . . . . 13
2213, 20, 21oveq123d 6317 . . . . . . . . . . . 12
23 eqidd 2458 . . . . . . . . . . . . 13
2413oveqd 6313 . . . . . . . . . . . . 13
2513, 23, 24oveq123d 6317 . . . . . . . . . . . 12
2622, 25eqeq12d 2479 . . . . . . . . . . 11
2711oveqd 6313 . . . . . . . . . . . . 13
2813, 23, 27oveq123d 6317 . . . . . . . . . . . 12
2913oveqd 6313 . . . . . . . . . . . . 13
3011, 20, 29oveq123d 6317 . . . . . . . . . . . 12
3128, 30eqeq12d 2479 . . . . . . . . . . 11
3211oveqd 6313 . . . . . . . . . . . . 13
3313, 32, 21oveq123d 6317 . . . . . . . . . . . 12
3411, 29, 24oveq123d 6317 . . . . . . . . . . . 12
3533, 34eqeq12d 2479 . . . . . . . . . . 11
3626, 31, 353anbi123d 1299 . . . . . . . . . 10
3716, 36raleqbidv 3068 . . . . . . . . 9
3816, 37raleqbidv 3068 . . . . . . . 8
3916, 38raleqbidv 3068 . . . . . . 7
4020eqeq1d 2459 . . . . . . . . . 10
4113oveqd 6313 . . . . . . . . . . 11
4241eqeq1d 2459 . . . . . . . . . 10
4340, 42anbi12d 710 . . . . . . . . 9
4416, 43raleqbidv 3068 . . . . . . . 8
4516, 44rexeqbidv 3069 . . . . . . 7
4639, 45anbi12d 710 . . . . . 6
4719, 46anbi12d 710 . . . . 5
4847opelopabga 4769 . . . 4
4910, 48syl5bb 257 . . 3
5049expcom 435 . 2
515, 8, 50pm5.21ndd 354 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 973   wceq 1395   wcel 1819  wral 2807  wrex 2808  cvv 3109  cop 4038   class class class wbr 4456  copab 4514   cxp 5006   crn 5009  wf 5590  (class class class)co 6296  cablo 25409  crngo 25503 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-rngo 25504 This theorem is referenced by:  isrngod  25507  rngoi  25508  cnrngo  25531
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