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Theorem rngohomval 30335
 Description: The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
rnghomval.1
rnghomval.2
rnghomval.3
rnghomval.4 GId
rnghomval.5
rnghomval.6
rnghomval.7
rnghomval.8 GId
Assertion
Ref Expression
rngohomval
Distinct variable groups:   ,,   ,   ,   ,   ,,   ,   ,,,   ,,,   ,,,   ,   ,
Allowed substitution hints:   (,)   (,)   (,)   (,)   (,)   (,)   ()

Proof of Theorem rngohomval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . . . . 8
21fveq2d 5856 . . . . . . 7
3 rnghomval.5 . . . . . . 7
42, 3syl6eqr 2500 . . . . . 6
54rneqd 5216 . . . . 5
6 rnghomval.7 . . . . 5
75, 6syl6eqr 2500 . . . 4
8 simpl 457 . . . . . . . 8
98fveq2d 5856 . . . . . . 7
10 rnghomval.1 . . . . . . 7
119, 10syl6eqr 2500 . . . . . 6
1211rneqd 5216 . . . . 5
13 rnghomval.3 . . . . 5
1412, 13syl6eqr 2500 . . . 4
157, 14oveq12d 6295 . . 3
168fveq2d 5856 . . . . . . . . 9
17 rnghomval.2 . . . . . . . . 9
1816, 17syl6eqr 2500 . . . . . . . 8
1918fveq2d 5856 . . . . . . 7 GId GId
20 rnghomval.4 . . . . . . 7 GId
2119, 20syl6eqr 2500 . . . . . 6 GId
2221fveq2d 5856 . . . . 5 GId
231fveq2d 5856 . . . . . . . 8
24 rnghomval.6 . . . . . . . 8
2523, 24syl6eqr 2500 . . . . . . 7
2625fveq2d 5856 . . . . . 6 GId GId
27 rnghomval.8 . . . . . 6 GId
2826, 27syl6eqr 2500 . . . . 5 GId
2922, 28eqeq12d 2463 . . . 4 GId GId
3011oveqd 6294 . . . . . . . . 9
3130fveq2d 5856 . . . . . . . 8
324oveqd 6294 . . . . . . . 8
3331, 32eqeq12d 2463 . . . . . . 7
3418oveqd 6294 . . . . . . . . 9
3534fveq2d 5856 . . . . . . . 8
3625oveqd 6294 . . . . . . . 8
3735, 36eqeq12d 2463 . . . . . . 7
3833, 37anbi12d 710 . . . . . 6
3914, 38raleqbidv 3052 . . . . 5
4014, 39raleqbidv 3052 . . . 4
4129, 40anbi12d 710 . . 3 GId GId
4215, 41rabeqbidv 3088 . 2 GId GId
43 df-rngohom 30334 . 2 GId GId
44 ovex 6305 . . 3
4544rabex 4584 . 2
4642, 43, 45ovmpt2a 6414 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1381   wcel 1802  wral 2791  crab 2795   crn 4986  cfv 5574  (class class class)co 6277  c1st 6779  c2nd 6780   cmap 7418  GIdcgi 25054  crngo 25242   crnghom 30331 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-rngohom 30334 This theorem is referenced by:  isrngohom  30336
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