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Theorem isrngohom 26535
 Description: The predicate "is a ring homomorphism from to ." (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
rnghomval.1
rnghomval.2
rnghomval.3
rnghomval.4 GId
rnghomval.5
rnghomval.6
rnghomval.7
rnghomval.8 GId
Assertion
Ref Expression
isrngohom
Distinct variable groups:   ,,   ,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)   (,)   (,)   (,)   ()

Proof of Theorem isrngohom
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rnghomval.1 . . . 4
2 rnghomval.2 . . . 4
3 rnghomval.3 . . . 4
4 rnghomval.4 . . . 4 GId
5 rnghomval.5 . . . 4
6 rnghomval.6 . . . 4
7 rnghomval.7 . . . 4
8 rnghomval.8 . . . 4 GId
91, 2, 3, 4, 5, 6, 7, 8rngohomval 26534 . . 3
109eleq2d 2502 . 2
11 fvex 5734 . . . . . . . 8
125, 11eqeltri 2505 . . . . . . 7
1312rnex 5125 . . . . . 6
147, 13eqeltri 2505 . . . . 5
15 fvex 5734 . . . . . . . 8
161, 15eqeltri 2505 . . . . . . 7
1716rnex 5125 . . . . . 6
183, 17eqeltri 2505 . . . . 5
1914, 18elmap 7034 . . . 4
2019anbi1i 677 . . 3
21 fveq1 5719 . . . . . 6
2221eqeq1d 2443 . . . . 5
23 fveq1 5719 . . . . . . . 8
24 fveq1 5719 . . . . . . . . 9
25 fveq1 5719 . . . . . . . . 9
2624, 25oveq12d 6091 . . . . . . . 8
2723, 26eqeq12d 2449 . . . . . . 7
28 fveq1 5719 . . . . . . . 8
2924, 25oveq12d 6091 . . . . . . . 8
3028, 29eqeq12d 2449 . . . . . . 7
3127, 30anbi12d 692 . . . . . 6
32312ralbidv 2739 . . . . 5
3322, 32anbi12d 692 . . . 4
3433elrab 3084 . . 3
35 3anass 940 . . 3
3620, 34, 353bitr4i 269 . 2
3710, 36syl6bb 253 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  crab 2701  cvv 2948   crn 4871  wf 5442  cfv 5446  (class class class)co 6073  c1st 6339  c2nd 6340   cmap 7010  GIdcgi 21765  crngo 21953   crnghom 26530 This theorem is referenced by:  rngohomf  26536  rngohom1  26538  rngohomadd  26539  rngohommul  26540  rngohomco  26544  rngoisocnv  26551 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-rngohom 26533
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