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Theorem rngohom1 26599
Description: A ring homomorphism preserves  1. (Contributed by Jeff Madsen, 24-Jun-2011.)
Hypotheses
Ref Expression
rnghom1.1  |-  H  =  ( 2nd `  R
)
rnghom1.2  |-  U  =  (GId `  H )
rnghom1.3  |-  K  =  ( 2nd `  S
)
rnghom1.4  |-  V  =  (GId `  K )
Assertion
Ref Expression
rngohom1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  U )  =  V )

Proof of Theorem rngohom1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
2 rnghom1.1 . . . . 5  |-  H  =  ( 2nd `  R
)
3 eqid 2283 . . . . 5  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
4 rnghom1.2 . . . . 5  |-  U  =  (GId `  H )
5 eqid 2283 . . . . 5  |-  ( 1st `  S )  =  ( 1st `  S )
6 rnghom1.3 . . . . 5  |-  K  =  ( 2nd `  S
)
7 eqid 2283 . . . . 5  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
8 rnghom1.4 . . . . 5  |-  V  =  (GId `  K )
91, 2, 3, 4, 5, 6, 7, 8isrngohom 26596 . . . 4  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngHom  S )  <->  ( F : ran  ( 1st `  R
) --> ran  ( 1st `  S )  /\  ( F `  U )  =  V  /\  A. x  e.  ran  ( 1st `  R
) A. y  e. 
ran  ( 1st `  R
) ( ( F `
 ( x ( 1st `  R ) y ) )  =  ( ( F `  x ) ( 1st `  S ) ( F `
 y ) )  /\  ( F `  ( x H y ) )  =  ( ( F `  x
) K ( F `
 y ) ) ) ) ) )
109biimpa 470 . . 3  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F : ran  ( 1st `  R ) --> ran  ( 1st `  S
)  /\  ( F `  U )  =  V  /\  A. x  e. 
ran  ( 1st `  R
) A. y  e. 
ran  ( 1st `  R
) ( ( F `
 ( x ( 1st `  R ) y ) )  =  ( ( F `  x ) ( 1st `  S ) ( F `
 y ) )  /\  ( F `  ( x H y ) )  =  ( ( F `  x
) K ( F `
 y ) ) ) ) )
1110simp2d 968 . 2  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F `  U
)  =  V )
12113impa 1146 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  U )  =  V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   RingOpscrngo 21042    RngHom crnghom 26591
This theorem is referenced by:  rngohomco  26605  rngoisocnv  26612
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-rngohom 26594
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