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Theorem rngohom1 26584
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 2436 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
2 rnghom1.1 . . . . 5  |-  H  =  ( 2nd `  R
)
3 eqid 2436 . . . . 5  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
4 rnghom1.2 . . . . 5  |-  U  =  (GId `  H )
5 eqid 2436 . . . . 5  |-  ( 1st `  S )  =  ( 1st `  S )
6 rnghom1.3 . . . . 5  |-  K  =  ( 2nd `  S
)
7 eqid 2436 . . . . 5  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
8 rnghom1.4 . . . . 5  |-  V  =  (GId `  K )
91, 2, 3, 4, 5, 6, 7, 8isrngohom 26581 . . . 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 471 . . 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 970 . 2  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F `  U
)  =  V )
12113impa 1148 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   ran crn 4879   -->wf 5450   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348  GIdcgi 21775   RingOpscrngo 21963    RngHom crnghom 26576
This theorem is referenced by:  rngohomco  26590  rngoisocnv  26597
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-rngohom 26579
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