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Theorem rngohom1 26702
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 2296 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
2 rnghom1.1 . . . . 5  |-  H  =  ( 2nd `  R
)
3 eqid 2296 . . . . 5  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
4 rnghom1.2 . . . . 5  |-  U  =  (GId `  H )
5 eqid 2296 . . . . 5  |-  ( 1st `  S )  =  ( 1st `  S )
6 rnghom1.3 . . . . 5  |-  K  =  ( 2nd `  S
)
7 eqid 2296 . . . . 5  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
8 rnghom1.4 . . . . 5  |-  V  =  (GId `  K )
91, 2, 3, 4, 5, 6, 7, 8isrngohom 26699 . . . 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 1632    e. wcel 1696   A.wral 2556   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870   RingOpscrngo 21058    RngHom crnghom 26694
This theorem is referenced by:  rngohomco  26708  rngoisocnv  26715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-rngohom 26697
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