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Theorem rngohommul 28774
Description: Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.)
Hypotheses
Ref Expression
rnghommul.1  |-  G  =  ( 1st `  R
)
rnghommul.2  |-  X  =  ran  G
rnghommul.3  |-  H  =  ( 2nd `  R
)
rnghommul.4  |-  K  =  ( 2nd `  S
)
Assertion
Ref Expression
rngohommul  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  ( A H B ) )  =  ( ( F `
 A ) K ( F `  B
) ) )

Proof of Theorem rngohommul
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnghommul.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
2 rnghommul.3 . . . . . . 7  |-  H  =  ( 2nd `  R
)
3 rnghommul.2 . . . . . . 7  |-  X  =  ran  G
4 eqid 2442 . . . . . . 7  |-  (GId `  H )  =  (GId
`  H )
5 eqid 2442 . . . . . . 7  |-  ( 1st `  S )  =  ( 1st `  S )
6 rnghommul.4 . . . . . . 7  |-  K  =  ( 2nd `  S
)
7 eqid 2442 . . . . . . 7  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
8 eqid 2442 . . . . . . 7  |-  (GId `  K )  =  (GId
`  K )
91, 2, 3, 4, 5, 6, 7, 8isrngohom 28769 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngHom  S )  <->  ( F : X --> ran  ( 1st `  S )  /\  ( F `  (GId `  H
) )  =  (GId
`  K )  /\  A. x  e.  X  A. y  e.  X  (
( F `  (
x G y ) )  =  ( ( F `  x ) ( 1st `  S
) ( F `  y ) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) ) )
109biimpa 484 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F : X --> ran  ( 1st `  S
)  /\  ( F `  (GId `  H )
)  =  (GId `  K )  /\  A. x  e.  X  A. y  e.  X  (
( F `  (
x G y ) )  =  ( ( F `  x ) ( 1st `  S
) ( F `  y ) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) )
1110simp3d 1002 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `  x
) ( 1st `  S
) ( F `  y ) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) )
12113impa 1182 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) ( 1st `  S ) ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) )
13 simpr 461 . . . . 5  |-  ( ( ( F `  (
x G y ) )  =  ( ( F `  x ) ( 1st `  S
) ( F `  y ) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) )  -> 
( F `  (
x H y ) )  =  ( ( F `  x ) K ( F `  y ) ) )
1413ralimi 2790 . . . 4  |-  ( A. y  e.  X  (
( F `  (
x G y ) )  =  ( ( F `  x ) ( 1st `  S
) ( F `  y ) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) )  ->  A. y  e.  X  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) )
1514ralimi 2790 . . 3  |-  ( A. x  e.  X  A. y  e.  X  (
( F `  (
x G y ) )  =  ( ( F `  x ) ( 1st `  S
) ( F `  y ) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) )  ->  A. x  e.  X  A. y  e.  X  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) )
1612, 15syl 16 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  A. x  e.  X  A. y  e.  X  ( F `  ( x H y ) )  =  ( ( F `  x
) K ( F `
 y ) ) )
17 oveq1 6097 . . . . 5  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
1817fveq2d 5694 . . . 4  |-  ( x  =  A  ->  ( F `  ( x H y ) )  =  ( F `  ( A H y ) ) )
19 fveq2 5690 . . . . 5  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
2019oveq1d 6105 . . . 4  |-  ( x  =  A  ->  (
( F `  x
) K ( F `
 y ) )  =  ( ( F `
 A ) K ( F `  y
) ) )
2118, 20eqeq12d 2456 . . 3  |-  ( x  =  A  ->  (
( F `  (
x H y ) )  =  ( ( F `  x ) K ( F `  y ) )  <->  ( F `  ( A H y ) )  =  ( ( F `  A
) K ( F `
 y ) ) ) )
22 oveq2 6098 . . . . 5  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
2322fveq2d 5694 . . . 4  |-  ( y  =  B  ->  ( F `  ( A H y ) )  =  ( F `  ( A H B ) ) )
24 fveq2 5690 . . . . 5  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
2524oveq2d 6106 . . . 4  |-  ( y  =  B  ->  (
( F `  A
) K ( F `
 y ) )  =  ( ( F `
 A ) K ( F `  B
) ) )
2623, 25eqeq12d 2456 . . 3  |-  ( y  =  B  ->  (
( F `  ( A H y ) )  =  ( ( F `
 A ) K ( F `  y
) )  <->  ( F `  ( A H B ) )  =  ( ( F `  A
) K ( F `
 B ) ) ) )
2721, 26rspc2v 3078 . 2  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( F `  ( x H y ) )  =  ( ( F `  x
) K ( F `
 y ) )  ->  ( F `  ( A H B ) )  =  ( ( F `  A ) K ( F `  B ) ) ) )
2816, 27mpan9 469 1  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  ( A H B ) )  =  ( ( F `
 A ) K ( F `  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2714   ran crn 4840   -->wf 5413   ` cfv 5417  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575  GIdcgi 23673   RingOpscrngo 23861    RngHom crnghom 28764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7215  df-rngohom 28767
This theorem is referenced by:  rngohomco  28778  rngoisocnv  28785  crngohomfo  28804  keridl  28830
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