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Theorem rngohommul 26578
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 2436 . . . . . . 7  |-  (GId `  H )  =  (GId
`  H )
5 eqid 2436 . . . . . . 7  |-  ( 1st `  S )  =  ( 1st `  S )
6 rnghommul.4 . . . . . . 7  |-  K  =  ( 2nd `  S
)
7 eqid 2436 . . . . . . 7  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
8 eqid 2436 . . . . . . 7  |-  (GId `  K )  =  (GId
`  K )
91, 2, 3, 4, 5, 6, 7, 8isrngohom 26573 . . . . . 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 471 . . . . 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 971 . . . 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 1148 . . 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 448 . . . . 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 2774 . . . 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 2774 . . 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 6081 . . . . 5  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
1817fveq2d 5725 . . . 4  |-  ( x  =  A  ->  ( F `  ( x H y ) )  =  ( F `  ( A H y ) ) )
19 fveq2 5721 . . . . 5  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
2019oveq1d 6089 . . . 4  |-  ( x  =  A  ->  (
( F `  x
) K ( F `
 y ) )  =  ( ( F `
 A ) K ( F `  y
) ) )
2118, 20eqeq12d 2450 . . 3  |-  ( x  =  A  ->  (
( F `  (
x H y ) )  =  ( ( F `  x ) K ( F `  y ) )  <->  ( F `  ( A H y ) )  =  ( ( F `  A
) K ( F `
 y ) ) ) )
22 oveq2 6082 . . . . 5  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
2322fveq2d 5725 . . . 4  |-  ( y  =  B  ->  ( F `  ( A H y ) )  =  ( F `  ( A H B ) ) )
24 fveq2 5721 . . . . 5  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
2524oveq2d 6090 . . . 4  |-  ( y  =  B  ->  (
( F `  A
) K ( F `
 y ) )  =  ( ( F `
 A ) K ( F `  B
) ) )
2623, 25eqeq12d 2450 . . 3  |-  ( y  =  B  ->  (
( F `  ( A H y ) )  =  ( ( F `
 A ) K ( F `  y
) )  <->  ( F `  ( A H B ) )  =  ( ( F `  A
) K ( F `
 B ) ) ) )
2721, 26rspc2v 3051 . 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 456 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2698   ran crn 4872   -->wf 5443   ` cfv 5447  (class class class)co 6074   1stc1st 6340   2ndc2nd 6341  GIdcgi 21768   RingOpscrngo 21956    RngHom crnghom 26568
This theorem is referenced by:  rngohomco  26582  rngoisocnv  26589  crngohomfo  26608  keridl  26634
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 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
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 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-map 7013  df-rngohom 26571
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