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Theorem rngohommul 29965
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 2462 . . . . . . 7  |-  (GId `  H )  =  (GId
`  H )
5 eqid 2462 . . . . . . 7  |-  ( 1st `  S )  =  ( 1st `  S )
6 rnghommul.4 . . . . . . 7  |-  K  =  ( 2nd `  S
)
7 eqid 2462 . . . . . . 7  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
8 eqid 2462 . . . . . . 7  |-  (GId `  K )  =  (GId
`  K )
91, 2, 3, 4, 5, 6, 7, 8isrngohom 29960 . . . . . 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 1005 . . . 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 1186 . . 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 2852 . . . 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 2852 . . 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 6284 . . . . 5  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
1817fveq2d 5863 . . . 4  |-  ( x  =  A  ->  ( F `  ( x H y ) )  =  ( F `  ( A H y ) ) )
19 fveq2 5859 . . . . 5  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
2019oveq1d 6292 . . . 4  |-  ( x  =  A  ->  (
( F `  x
) K ( F `
 y ) )  =  ( ( F `
 A ) K ( F `  y
) ) )
2118, 20eqeq12d 2484 . . 3  |-  ( x  =  A  ->  (
( F `  (
x H y ) )  =  ( ( F `  x ) K ( F `  y ) )  <->  ( F `  ( A H y ) )  =  ( ( F `  A
) K ( F `
 y ) ) ) )
22 oveq2 6285 . . . . 5  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
2322fveq2d 5863 . . . 4  |-  ( y  =  B  ->  ( F `  ( A H y ) )  =  ( F `  ( A H B ) ) )
24 fveq2 5859 . . . . 5  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
2524oveq2d 6293 . . . 4  |-  ( y  =  B  ->  (
( F `  A
) K ( F `
 y ) )  =  ( ( F `
 A ) K ( F `  B
) ) )
2623, 25eqeq12d 2484 . . 3  |-  ( y  =  B  ->  (
( F `  ( A H y ) )  =  ( ( F `
 A ) K ( F `  y
) )  <->  ( F `  ( A H B ) )  =  ( ( F `  A
) K ( F `
 B ) ) ) )
2721, 26rspc2v 3218 . 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 968    = wceq 1374    e. wcel 1762   A.wral 2809   ran crn 4995   -->wf 5577   ` cfv 5581  (class class class)co 6277   1stc1st 6774   2ndc2nd 6775  GIdcgi 24853   RingOpscrngo 25041    RngHom crnghom 29955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7414  df-rngohom 29958
This theorem is referenced by:  rngohomco  29969  rngoisocnv  29976  crngohomfo  29995  keridl  30021
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