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Theorem rngohommul 31655
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 2402 . . . . . . 7  |-  (GId `  H )  =  (GId
`  H )
5 eqid 2402 . . . . . . 7  |-  ( 1st `  S )  =  ( 1st `  S )
6 rnghommul.4 . . . . . . 7  |-  K  =  ( 2nd `  S
)
7 eqid 2402 . . . . . . 7  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
8 eqid 2402 . . . . . . 7  |-  (GId `  K )  =  (GId
`  K )
91, 2, 3, 4, 5, 6, 7, 8isrngohom 31650 . . . . . 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 482 . . . . 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 1011 . . . 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 1192 . . 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 459 . . . . 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 2797 . . . 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 2797 . . 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 17 . 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 6285 . . . . 5  |-  ( x  =  A  ->  (
x H y )  =  ( A H y ) )
1817fveq2d 5853 . . . 4  |-  ( x  =  A  ->  ( F `  ( x H y ) )  =  ( F `  ( A H y ) ) )
19 fveq2 5849 . . . . 5  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
2019oveq1d 6293 . . . 4  |-  ( x  =  A  ->  (
( F `  x
) K ( F `
 y ) )  =  ( ( F `
 A ) K ( F `  y
) ) )
2118, 20eqeq12d 2424 . . 3  |-  ( x  =  A  ->  (
( F `  (
x H y ) )  =  ( ( F `  x ) K ( F `  y ) )  <->  ( F `  ( A H y ) )  =  ( ( F `  A
) K ( F `
 y ) ) ) )
22 oveq2 6286 . . . . 5  |-  ( y  =  B  ->  ( A H y )  =  ( A H B ) )
2322fveq2d 5853 . . . 4  |-  ( y  =  B  ->  ( F `  ( A H y ) )  =  ( F `  ( A H B ) ) )
24 fveq2 5849 . . . . 5  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
2524oveq2d 6294 . . . 4  |-  ( y  =  B  ->  (
( F `  A
) K ( F `
 y ) )  =  ( ( F `
 A ) K ( F `  B
) ) )
2623, 25eqeq12d 2424 . . 3  |-  ( y  =  B  ->  (
( F `  ( A H y ) )  =  ( ( F `
 A ) K ( F `  y
) )  <->  ( F `  ( A H B ) )  =  ( ( F `  A
) K ( F `
 B ) ) ) )
2721, 26rspc2v 3169 . 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 467 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   ran crn 4824   -->wf 5565   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783  GIdcgi 25603   RingOpscrngo 25791    RngHom crnghom 31645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-map 7459  df-rngohom 31648
This theorem is referenced by:  rngohomco  31659  rngoisocnv  31666  crngohomfo  31685  keridl  31711
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