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Theorem rngohomadd 26600
Description: Ring homomorphisms preserve addition. (Contributed by Jeff Madsen, 3-Jan-2011.)
Hypotheses
Ref Expression
rnghomadd.1  |-  G  =  ( 1st `  R
)
rnghomadd.2  |-  X  =  ran  G
rnghomadd.3  |-  J  =  ( 1st `  S
)
Assertion
Ref Expression
rngohomadd  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  ( A G B ) )  =  ( ( F `
 A ) J ( F `  B
) ) )

Proof of Theorem rngohomadd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnghomadd.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
2 eqid 2283 . . . . . . 7  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 rnghomadd.2 . . . . . . 7  |-  X  =  ran  G
4 eqid 2283 . . . . . . 7  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
5 rnghomadd.3 . . . . . . 7  |-  J  =  ( 1st `  S
)
6 eqid 2283 . . . . . . 7  |-  ( 2nd `  S )  =  ( 2nd `  S )
7 eqid 2283 . . . . . . 7  |-  ran  J  =  ran  J
8 eqid 2283 . . . . . . 7  |-  (GId `  ( 2nd `  S ) )  =  (GId `  ( 2nd `  S ) )
91, 2, 3, 4, 5, 6, 7, 8isrngohom 26596 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngHom  S )  <->  ( F : X --> ran  J  /\  ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId
`  ( 2nd `  S
) )  /\  A. x  e.  X  A. y  e.  X  (
( F `  (
x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  /\  ( F `  ( x ( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) ) ) ) )
109biimpa 470 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F : X --> ran  J  /\  ( F `
 (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) )  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x
( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) ) ) )
1110simp3d 969 . . . 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
) J ( F `
 y ) )  /\  ( F `  ( x ( 2nd `  R ) y ) )  =  ( ( F `  x ) ( 2nd `  S
) ( F `  y ) ) ) )
12113impa 1146 . . 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 ) J ( F `  y
) )  /\  ( F `  ( x
( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) ) )
13 simpl 443 . . . . 5  |-  ( ( ( F `  (
x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  /\  ( F `  ( x ( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) J ( F `  y ) ) )
1413ralimi 2618 . . . 4  |-  ( A. y  e.  X  (
( F `  (
x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  /\  ( F `  ( x ( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) )  ->  A. y  e.  X  ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) ) )
1514ralimi 2618 . . 3  |-  ( A. x  e.  X  A. y  e.  X  (
( F `  (
x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  /\  ( F `  ( x ( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) )  ->  A. x  e.  X  A. y  e.  X  ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) ) )
1612, 15syl 15 . 2  |-  ( ( 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
) J ( F `
 y ) ) )
17 oveq1 5865 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
1817fveq2d 5529 . . . 4  |-  ( x  =  A  ->  ( F `  ( x G y ) )  =  ( F `  ( A G y ) ) )
19 fveq2 5525 . . . . 5  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
2019oveq1d 5873 . . . 4  |-  ( x  =  A  ->  (
( F `  x
) J ( F `
 y ) )  =  ( ( F `
 A ) J ( F `  y
) ) )
2118, 20eqeq12d 2297 . . 3  |-  ( x  =  A  ->  (
( F `  (
x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  <->  ( F `  ( A G y ) )  =  ( ( F `  A
) J ( F `
 y ) ) ) )
22 oveq2 5866 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
2322fveq2d 5529 . . . 4  |-  ( y  =  B  ->  ( F `  ( A G y ) )  =  ( F `  ( A G B ) ) )
24 fveq2 5525 . . . . 5  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
2524oveq2d 5874 . . . 4  |-  ( y  =  B  ->  (
( F `  A
) J ( F `
 y ) )  =  ( ( F `
 A ) J ( F `  B
) ) )
2623, 25eqeq12d 2297 . . 3  |-  ( y  =  B  ->  (
( F `  ( A G y ) )  =  ( ( F `
 A ) J ( F `  y
) )  <->  ( F `  ( A G B ) )  =  ( ( F `  A
) J ( F `
 B ) ) ) )
2721, 26rspc2v 2890 . 2  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  ( F `  ( x G y ) )  =  ( ( F `  x
) J ( F `
 y ) )  ->  ( F `  ( A G B ) )  =  ( ( F `  A ) J ( F `  B ) ) ) )
2816, 27mpan9 455 1  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  ( A G B ) )  =  ( ( F `
 A ) J ( F `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   RingOpscrngo 21042    RngHom crnghom 26591
This theorem is referenced by:  rngogrphom  26602  rngohomco  26605  rngoisocnv  26612  keridl  26657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-rngohom 26594
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