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Theorem rngohomadd 26576
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 2435 . . . . . . 7  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 rnghomadd.2 . . . . . . 7  |-  X  =  ran  G
4 eqid 2435 . . . . . . 7  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
5 rnghomadd.3 . . . . . . 7  |-  J  =  ( 1st `  S
)
6 eqid 2435 . . . . . . 7  |-  ( 2nd `  S )  =  ( 2nd `  S )
7 eqid 2435 . . . . . . 7  |-  ran  J  =  ran  J
8 eqid 2435 . . . . . . 7  |-  (GId `  ( 2nd `  S ) )  =  (GId `  ( 2nd `  S ) )
91, 2, 3, 4, 5, 6, 7, 8isrngohom 26572 . . . . . 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 471 . . . . 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 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
) J ( F `
 y ) )  /\  ( F `  ( x ( 2nd `  R ) y ) )  =  ( ( F `  x ) ( 2nd `  S
) ( 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 ) J ( F `  y
) )  /\  ( F `  ( x
( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) ) )
13 simpl 444 . . . . 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 2773 . . . 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 2773 . . 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 16 . 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 6080 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
1817fveq2d 5724 . . . 4  |-  ( x  =  A  ->  ( F `  ( x G y ) )  =  ( F `  ( A G y ) ) )
19 fveq2 5720 . . . . 5  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
2019oveq1d 6088 . . . 4  |-  ( x  =  A  ->  (
( F `  x
) J ( F `
 y ) )  =  ( ( F `
 A ) J ( F `  y
) ) )
2118, 20eqeq12d 2449 . . 3  |-  ( x  =  A  ->  (
( F `  (
x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  <->  ( F `  ( A G y ) )  =  ( ( F `  A
) J ( F `
 y ) ) ) )
22 oveq2 6081 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
2322fveq2d 5724 . . . 4  |-  ( y  =  B  ->  ( F `  ( A G y ) )  =  ( F `  ( A G B ) ) )
24 fveq2 5720 . . . . 5  |-  ( y  =  B  ->  ( F `  y )  =  ( F `  B ) )
2524oveq2d 6089 . . . 4  |-  ( y  =  B  ->  (
( F `  A
) J ( F `
 y ) )  =  ( ( F `
 A ) J ( F `  B
) ) )
2623, 25eqeq12d 2449 . . 3  |-  ( y  =  B  ->  (
( F `  ( A G y ) )  =  ( ( F `
 A ) J ( F `  y
) )  <->  ( F `  ( A G B ) )  =  ( ( F `  A
) J ( F `
 B ) ) ) )
2721, 26rspc2v 3050 . 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 456 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340  GIdcgi 21767   RingOpscrngo 21955    RngHom crnghom 26567
This theorem is referenced by:  rngogrphom  26578  rngohomco  26581  rngoisocnv  26588  keridl  26633
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-rngohom 26570
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