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Theorem isrngohom 30336
Description: The predicate "is a ring homomorphism from  R to  S." (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
rnghomval.1  |-  G  =  ( 1st `  R
)
rnghomval.2  |-  H  =  ( 2nd `  R
)
rnghomval.3  |-  X  =  ran  G
rnghomval.4  |-  U  =  (GId `  H )
rnghomval.5  |-  J  =  ( 1st `  S
)
rnghomval.6  |-  K  =  ( 2nd `  S
)
rnghomval.7  |-  Y  =  ran  J
rnghomval.8  |-  V  =  (GId `  K )
Assertion
Ref Expression
isrngohom  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngHom  S )  <->  ( F : X --> Y  /\  ( F `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) ) )
Distinct variable groups:    x, y, F    y, Y    x, R, y    x, S, y    x, X, y
Allowed substitution hints:    U( x, y)    G( x, y)    H( x, y)    J( x, y)    K( x, y)    V( x, y)    Y( x)

Proof of Theorem isrngohom
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 rnghomval.1 . . . 4  |-  G  =  ( 1st `  R
)
2 rnghomval.2 . . . 4  |-  H  =  ( 2nd `  R
)
3 rnghomval.3 . . . 4  |-  X  =  ran  G
4 rnghomval.4 . . . 4  |-  U  =  (GId `  H )
5 rnghomval.5 . . . 4  |-  J  =  ( 1st `  S
)
6 rnghomval.6 . . . 4  |-  K  =  ( 2nd `  S
)
7 rnghomval.7 . . . 4  |-  Y  =  ran  J
8 rnghomval.8 . . . 4  |-  V  =  (GId `  K )
91, 2, 3, 4, 5, 6, 7, 8rngohomval 30335 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  RngHom  S )  =  { f  e.  ( Y  ^m  X )  |  ( ( f `
 U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( (
f `  ( x G y ) )  =  ( ( f `
 x ) J ( f `  y
) )  /\  (
f `  ( x H y ) )  =  ( ( f `
 x ) K ( f `  y
) ) ) ) } )
109eleq2d 2511 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngHom  S )  <->  F  e.  { f  e.  ( Y  ^m  X )  |  ( ( f `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( f `
 ( x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  /\  ( f `
 ( x H y ) )  =  ( ( f `  x ) K ( f `  y ) ) ) ) } ) )
11 fvex 5862 . . . . . . . 8  |-  ( 1st `  S )  e.  _V
125, 11eqeltri 2525 . . . . . . 7  |-  J  e. 
_V
1312rnex 6715 . . . . . 6  |-  ran  J  e.  _V
147, 13eqeltri 2525 . . . . 5  |-  Y  e. 
_V
15 fvex 5862 . . . . . . . 8  |-  ( 1st `  R )  e.  _V
161, 15eqeltri 2525 . . . . . . 7  |-  G  e. 
_V
1716rnex 6715 . . . . . 6  |-  ran  G  e.  _V
183, 17eqeltri 2525 . . . . 5  |-  X  e. 
_V
1914, 18elmap 7445 . . . 4  |-  ( F  e.  ( Y  ^m  X )  <->  F : X
--> Y )
2019anbi1i 695 . . 3  |-  ( ( F  e.  ( Y  ^m  X )  /\  ( ( F `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `
 ( x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  /\  ( F `
 ( x H y ) )  =  ( ( F `  x ) K ( F `  y ) ) ) ) )  <-> 
( F : X --> Y  /\  ( ( F `
 U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) ) )
21 fveq1 5851 . . . . . 6  |-  ( f  =  F  ->  (
f `  U )  =  ( F `  U ) )
2221eqeq1d 2443 . . . . 5  |-  ( f  =  F  ->  (
( f `  U
)  =  V  <->  ( F `  U )  =  V ) )
23 fveq1 5851 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  ( x G y ) )  =  ( F `  ( x G y ) ) )
24 fveq1 5851 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
25 fveq1 5851 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
2624, 25oveq12d 6295 . . . . . . . 8  |-  ( f  =  F  ->  (
( f `  x
) J ( f `
 y ) )  =  ( ( F `
 x ) J ( F `  y
) ) )
2723, 26eqeq12d 2463 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  (
x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  <->  ( F `  ( x G y ) )  =  ( ( F `  x
) J ( F `
 y ) ) ) )
28 fveq1 5851 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  ( x H y ) )  =  ( F `  ( x H y ) ) )
2924, 25oveq12d 6295 . . . . . . . 8  |-  ( f  =  F  ->  (
( f `  x
) K ( f `
 y ) )  =  ( ( F `
 x ) K ( F `  y
) ) )
3028, 29eqeq12d 2463 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  (
x H y ) )  =  ( ( f `  x ) K ( f `  y ) )  <->  ( F `  ( x H y ) )  =  ( ( F `  x
) K ( F `
 y ) ) ) )
3127, 30anbi12d 710 . . . . . 6  |-  ( f  =  F  ->  (
( ( f `  ( x G y ) )  =  ( ( f `  x
) J ( f `
 y ) )  /\  ( f `  ( x H y ) )  =  ( ( f `  x
) K ( f `
 y ) ) )  <->  ( ( F `
 ( x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  /\  ( F `
 ( x H y ) )  =  ( ( F `  x ) K ( F `  y ) ) ) ) )
32312ralbidv 2885 . . . . 5  |-  ( f  =  F  ->  ( A. x  e.  X  A. y  e.  X  ( ( f `  ( x G y ) )  =  ( ( f `  x
) J ( f `
 y ) )  /\  ( f `  ( x H y ) )  =  ( ( f `  x
) K ( f `
 y ) ) )  <->  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `  x
) J ( F `
 y ) )  /\  ( F `  ( x H y ) )  =  ( ( F `  x
) K ( F `
 y ) ) ) ) )
3322, 32anbi12d 710 . . . 4  |-  ( f  =  F  ->  (
( ( f `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( f `
 ( x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  /\  ( f `
 ( x H y ) )  =  ( ( f `  x ) K ( f `  y ) ) ) )  <->  ( ( F `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) ) )
3433elrab 3241 . . 3  |-  ( F  e.  { f  e.  ( Y  ^m  X
)  |  ( ( f `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  (
( f `  (
x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  /\  ( f `  (
x H y ) )  =  ( ( f `  x ) K ( f `  y ) ) ) ) }  <->  ( F  e.  ( Y  ^m  X
)  /\  ( ( F `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) ) )
35 3anass 976 . . 3  |-  ( ( F : X --> Y  /\  ( F `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  (
( F `  (
x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) )  <-> 
( F : X --> Y  /\  ( ( F `
 U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) ) )
3620, 34, 353bitr4i 277 . 2  |-  ( F  e.  { f  e.  ( Y  ^m  X
)  |  ( ( f `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  (
( f `  (
x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  /\  ( f `  (
x H y ) )  =  ( ( f `  x ) K ( f `  y ) ) ) ) }  <->  ( F : X --> Y  /\  ( F `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) )
3710, 36syl6bb 261 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngHom  S )  <->  ( F : X --> Y  /\  ( F `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x H y ) )  =  ( ( F `
 x ) K ( F `  y
) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   {crab 2795   _Vcvv 3093   ran crn 4986   -->wf 5570   ` cfv 5574  (class class class)co 6277   1stc1st 6779   2ndc2nd 6780    ^m cmap 7418  GIdcgi 25054   RingOpscrngo 25242    RngHom crnghom 30331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7420  df-rngohom 30334
This theorem is referenced by:  rngohomf  30337  rngohom1  30339  rngohomadd  30340  rngohommul  30341  rngohomco  30345  rngoisocnv  30352
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