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Theorem isrngohom 26535
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 26534 . . 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 2502 . 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 5734 . . . . . . . 8  |-  ( 1st `  S )  e.  _V
125, 11eqeltri 2505 . . . . . . 7  |-  J  e. 
_V
1312rnex 5125 . . . . . 6  |-  ran  J  e.  _V
147, 13eqeltri 2505 . . . . 5  |-  Y  e. 
_V
15 fvex 5734 . . . . . . . 8  |-  ( 1st `  R )  e.  _V
161, 15eqeltri 2505 . . . . . . 7  |-  G  e. 
_V
1716rnex 5125 . . . . . 6  |-  ran  G  e.  _V
183, 17eqeltri 2505 . . . . 5  |-  X  e. 
_V
1914, 18elmap 7034 . . . 4  |-  ( F  e.  ( Y  ^m  X )  <->  F : X
--> Y )
2019anbi1i 677 . . 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 5719 . . . . . 6  |-  ( f  =  F  ->  (
f `  U )  =  ( F `  U ) )
2221eqeq1d 2443 . . . . 5  |-  ( f  =  F  ->  (
( f `  U
)  =  V  <->  ( F `  U )  =  V ) )
23 fveq1 5719 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  ( x G y ) )  =  ( F `  ( x G y ) ) )
24 fveq1 5719 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
25 fveq1 5719 . . . . . . . . 9  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
2624, 25oveq12d 6091 . . . . . . . 8  |-  ( f  =  F  ->  (
( f `  x
) J ( f `
 y ) )  =  ( ( F `
 x ) J ( F `  y
) ) )
2723, 26eqeq12d 2449 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  (
x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  <->  ( F `  ( x G y ) )  =  ( ( F `  x
) J ( F `
 y ) ) ) )
28 fveq1 5719 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  ( x H y ) )  =  ( F `  ( x H y ) ) )
2924, 25oveq12d 6091 . . . . . . . 8  |-  ( f  =  F  ->  (
( f `  x
) K ( f `
 y ) )  =  ( ( F `
 x ) K ( F `  y
) ) )
3028, 29eqeq12d 2449 . . . . . . 7  |-  ( f  =  F  ->  (
( f `  (
x H y ) )  =  ( ( f `  x ) K ( f `  y ) )  <->  ( F `  ( x H y ) )  =  ( ( F `  x
) K ( F `
 y ) ) ) )
3127, 30anbi12d 692 . . . . . 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 2739 . . . . 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 692 . . . 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 3084 . . 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 940 . . 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 269 . 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 253 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340    ^m cmap 7010  GIdcgi 21765   RingOpscrngo 21953    RngHom crnghom 26530
This theorem is referenced by:  rngohomf  26536  rngohom1  26538  rngohomadd  26539  rngohommul  26540  rngohomco  26544  rngoisocnv  26551
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 26533
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