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Theorem isrngohom 33764
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 33763 . . 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 2687 . 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 6201 . . . . . . . 8 |- ( 1st ` S ) e. _V
125, 11eqeltri 2697 . . . . . . 7 |- J e. _V
1312rnex 7100 . . . . . 6 |- ran J e. _V
147, 13eqeltri 2697 . . . . 5 |- Y e. _V
15 fvex 6201 . . . . . . . 8 |- ( 1st ` R ) e. _V
161, 15eqeltri 2697 . . . . . . 7 |- G e. _V
1716rnex 7100 . . . . . 6 |- ran G e. _V
183, 17eqeltri 2697 . . . . 5 |- X e. _V
1914, 18elmap 7886 . . . 4 |- ( F e. ( Y ^m X ) <-> F : X --> Y )
2019anbi1i 731 . . 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 6190 . . . . . 6 |- ( f = F -> ( f ` U ) = ( F ` U ) )
2221eqeq1d 2624 . . . . 5 |- ( f = F -> ( ( f ` U ) = V <-> ( F ` U ) = V ) )
23 fveq1 6190 . . . . . . . 8 |- ( f = F -> ( f ` ( x G y ) ) = ( F ` ( x G y ) ) )
24 fveq1 6190 . . . . . . . . 9 |- ( f = F -> ( f ` x ) = ( F ` x ) )
25 fveq1 6190 . . . . . . . . 9 |- ( f = F -> ( f ` y ) = ( F ` y ) )
2624, 25oveq12d 6668 . . . . . . . 8 |- ( f = F -> ( ( f ` x ) J ( f ` y ) ) = ( ( F ` x ) J ( F ` y ) ) )
2723, 26eqeq12d 2637 . . . . . . 7 |- ( f = F -> ( ( f ` ( x G y ) ) = ( ( f ` x ) J ( f ` y ) ) <-> ( F ` ( x G y ) ) = ( ( F ` x ) J ( F ` y ) ) ) )
28 fveq1 6190 . . . . . . . 8 |- ( f = F -> ( f ` ( x H y ) ) = ( F ` ( x H y ) ) )
2924, 25oveq12d 6668 . . . . . . . 8 |- ( f = F -> ( ( f ` x ) K ( f ` y ) ) = ( ( F ` x ) K ( F ` y ) ) )
3028, 29eqeq12d 2637 . . . . . . 7 |- ( f = F -> ( ( f ` ( x H y ) ) = ( ( f ` x ) K ( f ` y ) ) <-> ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) )
3127, 30anbi12d 747 . . . . . 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 2989 . . . . 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 747 . . . 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 3363 . . 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 1042 . . 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 292 . 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 276 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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857  GIdcgi 27344   RingOpscrngo 33693   RngHom crnghom 33759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-rngohom 33762
This theorem is referenced by:  rngohomf  33765  rngohom1  33767  rngohomadd  33768  rngohommul  33769  rngohomco  33773  rngoisocnv  33780
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