Theorem rngohomval 33763

Description: The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.)

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
rngohomval	|- ( ( 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 ) ) ) ) } )

Distinct variable groups:   x, f, y   f, G   f, H   f, J   f, Y, y   f, K   R, f, x, y   S, f, x, y   f, X, x, y   U, f   f, V

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 rngohomval

Dummy variables r s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> s = S )
2	1	fveq2d 6195	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( 1st ` s ) = ( 1st ` S ) )
3		rnghomval.5	. . . . . . 7 |- J = ( 1st ` S )
4	2, 3	syl6eqr 2674	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( 1st ` s ) = J )
5	4	rneqd 5353	. . . . 5 |- ( ( r = R /\ s = S ) -> ran ( 1st ` s ) = ran J )
6		rnghomval.7	. . . . 5 |- Y = ran J
7	5, 6	syl6eqr 2674	. . . 4 |- ( ( r = R /\ s = S ) -> ran ( 1st ` s ) = Y )
8		simpl 473	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> r = R )
9	8	fveq2d 6195	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( 1st ` r ) = ( 1st ` R ) )
10		rnghomval.1	. . . . . . 7 |- G = ( 1st ` R )
11	9, 10	syl6eqr 2674	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( 1st ` r ) = G )
12	11	rneqd 5353	. . . . 5 |- ( ( r = R /\ s = S ) -> ran ( 1st ` r ) = ran G )
13		rnghomval.3	. . . . 5 |- X = ran G
14	12, 13	syl6eqr 2674	. . . 4 |- ( ( r = R /\ s = S ) -> ran ( 1st ` r ) = X )
15	7, 14	oveq12d 6668	. . 3 |- ( ( r = R /\ s = S ) -> ( ran ( 1st ` s ) ^m ran ( 1st ` r ) ) = ( Y ^m X ) )
16	8	fveq2d 6195	. . . . . . . . 9 |- ( ( r = R /\ s = S ) -> ( 2nd ` r ) = ( 2nd ` R ) )
17		rnghomval.2	. . . . . . . . 9 |- H = ( 2nd ` R )
18	16, 17	syl6eqr 2674	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> ( 2nd ` r ) = H )
19	18	fveq2d 6195	. . . . . . 7 |- ( ( r = R /\ s = S ) -> (GId ` ( 2nd ` r ) ) = (GId ` H ) )
20		rnghomval.4	. . . . . . 7 |- U = (GId ` H )
21	19, 20	syl6eqr 2674	. . . . . 6 |- ( ( r = R /\ s = S ) -> (GId ` ( 2nd ` r ) ) = U )
22	21	fveq2d 6195	. . . . 5 |- ( ( r = R /\ s = S ) -> ( f ` (GId ` ( 2nd ` r ) ) ) = ( f ` U ) )
23	1	fveq2d 6195	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> ( 2nd ` s ) = ( 2nd ` S ) )
24		rnghomval.6	. . . . . . . 8 |- K = ( 2nd ` S )
25	23, 24	syl6eqr 2674	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( 2nd ` s ) = K )
26	25	fveq2d 6195	. . . . . 6 |- ( ( r = R /\ s = S ) -> (GId ` ( 2nd ` s ) ) = (GId ` K ) )
27		rnghomval.8	. . . . . 6 |- V = (GId ` K )
28	26, 27	syl6eqr 2674	. . . . 5 |- ( ( r = R /\ s = S ) -> (GId ` ( 2nd ` s ) ) = V )
29	22, 28	eqeq12d 2637	. . . 4 |- ( ( r = R /\ s = S ) -> ( ( f ` (GId ` ( 2nd ` r ) ) ) = (GId ` ( 2nd ` s ) ) <-> ( f ` U ) = V ) )
30	11	oveqd 6667	. . . . . . . . 9 |- ( ( r = R /\ s = S ) -> ( x ( 1st ` r ) y ) = ( x G y ) )
31	30	fveq2d 6195	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> ( f ` ( x ( 1st ` r ) y ) ) = ( f ` ( x G y ) ) )
32	4	oveqd 6667	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> ( ( f ` x ) ( 1st ` s ) ( f ` y ) ) = ( ( f ` x ) J ( f ` y ) ) )
33	31, 32	eqeq12d 2637	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( ( f ` ( x ( 1st ` r ) y ) ) = ( ( f ` x ) ( 1st ` s ) ( f ` y ) ) <-> ( f ` ( x G y ) ) = ( ( f ` x ) J ( f ` y ) ) ) )
34	18	oveqd 6667	. . . . . . . . 9 |- ( ( r = R /\ s = S ) -> ( x ( 2nd ` r ) y ) = ( x H y ) )
35	34	fveq2d 6195	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> ( f ` ( x ( 2nd ` r ) y ) ) = ( f ` ( x H y ) ) )
36	25	oveqd 6667	. . . . . . . 8 |- ( ( r = R /\ s = S ) -> ( ( f ` x ) ( 2nd ` s ) ( f ` y ) ) = ( ( f ` x ) K ( f ` y ) ) )
37	35, 36	eqeq12d 2637	. . . . . . 7 |- ( ( r = R /\ s = S ) -> ( ( f ` ( x ( 2nd ` r ) y ) ) = ( ( f ` x ) ( 2nd ` s ) ( f ` y ) ) <-> ( f ` ( x H y ) ) = ( ( f ` x ) K ( f ` y ) ) ) )
38	33, 37	anbi12d 747	. . . . . 6 |- ( ( r = R /\ s = S ) -> ( ( ( f ` ( x ( 1st ` r ) y ) ) = ( ( f ` x ) ( 1st ` s ) ( f ` y ) ) /\ ( f ` ( x ( 2nd ` r ) y ) ) = ( ( f ` x ) ( 2nd ` s ) ( f ` y ) ) ) <-> ( ( f ` ( x G y ) ) = ( ( f ` x ) J ( f ` y ) ) /\ ( f ` ( x H y ) ) = ( ( f ` x ) K ( f ` y ) ) ) ) )
39	14, 38	raleqbidv 3152	. . . . 5 |- ( ( r = R /\ s = S ) -> ( A. y e. ran ( 1st ` r ) ( ( f ` ( x ( 1st ` r ) y ) ) = ( ( f ` x ) ( 1st ` s ) ( 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 ) ) /\ ( f ` ( x H y ) ) = ( ( f ` x ) K ( f ` y ) ) ) ) )
40	14, 39	raleqbidv 3152	. . . 4 |- ( ( r = R /\ s = S ) -> ( A. x e. ran ( 1st ` r ) A. y e. ran ( 1st ` r ) ( ( f ` ( x ( 1st ` r ) y ) ) = ( ( f ` x ) ( 1st ` s ) ( 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 ) ) /\ ( f ` ( x H y ) ) = ( ( f ` x ) K ( f ` y ) ) ) ) )
41	29, 40	anbi12d 747	. . 3 |- ( ( r = R /\ s = S ) -> ( ( ( f ` (GId ` ( 2nd ` r ) ) ) = (GId ` ( 2nd ` s ) ) /\ A. x e. ran ( 1st ` r ) A. y e. ran ( 1st ` r ) ( ( f ` ( x ( 1st ` r ) y ) ) = ( ( f ` x ) ( 1st ` s ) ( f ` y ) ) /\ ( f ` ( x ( 2nd ` r ) y ) ) = ( ( f ` x ) ( 2nd ` s ) ( 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 ) ) ) ) ) )
42	15, 41	rabeqbidv 3195	. 2 |- ( ( r = R /\ s = S ) -> { f e. ( ran ( 1st ` s ) ^m ran ( 1st ` r ) ) | ( ( f ` (GId ` ( 2nd ` r ) ) ) = (GId ` ( 2nd ` s ) ) /\ A. x e. ran ( 1st ` r ) A. y e. ran ( 1st ` r ) ( ( f ` ( x ( 1st ` r ) y ) ) = ( ( f ` x ) ( 1st ` s ) ( f ` y ) ) /\ ( f ` ( x ( 2nd ` r ) y ) ) = ( ( f ` x ) ( 2nd ` s ) ( 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 ) ) ) ) } )
43		df-rngohom 33762	. 2 |- RngHom = ( r e. RingOps , s e. RingOps |-> { f e. ( ran ( 1st ` s ) ^m ran ( 1st ` r ) ) | ( ( f ` (GId ` ( 2nd ` r ) ) ) = (GId ` ( 2nd ` s ) ) /\ A. x e. ran ( 1st ` r ) A. y e. ran ( 1st ` r ) ( ( f ` ( x ( 1st ` r ) y ) ) = ( ( f ` x ) ( 1st ` s ) ( f ` y ) ) /\ ( f ` ( x ( 2nd ` r ) y ) ) = ( ( f ` x ) ( 2nd ` s ) ( f ` y ) ) ) ) } )
44		ovex 6678	. . 3 |- ( Y ^m X ) e. _V
45	44	rabex 4813	. 2 |- { 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 ) ) ) ) } e. _V
46	42, 43, 45	ovmpt2a 6791	1 |- ( ( 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 ) ) ) ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   ran crn 5115   ` 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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rngohom 33762

This theorem is referenced by:  isrngohom  33764