Theorem rnghomval 16118

Description: The set of ring homomorphisms.

Hypotheses

Ref	Expression
rnghomval.1	|- G = (1st` R)
rnghomval.2	|- H = (2nd` R)
rnghomval.3	|- X = ran G
rnghomval.4	|- U = (Id` H)
rnghomval.5	|- J = (1st` S)
rnghomval.6	|- K = (2nd` S)
rnghomval.7	|- Y = ran J
rnghomval.8	|- V = (Id` K)

Assertion

Ref	Expression
rnghomval	|- ((R e. Ring /\ S e. Ring) -> (R RngHom S) = {f | (f:X-->Y /\ (f` U) = V /\ A.x e. X A.y e. X ((f` (xGy)) = ((f` x)J(f` y)) /\ (f` (xHy)) = ((f` x)K(f` y))))})

Distinct variable groups:   R,f,x,y   S,f,x,y   f,X,x,y   f,Y,y

Proof of Theorem rnghomval

Step	Hyp	Ref	Expression
1		rnghomval.3	. . . . 5 |- X = ran G
2		rnghomval.1	. . . . . . 7 |- G = (1st` R)
3		fvex 4689	. . . . . . 7 |- (1st` R) e. _V
4	2, 3	eqeltri 1967	. . . . . 6 |- G e. _V
5	4	rnex 4209	. . . . 5 |- ran G e. _V
6	1, 5	eqeltri 1967	. . . 4 |- X e. _V
7		rnghomval.7	. . . . 5 |- Y = ran J
8		rnghomval.5	. . . . . . 7 |- J = (1st` S)
9		fvex 4689	. . . . . . 7 |- (1st` S) e. _V
10	8, 9	eqeltri 1967	. . . . . 6 |- J e. _V
11	10	rnex 4209	. . . . 5 |- ran J e. _V
12	7, 11	eqeltri 1967	. . . 4 |- Y e. _V
13		mapex 5387	. . . 4 |- ((X e. _V /\ Y e. _V) -> {f | f:X-->Y} e. _V)
14	6, 12, 13	mp2an 761	. . 3 |- {f | f:X-->Y} e. _V
15		simp1 876	. . . 4 |- ((f:X-->Y /\ (f` U) = V /\ A.x e. X A.y e. X ((f` (xGy)) = ((f` x)J(f` y)) /\ (f` (xHy)) = ((f` x)K(f` y)))) -> f:X-->Y)
16	15	ss2abi 2679	. . 3 |- {f | (f:X-->Y /\ (f` U) = V /\ A.x e. X A.y e. X ((f` (xGy)) = ((f` x)J(f` y)) /\ (f` (xHy)) = ((f` x)K(f` y))))} C_ {f | f:X-->Y}
17	14, 16	ssexi 3456	. 2 |- {f | (f:X-->Y /\ (f` U) = V /\ A.x e. X A.y e. X ((f` (xGy)) = ((f` x)J(f` y)) /\ (f` (xHy)) = ((f` x)K(f` y))))} e. _V
18		fveq2 4681	. . . . . . . 8 |- (r = R -> (1st` r) = (1st` R))
19	18, 2	syl6eqr 1946	. . . . . . 7 |- (r = R -> (1st` r) = G)
20	19	rneqd 4188	. . . . . 6 |- (r = R -> ran (1st` r) = ran G)
21	20, 1	syl6eqr 1946	. . . . 5 |- (r = R -> ran (1st` r) = X)
22	21	feq2d 4557	. . . 4 |- (r = R -> (f:ran (1st` r)-->ran (1st` s) <-> f:X-->ran (1st` s)))
23		fveq2 4681	. . . . . . . . 9 |- (r = R -> (2nd` r) = (2nd` R))
24		rnghomval.2	. . . . . . . . 9 |- H = (2nd` R)
25	23, 24	syl6eqr 1946	. . . . . . . 8 |- (r = R -> (2nd` r) = H)
26	25	fveq2d 4685	. . . . . . 7 |- (r = R -> (Id` (2nd` r)) = (Id` H))
27		rnghomval.4	. . . . . . 7 |- U = (Id` H)
28	26, 27	syl6eqr 1946	. . . . . 6 |- (r = R -> (Id` (2nd` r)) = U)
29	28	fveq2d 4685	. . . . 5 |- (r = R -> (f` (Id` (2nd` r))) = (f` U))
30	29	eqeq1d 1892	. . . 4 |- (r = R -> ((f` (Id` (2nd` r))) = (Id` (2nd` s)) <-> (f` U) = (Id` (2nd` s))))
31	19	opreqd 4899	. . . . . . . . 9 |- (r = R -> (x(1st` r)y) = (xGy))
32	31	fveq2d 4685	. . . . . . . 8 |- (r = R -> (f` (x(1st` r)y)) = (f` (xGy)))
33	32	eqeq1d 1892	. . . . . . 7 |- (r = R -> ((f` (x(1st` r)y)) = ((f` x)(1st` s)(f` y)) <-> (f` (xGy)) = ((f` x)(1st` s)(f` y))))
34	25	opreqd 4899	. . . . . . . . 9 |- (r = R -> (x(2nd` r)y) = (xHy))
35	34	fveq2d 4685	. . . . . . . 8 |- (r = R -> (f` (x(2nd` r)y)) = (f` (xHy)))
36	35	eqeq1d 1892	. . . . . . 7 |- (r = R -> ((f` (x(2nd` r)y)) = ((f` x)(2nd` s)(f` y)) <-> (f` (xHy)) = ((f` x)(2nd` s)(f` y))))
37	33, 36	anbi12d 690	. . . . . 6 |- (r = R -> (((f` (x(1st` r)y)) = ((f` x)(1st` s)(f` y)) /\ (f` (x(2nd` r)y)) = ((f` x)(2nd` s)(f` y))) <-> ((f` (xGy)) = ((f` x)(1st` s)(f` y)) /\ (f` (xHy)) = ((f` x)(2nd` s)(f` y)))))
38	21, 37	raleqbidv 2274	. . . . 5 |- (r = 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.y e. X ((f` (xGy)) = ((f` x)(1st` s)(f` y)) /\ (f` (xHy)) = ((f` x)(2nd` s)(f` y)))))
39	21, 38	raleqbidv 2274	. . . 4 |- (r = R -> (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` (xGy)) = ((f` x)(1st` s)(f` y)) /\ (f` (xHy)) = ((f` x)(2nd` s)(f` y)))))
40	22, 30, 39	3anbi123d 1168	. . 3 |- (r = R -> ((f:ran (1st` r)-->ran (1st` s) /\ (f` (Id` (2nd` r))) = (Id` (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:X-->ran (1st` s) /\ (f` U) = (Id` (2nd` s)) /\ A.x e. X A.y e. X ((f` (xGy)) = ((f` x)(1st` s)(f` y)) /\ (f` (xHy)) = ((f` x)(2nd` s)(f` y))))))
41	40	abbidv 2008	. 2 |- (r = R -> {f | (f:ran (1st` r)-->ran (1st` s) /\ (f` (Id` (2nd` r))) = (Id` (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 | (f:X-->ran (1st` s) /\ (f` U) = (Id` (2nd` s)) /\ A.x e. X A.y e. X ((f` (xGy)) = ((f` x)(1st` s)(f` y)) /\ (f` (xHy)) = ((f` x)(2nd` s)(f` y))))})
42		fveq2 4681	. . . . . . . 8 |- (s = S -> (1st` s) = (1st` S))
43	42, 8	syl6eqr 1946	. . . . . . 7 |- (s = S -> (1st` s) = J)
44	43	rneqd 4188	. . . . . 6 |- (s = S -> ran (1st` s) = ran J)
45	44, 7	syl6eqr 1946	. . . . 5 |- (s = S -> ran (1st` s) = Y)
46		feq3 4553	. . . . 5 |- (ran (1st` s) = Y -> (f:X-->ran (1st` s) <-> f:X-->Y))
47	45, 46	syl 12	. . . 4 |- (s = S -> (f:X-->ran (1st` s) <-> f:X-->Y))
48		fveq2 4681	. . . . . . . 8 |- (s = S -> (2nd` s) = (2nd` S))
49		rnghomval.6	. . . . . . . 8 |- K = (2nd` S)
50	48, 49	syl6eqr 1946	. . . . . . 7 |- (s = S -> (2nd` s) = K)
51	50	fveq2d 4685	. . . . . 6 |- (s = S -> (Id` (2nd` s)) = (Id` K))
52		rnghomval.8	. . . . . 6 |- V = (Id` K)
53	51, 52	syl6eqr 1946	. . . . 5 |- (s = S -> (Id` (2nd` s)) = V)
54	53	eqeq2d 1895	. . . 4 |- (s = S -> ((f` U) = (Id` (2nd` s)) <-> (f` U) = V))
55	43	opreqd 4899	. . . . . . 7 |- (s = S -> ((f` x)(1st` s)(f` y)) = ((f` x)J(f` y)))
56	55	eqeq2d 1895	. . . . . 6 |- (s = S -> ((f` (xGy)) = ((f` x)(1st` s)(f` y)) <-> (f` (xGy)) = ((f` x)J(f` y))))
57	50	opreqd 4899	. . . . . . 7 |- (s = S -> ((f` x)(2nd` s)(f` y)) = ((f` x)K(f` y)))
58	57	eqeq2d 1895	. . . . . 6 |- (s = S -> ((f` (xHy)) = ((f` x)(2nd` s)(f` y)) <-> (f` (xHy)) = ((f` x)K(f` y))))
59	56, 58	anbi12d 690	. . . . 5 |- (s = S -> (((f` (xGy)) = ((f` x)(1st` s)(f` y)) /\ (f` (xHy)) = ((f` x)(2nd` s)(f` y))) <-> ((f` (xGy)) = ((f` x)J(f` y)) /\ (f` (xHy)) = ((f` x)K(f` y)))))
60	59	2ralbidv 2140	. . . 4 |- (s = S -> (A.x e. X A.y e. X ((f` (xGy)) = ((f` x)(1st` s)(f` y)) /\ (f` (xHy)) = ((f` x)(2nd` s)(f` y))) <-> A.x e. X A.y e. X ((f` (xGy)) = ((f` x)J(f` y)) /\ (f` (xHy)) = ((f` x)K(f` y)))))
61	47, 54, 60	3anbi123d 1168	. . 3 |- (s = S -> ((f:X-->ran (1st` s) /\ (f` U) = (Id` (2nd` s)) /\ A.x e. X A.y e. X ((f` (xGy)) = ((f` x)(1st` s)(f` y)) /\ (f` (xHy)) = ((f` x)(2nd` s)(f` y)))) <-> (f:X-->Y /\ (f` U) = V /\ A.x e. X A.y e. X ((f` (xGy)) = ((f` x)J(f` y)) /\ (f` (xHy)) = ((f` x)K(f` y))))))
62	61	abbidv 2008	. 2 |- (s = S -> {f | (f:X-->ran (1st` s) /\ (f` U) = (Id` (2nd` s)) /\ A.x e. X A.y e. X ((f` (xGy)) = ((f` x)(1st` s)(f` y)) /\ (f` (xHy)) = ((f` x)(2nd` s)(f` y))))} = {f | (f:X-->Y /\ (f` U) = V /\ A.x e. X A.y e. X ((f` (xGy)) = ((f` x)J(f` y)) /\ (f` (xHy)) = ((f` x)K(f` y))))})
63		df-rnghom 16117	. 2 |- RngHom = {<.<.r, s>., t>. | ((r e. Ring /\ s e. Ring) /\ t = {f | (f:ran (1st` r)-->ran (1st` s) /\ (f` (Id` (2nd` r))) = (Id` (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))))})}
64	17, 41, 62, 63	oprabval2 4957	1 |- ((R e. Ring /\ S e. Ring) -> (R RngHom S) = {f | (f:X-->Y /\ (f` U) = V /\ A.x e. X A.y e. X ((f` (xGy)) = ((f` x)J(f` y)) /\ (f` (xHy)) = ((f` x)K(f` y))))})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  _Vcvv 2292  ran crn 3987  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Ringcring 9463   RngHom crnghom 16114

This theorem is referenced by:  isrnghom 16119

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-rnghom 16117

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