Theorem ghomco 16040

Description: The composition of two group homomorphisms is a group homomorphism.

Assertion

Ref	Expression
ghomco	|- (((G e. Grp /\ H e. Grp /\ K e. Grp) /\ (S e. (G GrpHom H) /\ T e. (H GrpHom K))) -> (T o. S) e. (G GrpHom K))

Proof of Theorem ghomco

Step	Hyp	Ref	Expression
1		fco 4573	. . . . . . 7 |- ((T:ran H-->ran K /\ S:ran G-->ran H) -> (T o. S):ran G-->ran K)
2	1	ancoms 484	. . . . . 6 |- ((S:ran G-->ran H /\ T:ran H-->ran K) -> (T o. S):ran G-->ran K)
3	2	ad2ant2r 445	. . . . 5 |- (((S:ran G-->ran H /\ A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy))) /\ (T:ran H-->ran K /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv)))) -> (T o. S):ran G-->ran K)
4	3	a1i 8	. . . 4 |- ((G e. Grp /\ H e. Grp /\ K e. Grp) -> (((S:ran G-->ran H /\ A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy))) /\ (T:ran H-->ran K /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv)))) -> (T o. S):ran G-->ran K))
5		fveq2 4681	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (u = (S` x) -> (T` u) = (T` (S` x)))
6	5	opreq1d 4897	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (u = (S` x) -> ((T` u)K(T` v)) = ((T` (S` x))K(T` v)))
7		opreq1 4889	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (u = (S` x) -> (uHv) = ((S` x)Hv))
8	7	fveq2d 4685	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (u = (S` x) -> (T` (uHv)) = (T` ((S` x)Hv)))
9	6, 8	eqeq12d 1899	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (u = (S` x) -> (((T` u)K(T` v)) = (T` (uHv)) <-> ((T` (S` x))K(T` v)) = (T` ((S` x)Hv))))
10		fveq2 4681	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (v = (S` y) -> (T` v) = (T` (S` y)))
11	10	opreq2d 4898	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (v = (S` y) -> ((T` (S` x))K(T` v)) = ((T` (S` x))K(T` (S` y))))
12		opreq2 4890	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (v = (S` y) -> ((S` x)Hv) = ((S` x)H(S` y)))
13	12	fveq2d 4685	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (v = (S` y) -> (T` ((S` x)Hv)) = (T` ((S` x)H(S` y))))
14	11, 13	eqeq12d 1899	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (v = (S` y) -> (((T` (S` x))K(T` v)) = (T` ((S` x)Hv)) <-> ((T` (S` x))K(T` (S` y))) = (T` ((S` x)H(S` y)))))
15	9, 14	rcla42v 2384	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((S` x) e. ran H /\ (S` y) e. ran H) -> (A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv)) -> ((T` (S` x))K(T` (S` y))) = (T` ((S` x)H(S` y)))))
16	15	imp 377	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((((S` x) e. ran H /\ (S` y) e. ran H) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) -> ((T` (S` x))K(T` (S` y))) = (T` ((S` x)H(S` y))))
17		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((S:ran G-->ran H /\ x e. ran G) -> (S` x) e. ran H)
18	17	adantrr 431	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((S:ran G-->ran H /\ (x e. ran G /\ y e. ran G)) -> (S` x) e. ran H)
19		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((S:ran G-->ran H /\ y e. ran G) -> (S` y) e. ran H)
20	19	adantrl 430	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((S:ran G-->ran H /\ (x e. ran G /\ y e. ran G)) -> (S` y) e. ran H)
21	18, 20	jca 310	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((S:ran G-->ran H /\ (x e. ran G /\ y e. ran G)) -> ((S` x) e. ran H /\ (S` y) e. ran H))
22	16, 21	sylan 497	. . . . . . . . . . . . . . . . . . . . . 22 |- (((S:ran G-->ran H /\ (x e. ran G /\ y e. ran G)) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) -> ((T` (S` x))K(T` (S` y))) = (T` ((S` x)H(S` y))))
23	22	an1rs 547	. . . . . . . . . . . . . . . . . . . . 21 |- (((S:ran G-->ran H /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) /\ (x e. ran G /\ y e. ran G)) -> ((T` (S` x))K(T` (S` y))) = (T` ((S` x)H(S` y))))
24	23	adantllr 433	. . . . . . . . . . . . . . . . . . . 20 |- ((((S:ran G-->ran H /\ T:ran H-->ran K) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) /\ (x e. ran G /\ y e. ran G)) -> ((T` (S` x))K(T` (S` y))) = (T` ((S` x)H(S` y))))
25	24	adantllr 433	. . . . . . . . . . . . . . . . . . 19 |- (((((S:ran G-->ran H /\ T:ran H-->ran K) /\ G e. Grp) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) /\ (x e. ran G /\ y e. ran G)) -> ((T` (S` x))K(T` (S` y))) = (T` ((S` x)H(S` y))))
26		fveq2 4681	. . . . . . . . . . . . . . . . . . 19 |- (((S` x)H(S` y)) = (S` (xGy)) -> (T` ((S` x)H(S` y))) = (T` (S` (xGy))))
27	25, 26	sylan9eq 1948	. . . . . . . . . . . . . . . . . 18 |- ((((((S:ran G-->ran H /\ T:ran H-->ran K) /\ G e. Grp) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) /\ (x e. ran G /\ y e. ran G)) /\ ((S` x)H(S` y)) = (S` (xGy))) -> ((T` (S` x))K(T` (S` y))) = (T` (S` (xGy))))
28	27	anasss 488	. . . . . . . . . . . . . . . . 17 |- (((((S:ran G-->ran H /\ T:ran H-->ran K) /\ G e. Grp) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) /\ ((x e. ran G /\ y e. ran G) /\ ((S` x)H(S` y)) = (S` (xGy)))) -> ((T` (S` x))K(T` (S` y))) = (T` (S` (xGy))))
29		ffun 4565	. . . . . . . . . . . . . . . . . . . . . . 23 |- (T:ran H-->ran K -> Fun T)
30	29	ad2antlr 441	. . . . . . . . . . . . . . . . . . . . . 22 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ x e. ran G) -> Fun T)
31		ffun 4565	. . . . . . . . . . . . . . . . . . . . . . 23 |- (S:ran G-->ran H -> Fun S)
32	31	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . 22 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ x e. ran G) -> Fun S)
33		fdm 4567	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (S:ran G-->ran H -> dom S = ran G)
34	33	eleq2d 1964	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (S:ran G-->ran H -> (x e. dom S <-> x e. ran G))
35	34	biimpar 461	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((S:ran G-->ran H /\ x e. ran G) -> x e. dom S)
36	35	adantlr 429	. . . . . . . . . . . . . . . . . . . . . 22 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ x e. ran G) -> x e. dom S)
37		fvco 4736	. . . . . . . . . . . . . . . . . . . . . 22 |- ((Fun T /\ Fun S /\ x e. dom S) -> ((T o. S)` x) = (T` (S` x)))
38	30, 32, 36, 37	syl111anc 1100	. . . . . . . . . . . . . . . . . . . . 21 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ x e. ran G) -> ((T o. S)` x) = (T` (S` x)))
39	38	adantrr 431	. . . . . . . . . . . . . . . . . . . 20 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ (x e. ran G /\ y e. ran G)) -> ((T o. S)` x) = (T` (S` x)))
40	29	ad2antlr 441	. . . . . . . . . . . . . . . . . . . . . 22 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ y e. ran G) -> Fun T)
41	31	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . 22 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ y e. ran G) -> Fun S)
42	33	eleq2d 1964	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (S:ran G-->ran H -> (y e. dom S <-> y e. ran G))
43	42	biimpar 461	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((S:ran G-->ran H /\ y e. ran G) -> y e. dom S)
44	43	adantlr 429	. . . . . . . . . . . . . . . . . . . . . 22 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ y e. ran G) -> y e. dom S)
45		fvco 4736	. . . . . . . . . . . . . . . . . . . . . 22 |- ((Fun T /\ Fun S /\ y e. dom S) -> ((T o. S)` y) = (T` (S` y)))
46	40, 41, 44, 45	syl111anc 1100	. . . . . . . . . . . . . . . . . . . . 21 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ y e. ran G) -> ((T o. S)` y) = (T` (S` y)))
47	46	adantrl 430	. . . . . . . . . . . . . . . . . . . 20 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ (x e. ran G /\ y e. ran G)) -> ((T o. S)` y) = (T` (S` y)))
48	39, 47	opreq12d 4900	. . . . . . . . . . . . . . . . . . 19 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ (x e. ran G /\ y e. ran G)) -> (((T o. S)` x)K((T o. S)` y)) = ((T` (S` x))K(T` (S` y))))
49	48	adantlr 429	. . . . . . . . . . . . . . . . . 18 |- ((((S:ran G-->ran H /\ T:ran H-->ran K) /\ G e. Grp) /\ (x e. ran G /\ y e. ran G)) -> (((T o. S)` x)K((T o. S)` y)) = ((T` (S` x))K(T` (S` y))))
50	49	ad2ant2r 445	. . . . . . . . . . . . . . . . 17 |- (((((S:ran G-->ran H /\ T:ran H-->ran K) /\ G e. Grp) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) /\ ((x e. ran G /\ y e. ran G) /\ ((S` x)H(S` y)) = (S` (xGy)))) -> (((T o. S)` x)K((T o. S)` y)) = ((T` (S` x))K(T` (S` y))))
51	29	ad2antlr 441	. . . . . . . . . . . . . . . . . . . . 21 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ (xGy) e. ran G) -> Fun T)
52	31	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . 21 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ (xGy) e. ran G) -> Fun S)
53	33	eleq2d 1964	. . . . . . . . . . . . . . . . . . . . . . 23 |- (S:ran G-->ran H -> ((xGy) e. dom S <-> (xGy) e. ran G))
54	53	biimpar 461	. . . . . . . . . . . . . . . . . . . . . 22 |- ((S:ran G-->ran H /\ (xGy) e. ran G) -> (xGy) e. dom S)
55	54	adantlr 429	. . . . . . . . . . . . . . . . . . . . 21 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ (xGy) e. ran G) -> (xGy) e. dom S)
56		fvco 4736	. . . . . . . . . . . . . . . . . . . . 21 |- ((Fun T /\ Fun S /\ (xGy) e. dom S) -> ((T o. S)` (xGy)) = (T` (S` (xGy))))
57	51, 52, 55, 56	syl111anc 1100	. . . . . . . . . . . . . . . . . . . 20 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ (xGy) e. ran G) -> ((T o. S)` (xGy)) = (T` (S` (xGy))))
58		eqid 1884	. . . . . . . . . . . . . . . . . . . . . 22 |- ran G = ran G
59	58	grpcl 9324	. . . . . . . . . . . . . . . . . . . . 21 |- ((G e. Grp /\ x e. ran G /\ y e. ran G) -> (xGy) e. ran G)
60	59	3expb 1068	. . . . . . . . . . . . . . . . . . . 20 |- ((G e. Grp /\ (x e. ran G /\ y e. ran G)) -> (xGy) e. ran G)
61	57, 60	sylan2 500	. . . . . . . . . . . . . . . . . . 19 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ (G e. Grp /\ (x e. ran G /\ y e. ran G))) -> ((T o. S)` (xGy)) = (T` (S` (xGy))))
62	61	anassrs 489	. . . . . . . . . . . . . . . . . 18 |- ((((S:ran G-->ran H /\ T:ran H-->ran K) /\ G e. Grp) /\ (x e. ran G /\ y e. ran G)) -> ((T o. S)` (xGy)) = (T` (S` (xGy))))
63	62	ad2ant2r 445	. . . . . . . . . . . . . . . . 17 |- (((((S:ran G-->ran H /\ T:ran H-->ran K) /\ G e. Grp) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) /\ ((x e. ran G /\ y e. ran G) /\ ((S` x)H(S` y)) = (S` (xGy)))) -> ((T o. S)` (xGy)) = (T` (S` (xGy))))
64	28, 50, 63	3eqtr4d 1937	. . . . . . . . . . . . . . . 16 |- (((((S:ran G-->ran H /\ T:ran H-->ran K) /\ G e. Grp) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) /\ ((x e. ran G /\ y e. ran G) /\ ((S` x)H(S` y)) = (S` (xGy)))) -> (((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy)))
65	64	expr 418	. . . . . . . . . . . . . . 15 |- (((((S:ran G-->ran H /\ T:ran H-->ran K) /\ G e. Grp) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) /\ (x e. ran G /\ y e. ran G)) -> (((S` x)H(S` y)) = (S` (xGy)) -> (((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy))))
66	65	anassrs 489	. . . . . . . . . . . . . 14 |- ((((((S:ran G-->ran H /\ T:ran H-->ran K) /\ G e. Grp) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) /\ x e. ran G) /\ y e. ran G) -> (((S` x)H(S` y)) = (S` (xGy)) -> (((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy))))
67	66	ralimdvaa 2171	. . . . . . . . . . . . 13 |- (((((S:ran G-->ran H /\ T:ran H-->ran K) /\ G e. Grp) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) /\ x e. ran G) -> (A.y e. ran G((S` x)H(S` y)) = (S` (xGy)) -> A.y e. ran G(((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy))))
68	67	ralimdvaa 2171	. . . . . . . . . . . 12 |- ((((S:ran G-->ran H /\ T:ran H-->ran K) /\ G e. Grp) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) -> (A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy)) -> A.x e. ran GA.y e. ran G(((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy))))
69	68	an1rs 547	. . . . . . . . . . 11 |- ((((S:ran G-->ran H /\ T:ran H-->ran K) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) /\ G e. Grp) -> (A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy)) -> A.x e. ran GA.y e. ran G(((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy))))
70	69	ex 402	. . . . . . . . . 10 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) -> (G e. Grp -> (A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy)) -> A.x e. ran GA.y e. ran G(((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy)))))
71	70	com23 36	. . . . . . . . 9 |- (((S:ran G-->ran H /\ T:ran H-->ran K) /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))) -> (A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy)) -> (G e. Grp -> A.x e. ran GA.y e. ran G(((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy)))))
72	71	anasss 488	. . . . . . . 8 |- ((S:ran G-->ran H /\ (T:ran H-->ran K /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv)))) -> (A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy)) -> (G e. Grp -> A.x e. ran GA.y e. ran G(((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy)))))
73	72	imp 377	. . . . . . 7 |- (((S:ran G-->ran H /\ (T:ran H-->ran K /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv)))) /\ A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy))) -> (G e. Grp -> A.x e. ran GA.y e. ran G(((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy))))
74	73	an1rs 547	. . . . . 6 |- (((S:ran G-->ran H /\ A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy))) /\ (T:ran H-->ran K /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv)))) -> (G e. Grp -> A.x e. ran GA.y e. ran G(((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy))))
75	74	com12 14	. . . . 5 |- (G e. Grp -> (((S:ran G-->ran H /\ A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy))) /\ (T:ran H-->ran K /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv)))) -> A.x e. ran GA.y e. ran G(((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy))))
76	75	3ad2ant1 897	. . . 4 |- ((G e. Grp /\ H e. Grp /\ K e. Grp) -> (((S:ran G-->ran H /\ A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy))) /\ (T:ran H-->ran K /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv)))) -> A.x e. ran GA.y e. ran G(((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy))))
77	4, 76	jcad 661	. . 3 |- ((G e. Grp /\ H e. Grp /\ K e. Grp) -> (((S:ran G-->ran H /\ A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy))) /\ (T:ran H-->ran K /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv)))) -> ((T o. S):ran G-->ran K /\ A.x e. ran GA.y e. ran G(((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy)))))
78		eqid 1884	. . . . . 6 |- ran H = ran H
79	58, 78	elghom 10195	. . . . 5 |- ((G e. Grp /\ H e. Grp) -> (S e. (G GrpHom H) <-> (S:ran G-->ran H /\ A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy)))))
80	79	3adant3 896	. . . 4 |- ((G e. Grp /\ H e. Grp /\ K e. Grp) -> (S e. (G GrpHom H) <-> (S:ran G-->ran H /\ A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy)))))
81		eqid 1884	. . . . . 6 |- ran K = ran K
82	78, 81	elghom 10195	. . . . 5 |- ((H e. Grp /\ K e. Grp) -> (T e. (H GrpHom K) <-> (T:ran H-->ran K /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv)))))
83	82	3adant1 894	. . . 4 |- ((G e. Grp /\ H e. Grp /\ K e. Grp) -> (T e. (H GrpHom K) <-> (T:ran H-->ran K /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv)))))
84	80, 83	anbi12d 690	. . 3 |- ((G e. Grp /\ H e. Grp /\ K e. Grp) -> ((S e. (G GrpHom H) /\ T e. (H GrpHom K)) <-> ((S:ran G-->ran H /\ A.x e. ran GA.y e. ran G((S` x)H(S` y)) = (S` (xGy))) /\ (T:ran H-->ran K /\ A.u e. ran HA.v e. ran H((T` u)K(T` v)) = (T` (uHv))))))
85	58, 81	elghom 10195	. . . 4 |- ((G e. Grp /\ K e. Grp) -> ((T o. S) e. (G GrpHom K) <-> ((T o. S):ran G-->ran K /\ A.x e. ran GA.y e. ran G(((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy)))))
86	85	3adant2 895	. . 3 |- ((G e. Grp /\ H e. Grp /\ K e. Grp) -> ((T o. S) e. (G GrpHom K) <-> ((T o. S):ran G-->ran K /\ A.x e. ran GA.y e. ran G(((T o. S)` x)K((T o. S)` y)) = ((T o. S)` (xGy)))))
87	77, 84, 86	3imtr4d 602	. 2 |- ((G e. Grp /\ H e. Grp /\ K e. Grp) -> ((S e. (G GrpHom H) /\ T e. (H GrpHom K)) -> (T o. S) e. (G GrpHom K)))
88	87	imp 377	1 |- (((G e. Grp /\ H e. Grp /\ K e. Grp) /\ (S e. (G GrpHom H) /\ T e. (H GrpHom K))) -> (T o. S) e. (G GrpHom K))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  dom cdm 3986  ran crn 3987   o. ccom 3990  Fun wfun 3992  -->wf 3994  ` cfv 3998  (class class class)co 4884  Grpcgr 9311   GrpHom cghom 10189

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-rep 3428  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-fo 4012  df-fv 4014  df-opr 4886  df-oprab 4887  df-grp 9316  df-ghom 10190

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