Theorem pi1val 16094

Description: The concatenation of two path-homotopy classes in the fundamental group.

Hypothesis

Ref	Expression
pi1fval.1	|- X = U.J

Assertion

Ref	Expression
pi1val	|- (((J e. Top /\ Y e. X) /\ (F e. (II Cn J) /\ (F` 0) = Y /\ (F` 1) = Y) /\ (G e. (II Cn J) /\ (G` 0) = Y /\ (G` 1) = Y)) -> ([F](~=ph` J)(pi1` <.J, Y>.)[G](~=ph` J)) = [(F(*p` J)G)](~=ph` J))

Proof of Theorem pi1val

Step	Hyp	Ref	Expression
1		pi1fval.1	. . . . . . . 8 |- X = U.J
2	1	pi1fval 16092	. . . . . . 7 |- ((J e. Top /\ Y e. X) -> (pi1` <.J, Y>.) = {<.<.f, g>., h>. | E.m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}E.n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} ((f = [m](~=ph` J) /\ g = [n](~=ph` J)) /\ h = [(m(*p` J)n)](~=ph` J))})
3	2	adantr 425	. . . . . 6 |- (((J e. Top /\ Y e. X) /\ (F e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ G e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)})) -> (pi1` <.J, Y>.) = {<.<.f, g>., h>. | E.m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}E.n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} ((f = [m](~=ph` J) /\ g = [n](~=ph` J)) /\ h = [(m(*p` J)n)](~=ph` J))})
4		oprvres 4963	. . . . . . . . . . 11 |- ((m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) -> (m((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))n) = (m(*p` J)n))
5		eceq2 5336	. . . . . . . . . . 11 |- ((m((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))n) = (m(*p` J)n) -> [(m((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))n)](~=ph` J) = [(m(*p` J)n)](~=ph` J))
6	4, 5	syl 12	. . . . . . . . . 10 |- ((m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) -> [(m((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))n)](~=ph` J) = [(m(*p` J)n)](~=ph` J))
7	6	eqeq2d 1895	. . . . . . . . 9 |- ((m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) -> (h = [(m((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))n)](~=ph` J) <-> h = [(m(*p` J)n)](~=ph` J)))
8	7	anbi2d 678	. . . . . . . 8 |- ((m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) -> (((f = [m](~=ph` J) /\ g = [n](~=ph` J)) /\ h = [(m((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))n)](~=ph` J)) <-> ((f = [m](~=ph` J) /\ g = [n](~=ph` J)) /\ h = [(m(*p` J)n)](~=ph` J))))
9	8	2rexbiia 2135	. . . . . . 7 |- (E.m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}E.n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} ((f = [m](~=ph` J) /\ g = [n](~=ph` J)) /\ h = [(m((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))n)](~=ph` J)) <-> E.m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}E.n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} ((f = [m](~=ph` J) /\ g = [n](~=ph` J)) /\ h = [(m(*p` J)n)](~=ph` J)))
10	9	oprabbii 4923	. . . . . 6 |- {<.<.f, g>., h>. | E.m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}E.n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} ((f = [m](~=ph` J) /\ g = [n](~=ph` J)) /\ h = [(m((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))n)](~=ph` J))} = {<.<.f, g>., h>. | E.m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}E.n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} ((f = [m](~=ph` J) /\ g = [n](~=ph` J)) /\ h = [(m(*p` J)n)](~=ph` J))}
11	3, 10	syl6eqr 1946	. . . . 5 |- (((J e. Top /\ Y e. X) /\ (F e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ G e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)})) -> (pi1` <.J, Y>.) = {<.<.f, g>., h>. | E.m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}E.n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} ((f = [m](~=ph` J) /\ g = [n](~=ph` J)) /\ h = [(m((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))n)](~=ph` J))})
12	11	opreqd 4899	. . . 4 |- (((J e. Top /\ Y e. X) /\ (F e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ G e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)})) -> ([F](~=ph` J)(pi1` <.J, Y>.)[G](~=ph` J)) = ([F](~=ph` J){<.<.f, g>., h>. | E.m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}E.n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} ((f = [m](~=ph` J) /\ g = [n](~=ph` J)) /\ h = [(m((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))n)](~=ph` J))}[G](~=ph` J)))
13		eqid 1884	. . . . . 6 |- ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}/.(~=ph` J)) = ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}/.(~=ph` J))
14		eqid 1884	. . . . . 6 |- {<.<.f, g>., h>. | E.m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}E.n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} ((f = [m](~=ph` J) /\ g = [n](~=ph` J)) /\ h = [(m((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))n)](~=ph` J))} = {<.<.f, g>., h>. | E.m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}E.n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} ((f = [m](~=ph` J) /\ g = [n](~=ph` J)) /\ h = [(m((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))n)](~=ph` J))}
15		fvex 4689	. . . . . . 7 |- (~=ph` J) e. _V
16	15	a1i 8	. . . . . 6 |- ((J e. Top /\ Y e. X) -> (~=ph` J) e. _V)
17		phtpcer 16062	. . . . . . 7 |- (J e. Top -> Er (~=ph` J))
18	17	adantr 425	. . . . . 6 |- ((J e. Top /\ Y e. X) -> Er (~=ph` J))
19		ssrab2 2692	. . . . . . . 8 |- {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} C_ (II Cn J)
20	19	a1i 8	. . . . . . 7 |- ((J e. Top /\ Y e. X) -> {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} C_ (II Cn J))
21		phtpcdm 16061	. . . . . . . 8 |- (J e. Top -> dom (~=ph` J) = (II Cn J))
22	21	adantr 425	. . . . . . 7 |- ((J e. Top /\ Y e. X) -> dom (~=ph` J) = (II Cn J))
23	20, 22	sseqtr4d 2654	. . . . . 6 |- ((J e. Top /\ Y e. X) -> {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} C_ dom (~=ph` J))
24		eqid 1884	. . . . . . 7 |- {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} = {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}
25	1, 24	pcoloopf 16079	. . . . . 6 |- ((J e. Top /\ Y e. X) -> ((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)})):({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)})-->{t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)})
26		visset 2295	. . . . . . . . . 10 |- q e. _V
27		visset 2295	. . . . . . . . . 10 |- s e. _V
28	26, 27	pm3.2i 307	. . . . . . . . 9 |- (q e. _V /\ s e. _V)
29		pcohtpy 16083	. . . . . . . . 9 |- (((J e. Top /\ Y e. X) /\ (q e. _V /\ s e. _V) /\ (p` 1) = (r` 0)) -> ((p(~=ph` J)q /\ r(~=ph` J)s) -> (p(*p` J)r)(~=ph` J)(q(*p` J)s)))
30	28, 29	mp3an2 1179	. . . . . . . 8 |- (((J e. Top /\ Y e. X) /\ (p` 1) = (r` 0)) -> ((p(~=ph` J)q /\ r(~=ph` J)s) -> (p(*p` J)r)(~=ph` J)(q(*p` J)s)))
31		eqtr3 1907	. . . . . . . . . . . 12 |- (((p` 1) = Y /\ (r` 0) = Y) -> (p` 1) = (r` 0))
32	31	ad2ant2lr 446	. . . . . . . . . . 11 |- ((((p` 0) = Y /\ (p` 1) = Y) /\ ((r` 0) = Y /\ (r` 1) = Y)) -> (p` 1) = (r` 0))
33	32	ad2ant2l 444	. . . . . . . . . 10 |- (((p e. (II Cn J) /\ ((p` 0) = Y /\ (p` 1) = Y)) /\ (r e. (II Cn J) /\ ((r` 0) = Y /\ (r` 1) = Y))) -> (p` 1) = (r` 0))
34		fveq1 4680	. . . . . . . . . . . . 13 |- (t = p -> (t` 0) = (p` 0))
35	34	eqeq1d 1892	. . . . . . . . . . . 12 |- (t = p -> ((t` 0) = Y <-> (p` 0) = Y))
36		fveq1 4680	. . . . . . . . . . . . 13 |- (t = p -> (t` 1) = (p` 1))
37	36	eqeq1d 1892	. . . . . . . . . . . 12 |- (t = p -> ((t` 1) = Y <-> (p` 1) = Y))
38	35, 37	anbi12d 690	. . . . . . . . . . 11 |- (t = p -> (((t` 0) = Y /\ (t` 1) = Y) <-> ((p` 0) = Y /\ (p` 1) = Y)))
39	38	elrab 2414	. . . . . . . . . 10 |- (p e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} <-> (p e. (II Cn J) /\ ((p` 0) = Y /\ (p` 1) = Y)))
40		fveq1 4680	. . . . . . . . . . . . 13 |- (t = r -> (t` 0) = (r` 0))
41	40	eqeq1d 1892	. . . . . . . . . . . 12 |- (t = r -> ((t` 0) = Y <-> (r` 0) = Y))
42		fveq1 4680	. . . . . . . . . . . . 13 |- (t = r -> (t` 1) = (r` 1))
43	42	eqeq1d 1892	. . . . . . . . . . . 12 |- (t = r -> ((t` 1) = Y <-> (r` 1) = Y))
44	41, 43	anbi12d 690	. . . . . . . . . . 11 |- (t = r -> (((t` 0) = Y /\ (t` 1) = Y) <-> ((r` 0) = Y /\ (r` 1) = Y)))
45	44	elrab 2414	. . . . . . . . . 10 |- (r e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} <-> (r e. (II Cn J) /\ ((r` 0) = Y /\ (r` 1) = Y)))
46	33, 39, 45	syl2anb 504	. . . . . . . . 9 |- ((p e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ r e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) -> (p` 1) = (r` 0))
47	46	ad2ant2r 445	. . . . . . . 8 |- (((p e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ q e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) /\ (r e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ s e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)})) -> (p` 1) = (r` 0))
48	30, 47	sylan2 500	. . . . . . 7 |- (((J e. Top /\ Y e. X) /\ ((p e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ q e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) /\ (r e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ s e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))) -> ((p(~=ph` J)q /\ r(~=ph` J)s) -> (p(*p` J)r)(~=ph` J)(q(*p` J)s)))
49		oprvres 4963	. . . . . . . . . . 11 |- ((p e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ r e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) -> (p((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))r) = (p(*p` J)r))
50		oprvres 4963	. . . . . . . . . . 11 |- ((q e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ s e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) -> (q((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))s) = (q(*p` J)s))
51	49, 50	breqan12d 3354	. . . . . . . . . 10 |- (((p e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ r e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) /\ (q e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ s e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)})) -> ((p((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))r)(~=ph` J)(q((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))s) <-> (p(*p` J)r)(~=ph` J)(q(*p` J)s)))
52	51	an4s 566	. . . . . . . . 9 |- (((p e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ q e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) /\ (r e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ s e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)})) -> ((p((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))r)(~=ph` J)(q((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))s) <-> (p(*p` J)r)(~=ph` J)(q(*p` J)s)))
53	52	imbi2d 674	. . . . . . . 8 |- (((p e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ q e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) /\ (r e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ s e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)})) -> (((p(~=ph` J)q /\ r(~=ph` J)s) -> (p((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))r)(~=ph` J)(q((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))s)) <-> ((p(~=ph` J)q /\ r(~=ph` J)s) -> (p(*p` J)r)(~=ph` J)(q(*p` J)s))))
54	53	adantl 424	. . . . . . 7 |- (((J e. Top /\ Y e. X) /\ ((p e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ q e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) /\ (r e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ s e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))) -> (((p(~=ph` J)q /\ r(~=ph` J)s) -> (p((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))r)(~=ph` J)(q((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))s)) <-> ((p(~=ph` J)q /\ r(~=ph` J)s) -> (p(*p` J)r)(~=ph` J)(q(*p` J)s))))
55	48, 54	mpbird 213	. . . . . 6 |- (((J e. Top /\ Y e. X) /\ ((p e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ q e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) /\ (r e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ s e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))) -> ((p(~=ph` J)q /\ r(~=ph` J)s) -> (p((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))r)(~=ph` J)(q((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))s)))
56	13, 14, 16, 18, 23, 25, 55	eroprv2 15736	. . . . 5 |- (((J e. Top /\ Y e. X) /\ F e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ G e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) -> ([F](~=ph` J){<.<.f, g>., h>. | E.m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}E.n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} ((f = [m](~=ph` J) /\ g = [n](~=ph` J)) /\ h = [(m((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))n)](~=ph` J))}[G](~=ph` J)) = [(F((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))G)](~=ph` J))
57	56	3expb 1068	. . . 4 |- (((J e. Top /\ Y e. X) /\ (F e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ G e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)})) -> ([F](~=ph` J){<.<.f, g>., h>. | E.m e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}E.n e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} ((f = [m](~=ph` J) /\ g = [n](~=ph` J)) /\ h = [(m((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))n)](~=ph` J))}[G](~=ph` J)) = [(F((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))G)](~=ph` J))
58		oprvres 4963	. . . . . 6 |- ((F e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ G e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) -> (F((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))G) = (F(*p` J)G))
59	58	adantl 424	. . . . 5 |- (((J e. Top /\ Y e. X) /\ (F e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ G e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)})) -> (F((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))G) = (F(*p` J)G))
60		eceq2 5336	. . . . 5 |- ((F((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))G) = (F(*p` J)G) -> [(F((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))G)](~=ph` J) = [(F(*p` J)G)](~=ph` J))
61	59, 60	syl 12	. . . 4 |- (((J e. Top /\ Y e. X) /\ (F e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ G e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)})) -> [(F((*p` J) |` ({t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} X. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}))G)](~=ph` J) = [(F(*p` J)G)](~=ph` J))
62	12, 57, 61	3eqtrd 1929	. . 3 |- (((J e. Top /\ Y e. X) /\ (F e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ G e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)})) -> ([F](~=ph` J)(pi1` <.J, Y>.)[G](~=ph` J)) = [(F(*p` J)G)](~=ph` J))
63		fveq1 4680	. . . . . . . 8 |- (t = F -> (t` 0) = (F` 0))
64	63	eqeq1d 1892	. . . . . . 7 |- (t = F -> ((t` 0) = Y <-> (F` 0) = Y))
65		fveq1 4680	. . . . . . . 8 |- (t = F -> (t` 1) = (F` 1))
66	65	eqeq1d 1892	. . . . . . 7 |- (t = F -> ((t` 1) = Y <-> (F` 1) = Y))
67	64, 66	anbi12d 690	. . . . . 6 |- (t = F -> (((t` 0) = Y /\ (t` 1) = Y) <-> ((F` 0) = Y /\ (F` 1) = Y)))
68	67	elrab 2414	. . . . 5 |- (F e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} <-> (F e. (II Cn J) /\ ((F` 0) = Y /\ (F` 1) = Y)))
69		3anass 862	. . . . 5 |- ((F e. (II Cn J) /\ (F` 0) = Y /\ (F` 1) = Y) <-> (F e. (II Cn J) /\ ((F` 0) = Y /\ (F` 1) = Y)))
70	68, 69	bitr4i 193	. . . 4 |- (F e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} <-> (F e. (II Cn J) /\ (F` 0) = Y /\ (F` 1) = Y))
71		fveq1 4680	. . . . . . . 8 |- (t = G -> (t` 0) = (G` 0))
72	71	eqeq1d 1892	. . . . . . 7 |- (t = G -> ((t` 0) = Y <-> (G` 0) = Y))
73		fveq1 4680	. . . . . . . 8 |- (t = G -> (t` 1) = (G` 1))
74	73	eqeq1d 1892	. . . . . . 7 |- (t = G -> ((t` 1) = Y <-> (G` 1) = Y))
75	72, 74	anbi12d 690	. . . . . 6 |- (t = G -> (((t` 0) = Y /\ (t` 1) = Y) <-> ((G` 0) = Y /\ (G` 1) = Y)))
76	75	elrab 2414	. . . . 5 |- (G e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} <-> (G e. (II Cn J) /\ ((G` 0) = Y /\ (G` 1) = Y)))
77		3anass 862	. . . . 5 |- ((G e. (II Cn J) /\ (G` 0) = Y /\ (G` 1) = Y) <-> (G e. (II Cn J) /\ ((G` 0) = Y /\ (G` 1) = Y)))
78	76, 77	bitr4i 193	. . . 4 |- (G e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} <-> (G e. (II Cn J) /\ (G` 0) = Y /\ (G` 1) = Y))
79	70, 78	anbi12i 540	. . 3 |- ((F e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)} /\ G e. {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}) <-> ((F e. (II Cn J) /\ (F` 0) = Y /\ (F` 1) = Y) /\ (G e. (II Cn J) /\ (G` 0) = Y /\ (G` 1) = Y)))
80	62, 79	sylan2br 502	. 2 |- (((J e. Top /\ Y e. X) /\ ((F e. (II Cn J) /\ (F` 0) = Y /\ (F` 1) = Y) /\ (G e. (II Cn J) /\ (G` 0) = Y /\ (G` 1) = Y))) -> ([F](~=ph` J)(pi1` <.J, Y>.)[G](~=ph` J)) = [(F(*p` J)G)](~=ph` J))
81	80	3impb 1063	1 |- (((J e. Top /\ Y e. X) /\ (F e. (II Cn J) /\ (F` 0) = Y /\ (F` 1) = Y) /\ (G e. (II Cn J) /\ (G` 0) = Y /\ (G` 1) = Y)) -> ([F](~=ph` J)(pi1` <.J, Y>.)[G](~=ph` J)) = [(F(*p` J)G)](~=ph` J))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wrex 2106  {crab 2108  _Vcvv 2292   C_ wss 2593  <.cop 3046  U.cuni 3177   class class class wbr 3338   X. cxp 3984  dom cdm 3986   |` cres 3988  ` cfv 3998  (class class class)co 4884  {copab2 4885  Er wer 5315  [cec 5316  /.cqs 5317  0cc0 6386  1c1 6387  Topctop 8857   Cn ccn 9028  IIcii 15865  ~=phcphtpc 16044  *pcpco 16067  pi1cpi1 16068

This theorem is referenced by:  pi1gp 16095

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  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-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-rp 7232  df-n0 7309  df-z 7345  df-q 7436  df-fl 7463  df-ioo 7528  df-icc 7531  df-uz 7587  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-cncf 8525  df-top 8861  df-topsp 8862  df-bases 8863  df-topgen 8864  df-tx 8931  df-cld 8939  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-subsp 10243  df-ii 15866  df-phtpy 16045  df-phtpc 16057  df-pco 16069  df-pi1 16071

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