Theorem phtpyval 16047

Description: The class of path homotopies between two continuous functions.

Assertion

Ref	Expression
phtpyval	|- (((J e. Top /\ F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) -> (F(PHtpy` J)G) = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))})

Distinct variable groups:   h,J,s   h,F,s   h,G,s

Proof of Theorem phtpyval

Step	Hyp	Ref	Expression
1		phtpyfval 16046	. . . . 5 |- (J e. Top -> (PHtpy` J) = {<.<.f, g>., z>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})})
2	1	opreqd 4899	. . . 4 |- (J e. Top -> (F(PHtpy` J)G) = (F{<.<.f, g>., z>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})}G))
3	2	3ad2ant1 897	. . 3 |- ((J e. Top /\ F e. (II Cn J) /\ G e. (II Cn J)) -> (F(PHtpy` J)G) = (F{<.<.f, g>., z>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})}G))
4	3	adantr 425	. 2 |- (((J e. Top /\ F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) -> (F(PHtpy` J)G) = (F{<.<.f, g>., z>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})}G))
5		eqid 1884	. . . 4 |- {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))} = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))}
6		oprex 4907	. . . . . . 7 |- ((II X.t II) Cn J) e. _V
7	6	rabex 3461	. . . . . 6 |- {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))} e. _V
8		fveq1 4680	. . . . . . . . 9 |- (f = F -> (f` 0) = (F` 0))
9	8	eqeq1d 1892	. . . . . . . 8 |- (f = F -> ((f` 0) = (g` 0) <-> (F` 0) = (g` 0)))
10		fveq1 4680	. . . . . . . . 9 |- (f = F -> (f` 1) = (F` 1))
11	10	eqeq1d 1892	. . . . . . . 8 |- (f = F -> ((f` 1) = (g` 1) <-> (F` 1) = (g` 1)))
12	9, 11	anbi12d 690	. . . . . . 7 |- (f = F -> (((f` 0) = (g` 0) /\ (f` 1) = (g` 1)) <-> ((F` 0) = (g` 0) /\ (F` 1) = (g` 1))))
13		fveq1 4680	. . . . . . . . . . . . 13 |- (f = F -> (f` s) = (F` s))
14	13	eqeq2d 1895	. . . . . . . . . . . 12 |- (f = F -> ((sh0) = (f` s) <-> (sh0) = (F` s)))
15	14	anbi1d 679	. . . . . . . . . . 11 |- (f = F -> (((sh0) = (f` s) /\ (sh1) = (g` s)) <-> ((sh0) = (F` s) /\ (sh1) = (g` s))))
16	8	eqeq2d 1895	. . . . . . . . . . . 12 |- (f = F -> ((0hs) = (f` 0) <-> (0hs) = (F` 0)))
17	10	eqeq2d 1895	. . . . . . . . . . . 12 |- (f = F -> ((1hs) = (f` 1) <-> (1hs) = (F` 1)))
18	16, 17	anbi12d 690	. . . . . . . . . . 11 |- (f = F -> (((0hs) = (f` 0) /\ (1hs) = (f` 1)) <-> ((0hs) = (F` 0) /\ (1hs) = (F` 1))))
19	15, 18	anbi12d 690	. . . . . . . . . 10 |- (f = F -> ((((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1))) <-> (((sh0) = (F` s) /\ (sh1) = (g` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))))
20	19	ralbidv 2123	. . . . . . . . 9 |- (f = F -> (A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1))) <-> A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (g` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))))
21	20	rabbidv 2287	. . . . . . . 8 |- (f = F -> {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))} = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (g` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))})
22	21	eqeq2d 1895	. . . . . . 7 |- (f = F -> (z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))} <-> z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (g` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))}))
23	12, 22	anbi12d 690	. . . . . 6 |- (f = F -> ((((f` 0) = (g` 0) /\ (f` 1) = (g` 1)) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))}) <-> (((F` 0) = (g` 0) /\ (F` 1) = (g` 1)) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (g` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))})))
24		fveq1 4680	. . . . . . . . 9 |- (g = G -> (g` 0) = (G` 0))
25	24	eqeq2d 1895	. . . . . . . 8 |- (g = G -> ((F` 0) = (g` 0) <-> (F` 0) = (G` 0)))
26		fveq1 4680	. . . . . . . . 9 |- (g = G -> (g` 1) = (G` 1))
27	26	eqeq2d 1895	. . . . . . . 8 |- (g = G -> ((F` 1) = (g` 1) <-> (F` 1) = (G` 1)))
28	25, 27	anbi12d 690	. . . . . . 7 |- (g = G -> (((F` 0) = (g` 0) /\ (F` 1) = (g` 1)) <-> ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))))
29		fveq1 4680	. . . . . . . . . . . . 13 |- (g = G -> (g` s) = (G` s))
30	29	eqeq2d 1895	. . . . . . . . . . . 12 |- (g = G -> ((sh1) = (g` s) <-> (sh1) = (G` s)))
31	30	anbi2d 678	. . . . . . . . . . 11 |- (g = G -> (((sh0) = (F` s) /\ (sh1) = (g` s)) <-> ((sh0) = (F` s) /\ (sh1) = (G` s))))
32	31	anbi1d 679	. . . . . . . . . 10 |- (g = G -> ((((sh0) = (F` s) /\ (sh1) = (g` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1))) <-> (((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))))
33	32	ralbidv 2123	. . . . . . . . 9 |- (g = G -> (A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (g` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1))) <-> A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))))
34	33	rabbidv 2287	. . . . . . . 8 |- (g = G -> {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (g` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))} = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))})
35	34	eqeq2d 1895	. . . . . . 7 |- (g = G -> (z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (g` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))} <-> z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))}))
36	28, 35	anbi12d 690	. . . . . 6 |- (g = G -> ((((F` 0) = (g` 0) /\ (F` 1) = (g` 1)) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (g` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))}) <-> (((F` 0) = (G` 0) /\ (F` 1) = (G` 1)) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))})))
37		eqeq1 1890	. . . . . . 7 |- (z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))} -> (z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))} <-> {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))} = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))}))
38	37	anbi2d 678	. . . . . 6 |- (z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))} -> ((((F` 0) = (G` 0) /\ (F` 1) = (G` 1)) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))}) <-> (((F` 0) = (G` 0) /\ (F` 1) = (G` 1)) /\ {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))} = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))})))
39		moeq 2431	. . . . . . . 8 |- E*z z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))}
40	39	moani 1820	. . . . . . 7 |- E*z(((f` 0) = (g` 0) /\ (f` 1) = (g` 1)) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})
41	40	a1i 8	. . . . . 6 |- ((f e. (II Cn J) /\ g e. (II Cn J)) -> E*z(((f` 0) = (g` 0) /\ (f` 1) = (g` 1)) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))}))
42		anass 487	. . . . . . 7 |- ((((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))}) <-> ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (((f` 0) = (g` 0) /\ (f` 1) = (g` 1)) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})))
43	42	oprabbii 4923	. . . . . 6 |- {<.<.f, g>., z>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})} = {<.<.f, g>., z>. | ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (((f` 0) = (g` 0) /\ (f` 1) = (g` 1)) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))}))}
44	7, 23, 36, 38, 41, 43	oprabvali 4954	. . . . 5 |- ((F e. (II Cn J) /\ G e. (II Cn J)) -> ((((F` 0) = (G` 0) /\ (F` 1) = (G` 1)) /\ {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))} = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))}) -> (F{<.<.f, g>., z>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})}G) = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))}))
45	44	3adant1 894	. . . 4 |- ((J e. Top /\ F e. (II Cn J) /\ G e. (II Cn J)) -> ((((F` 0) = (G` 0) /\ (F` 1) = (G` 1)) /\ {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))} = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))}) -> (F{<.<.f, g>., z>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})}G) = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))}))
46	5, 45	mpan2i 763	. . 3 |- ((J e. Top /\ F e. (II Cn J) /\ G e. (II Cn J)) -> (((F` 0) = (G` 0) /\ (F` 1) = (G` 1)) -> (F{<.<.f, g>., z>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})}G) = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))}))
47	46	imp 377	. 2 |- (((J e. Top /\ F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) -> (F{<.<.f, g>., z>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ ((f` 0) = (g` 0) /\ (f` 1) = (g` 1))) /\ z = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (f` s) /\ (sh1) = (g` s)) /\ ((0hs) = (f` 0) /\ (1hs) = (f` 1)))})}G) = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))})
48	4, 47	eqtrd 1925	1 |- (((J e. Top /\ F e. (II Cn J) /\ G e. (II Cn J)) /\ ((F` 0) = (G` 0) /\ (F` 1) = (G` 1))) -> (F(PHtpy` J)G) = {h e. ((II X.t II) Cn J) | A.s e. (0[,]1)(((sh0) = (F` s) /\ (sh1) = (G` s)) /\ ((0hs) = (F` 0) /\ (1hs) = (F` 1)))})

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E*wmo 1772  A.wral 2105  {crab 2108  ` cfv 3998  (class class class)co 4884  {copab2 4885  0cc0 6386  1c1 6387  [,]cicc 7527  Topctop 8857   X.t ctx 8930   Cn ccn 9028  IIcii 15865  PHtpycphtpy 16043

This theorem is referenced by:  isphtpy 16048

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-rab 2112  df-v 2294  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-fv 4014  df-opr 4886  df-oprab 4887  df-phtpy 16045

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