Theorem phtpyid 16049

Description: A homotopy from a path to itself.

Hypothesis

Ref	Expression
phtpyid.1	|- H = {<.<.x, y>., z>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ z = (F` x))}

Assertion

Ref	Expression
phtpyid	|- ((J e. Top /\ F e. (II Cn J)) -> H e. (F(PHtpy` J)F))

Distinct variable groups:   x,J,y,z   x,F,y,z

Proof of Theorem phtpyid

Step	Hyp	Ref	Expression
1		simpl 346	. . 3 |- ((J e. Top /\ F e. (II Cn J)) -> J e. Top)
2		simpr 350	. . 3 |- ((J e. Top /\ F e. (II Cn J)) -> F e. (II Cn J))
3		eqidd 1885	. . 3 |- ((J e. Top /\ F e. (II Cn J)) -> (F` 0) = (F` 0))
4		eqidd 1885	. . 3 |- ((J e. Top /\ F e. (II Cn J)) -> (F` 1) = (F` 1))
5		isphtpy 16048	. . 3 |- (((J e. Top /\ F e. (II Cn J) /\ F e. (II Cn J)) /\ ((F` 0) = (F` 0) /\ (F` 1) = (F` 1))) -> (H e. (F(PHtpy` J)F) <-> (H e. ((II X.t II) Cn J) /\ A.s e. (0[,]1)(((sH0) = (F` s) /\ (sH1) = (F` s)) /\ ((0Hs) = (F` 0) /\ (1Hs) = (F` 1))))))
6	1, 2, 2, 3, 4, 5	syl32anc 1108	. 2 |- ((J e. Top /\ F e. (II Cn J)) -> (H e. (F(PHtpy` J)F) <-> (H e. ((II X.t II) Cn J) /\ A.s e. (0[,]1)(((sH0) = (F` s) /\ (sH1) = (F` s)) /\ ((0Hs) = (F` 0) /\ (1Hs) = (F` 1))))))
7		iitop 15867	. . . . 5 |- II e. Top
8		iiuni 15868	. . . . . 6 |- (0[,]1) = U.II
9		eqid 1884	. . . . . 6 |- U.J = U.J
10	8, 9	cnf 9038	. . . . 5 |- ((II e. Top /\ J e. Top /\ F e. (II Cn J)) -> F:(0[,]1)-->U.J)
11	7, 10	mp3an1 1178	. . . 4 |- ((J e. Top /\ F e. (II Cn J)) -> F:(0[,]1)-->U.J)
12		ffn 4562	. . . . 5 |- (F:(0[,]1)-->U.J -> F Fn (0[,]1))
13		simpl 346	. . . . . 6 |- ((x e. (0[,]1) /\ y e. (0[,]1)) -> x e. (0[,]1))
14		df1st2 5068	. . . . . . . 8 |- {<.<.x, y>., w>. | w = x} = (1st |` (_V X. _V))
15		reseq1 4218	. . . . . . . 8 |- ({<.<.x, y>., w>. | w = x} = (1st |` (_V X. _V)) -> ({<.<.x, y>., w>. | w = x} |` ((0[,]1) X. (0[,]1))) = ((1st |` (_V X. _V)) |` ((0[,]1) X. (0[,]1))))
16	14, 15	ax-mp 7	. . . . . . 7 |- ({<.<.x, y>., w>. | w = x} |` ((0[,]1) X. (0[,]1))) = ((1st |` (_V X. _V)) |` ((0[,]1) X. (0[,]1)))
17		resoprab 4938	. . . . . . 7 |- ({<.<.x, y>., w>. | w = x} |` ((0[,]1) X. (0[,]1))) = {<.<.x, y>., w>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ w = x)}
18		xpss 4056	. . . . . . . 8 |- ((0[,]1) X. (0[,]1)) C_ (_V X. _V)
19		resabs1 4244	. . . . . . . 8 |- (((0[,]1) X. (0[,]1)) C_ (_V X. _V) -> ((1st |` (_V X. _V)) |` ((0[,]1) X. (0[,]1))) = (1st |` ((0[,]1) X. (0[,]1))))
20	18, 19	ax-mp 7	. . . . . . 7 |- ((1st |` (_V X. _V)) |` ((0[,]1) X. (0[,]1))) = (1st |` ((0[,]1) X. (0[,]1)))
21	16, 17, 20	3eqtr3ri 1920	. . . . . 6 |- (1st |` ((0[,]1) X. (0[,]1))) = {<.<.x, y>., w>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ w = x)}
22		phtpyid.1	. . . . . 6 |- H = {<.<.x, y>., z>. | ((x e. (0[,]1) /\ y e. (0[,]1)) /\ z = (F` x))}
23	13, 21, 22	oprabco 10159	. . . . 5 |- (F Fn (0[,]1) -> H = (F o. (1st |` ((0[,]1) X. (0[,]1)))))
24	12, 23	syl 12	. . . 4 |- (F:(0[,]1)-->U.J -> H = (F o. (1st |` ((0[,]1) X. (0[,]1)))))
25	11, 24	syl 12	. . 3 |- ((J e. Top /\ F e. (II Cn J)) -> H = (F o. (1st |` ((0[,]1) X. (0[,]1)))))
26		eqid 1884	. . . . . . 7 |- (II X.t II) = (II X.t II)
27	26	txtop 8934	. . . . . 6 |- ((II e. Top /\ II e. Top) -> (II X.t II) e. Top)
28	7, 7, 27	mp2an 761	. . . . 5 |- (II X.t II) e. Top
29	28	a1i 8	. . . 4 |- ((J e. Top /\ F e. (II Cn J)) -> (II X.t II) e. Top)
30	7	a1i 8	. . . 4 |- ((J e. Top /\ F e. (II Cn J)) -> II e. Top)
31		eqid 1884	. . . . . . 7 |- ((0[,]1) X. (0[,]1)) = ((0[,]1) X. (0[,]1))
32	26, 8, 8, 31	tx1cn 10223	. . . . . 6 |- ((II e. Top /\ II e. Top) -> (1st |` ((0[,]1) X. (0[,]1))) e. ((II X.t II) Cn II))
33	7, 7, 32	mp2an 761	. . . . 5 |- (1st |` ((0[,]1) X. (0[,]1))) e. ((II X.t II) Cn II)
34	2, 33	jctil 316	. . . 4 |- ((J e. Top /\ F e. (II Cn J)) -> ((1st |` ((0[,]1) X. (0[,]1))) e. ((II X.t II) Cn II) /\ F e. (II Cn J)))
35		cnco 9045	. . . 4 |- ((((II X.t II) e. Top /\ II e. Top /\ J e. Top) /\ ((1st |` ((0[,]1) X. (0[,]1))) e. ((II X.t II) Cn II) /\ F e. (II Cn J))) -> (F o. (1st |` ((0[,]1) X. (0[,]1)))) e. ((II X.t II) Cn J))
36	29, 30, 1, 34, 35	syl31anc 1103	. . 3 |- ((J e. Top /\ F e. (II Cn J)) -> (F o. (1st |` ((0[,]1) X. (0[,]1)))) e. ((II X.t II) Cn J))
37	25, 36	eqeltrd 1971	. 2 |- ((J e. Top /\ F e. (II Cn J)) -> H e. ((II X.t II) Cn J))
38		0re 6603	. . . . . . 7 |- 0 e. RR
39		1re 6598	. . . . . . 7 |- 1 e. RR
40		lt01 6871	. . . . . . . 8 |- 0 < 1
41	38, 39, 40	ltleii 6756	. . . . . . 7 |- 0 <_ 1
42		lbicc2 7573	. . . . . . 7 |- ((0 e. RR /\ 1 e. RR /\ 0 <_ 1) -> 0 e. (0[,]1))
43	38, 39, 41, 42	mp3an 1191	. . . . . 6 |- 0 e. (0[,]1)
44		fvex 4689	. . . . . . 7 |- (F` s) e. _V
45		fveq2 4681	. . . . . . 7 |- (x = s -> (F` x) = (F` s))
46		eqidd 1885	. . . . . . 7 |- (y = 0 -> (F` s) = (F` s))
47	44, 45, 46, 22	oprabval2 4957	. . . . . 6 |- ((s e. (0[,]1) /\ 0 e. (0[,]1)) -> (sH0) = (F` s))
48	43, 47	mpan2 760	. . . . 5 |- (s e. (0[,]1) -> (sH0) = (F` s))
49		ubicc2 7574	. . . . . . 7 |- ((0 e. RR /\ 1 e. RR /\ 0 <_ 1) -> 1 e. (0[,]1))
50	38, 39, 41, 49	mp3an 1191	. . . . . 6 |- 1 e. (0[,]1)
51		eqidd 1885	. . . . . . 7 |- (y = 1 -> (F` s) = (F` s))
52	44, 45, 51, 22	oprabval2 4957	. . . . . 6 |- ((s e. (0[,]1) /\ 1 e. (0[,]1)) -> (sH1) = (F` s))
53	50, 52	mpan2 760	. . . . 5 |- (s e. (0[,]1) -> (sH1) = (F` s))
54		fvex 4689	. . . . . . . 8 |- (F` 0) e. _V
55		fveq2 4681	. . . . . . . 8 |- (x = 0 -> (F` x) = (F` 0))
56		eqidd 1885	. . . . . . . 8 |- (y = s -> (F` 0) = (F` 0))
57	54, 55, 56, 22	oprabval2 4957	. . . . . . 7 |- ((0 e. (0[,]1) /\ s e. (0[,]1)) -> (0Hs) = (F` 0))
58	43, 57	mpan 759	. . . . . 6 |- (s e. (0[,]1) -> (0Hs) = (F` 0))
59		fvex 4689	. . . . . . . 8 |- (F` 1) e. _V
60		fveq2 4681	. . . . . . . 8 |- (x = 1 -> (F` x) = (F` 1))
61		eqidd 1885	. . . . . . . 8 |- (y = s -> (F` 1) = (F` 1))
62	59, 60, 61, 22	oprabval2 4957	. . . . . . 7 |- ((1 e. (0[,]1) /\ s e. (0[,]1)) -> (1Hs) = (F` 1))
63	50, 62	mpan 759	. . . . . 6 |- (s e. (0[,]1) -> (1Hs) = (F` 1))
64	58, 63	jca 310	. . . . 5 |- (s e. (0[,]1) -> ((0Hs) = (F` 0) /\ (1Hs) = (F` 1)))
65	48, 53, 64	jca31 311	. . . 4 |- (s e. (0[,]1) -> (((sH0) = (F` s) /\ (sH1) = (F` s)) /\ ((0Hs) = (F` 0) /\ (1Hs) = (F` 1))))
66	65	adantl 424	. . 3 |- (((J e. Top /\ F e. (II Cn J)) /\ s e. (0[,]1)) -> (((sH0) = (F` s) /\ (sH1) = (F` s)) /\ ((0Hs) = (F` 0) /\ (1Hs) = (F` 1))))
67	66	r19.21aiva 2176	. 2 |- ((J e. Top /\ F e. (II Cn J)) -> A.s e. (0[,]1)(((sH0) = (F` s) /\ (sH1) = (F` s)) /\ ((0Hs) = (F` 0) /\ (1Hs) = (F` 1))))
68	6, 37, 67	mpbir2and 802	1 |- ((J e. Top /\ F e. (II Cn J)) -> H e. (F(PHtpy` J)F))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  U.cuni 3177   class class class wbr 3338   X. cxp 3984   |` cres 3988   o. ccom 3990   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  RRcr 6385  0cc0 6386  1c1 6387   <_ cle 6448  [,]cicc 7527  Topctop 8857   X.t ctx 8930   Cn ccn 9028  IIcii 15865  PHtpycphtpy 16043

This theorem is referenced by:  phtpcdm 16061

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-inf2 5731

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-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-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-n0 7309  df-z 7345  df-icc 7531  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-top 8861  df-bases 8863  df-topgen 8864  df-tx 8931  df-cn 9030  df-met 9070  df-bl 9072  df-opn 9073  df-ii 15866  df-phtpy 16045

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