Theorem pcoloopf 16079

Description: Path concatenation restricted to loops with a fixed basepoint.

Hypotheses

Ref	Expression
pcoloopf.1	|- X = U.J
pcoloopf.2	|- S = {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}

Assertion

Ref	Expression
pcoloopf	|- ((J e. Top /\ Y e. X) -> ((*p` J) |` (S X. S)):(S X. S)-->S)

Distinct variable groups:   t,J   t,Y

Proof of Theorem pcoloopf

Step	Hyp	Ref	Expression
1		ffnoprv 4943	. 2 |- (((*p` J) |` (S X. S)):(S X. S)-->S <-> (((*p` J) |` (S X. S)) Fn (S X. S) /\ A.f e. S A.g e. S (f((*p` J) |` (S X. S))g) e. S))
2		fnssres 4526	. . 3 |- (((*p` J) Fn {<.f, g>. | ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0))} /\ (S X. S) C_ {<.f, g>. | ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0))}) -> ((*p` J) |` (S X. S)) Fn (S X. S))
3		oprex 4907	. . . . . . . . 9 |- (0[,]1) e. _V
4	3	opabex2 4539	. . . . . . . 8 |- {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))} e. _V
5	4	eueq1 2428	. . . . . . 7 |- E!h h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))}
6	5	a1i 8	. . . . . 6 |- (((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)) -> E!h h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))})
7	6	fnoprab 4942	. . . . 5 |- {<.<.f, g>., h>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))})} Fn {<.f, g>. | ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0))}
8		pcofval 16072	. . . . . 6 |- (J e. Top -> (*p` J) = {<.<.f, g>., h>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))})})
9	8	fneq1d 4505	. . . . 5 |- (J e. Top -> ((*p` J) Fn {<.f, g>. | ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0))} <-> {<.<.f, g>., h>. | (((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)) /\ h = {<.x, y>. | (x e. (0[,]1) /\ y = if(x <_ (1 / 2), (f` (2 x. x)), (g` ((2 x. x) - 1))))})} Fn {<.f, g>. | ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0))}))
10	7, 9	mpbiri 211	. . . 4 |- (J e. Top -> (*p` J) Fn {<.f, g>. | ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0))})
11	10	adantr 425	. . 3 |- ((J e. Top /\ Y e. X) -> (*p` J) Fn {<.f, g>. | ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0))})
12		df-xp 4000	. . . 4 |- (S X. S) = {<.f, g>. | (f e. S /\ g e. S)}
13		eqtr3 1907	. . . . . . . . 9 |- (((f` 1) = Y /\ (g` 0) = Y) -> (f` 1) = (g` 0))
14	13	ad2ant2lr 446	. . . . . . . 8 |- ((((f` 0) = Y /\ (f` 1) = Y) /\ ((g` 0) = Y /\ (g` 1) = Y)) -> (f` 1) = (g` 0))
15	14	anim2i 362	. . . . . . 7 |- (((f e. (II Cn J) /\ g e. (II Cn J)) /\ (((f` 0) = Y /\ (f` 1) = Y) /\ ((g` 0) = Y /\ (g` 1) = Y))) -> ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)))
16	15	an4s 566	. . . . . 6 |- (((f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)) /\ (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y))) -> ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)))
17		fveq1 4680	. . . . . . . . 9 |- (t = f -> (t` 0) = (f` 0))
18	17	eqeq1d 1892	. . . . . . . 8 |- (t = f -> ((t` 0) = Y <-> (f` 0) = Y))
19		fveq1 4680	. . . . . . . . 9 |- (t = f -> (t` 1) = (f` 1))
20	19	eqeq1d 1892	. . . . . . . 8 |- (t = f -> ((t` 1) = Y <-> (f` 1) = Y))
21	18, 20	anbi12d 690	. . . . . . 7 |- (t = f -> (((t` 0) = Y /\ (t` 1) = Y) <-> ((f` 0) = Y /\ (f` 1) = Y)))
22		pcoloopf.2	. . . . . . 7 |- S = {t e. (II Cn J) | ((t` 0) = Y /\ (t` 1) = Y)}
23	21, 22	elrab2 2416	. . . . . 6 |- (f e. S <-> (f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)))
24		fveq1 4680	. . . . . . . . 9 |- (t = g -> (t` 0) = (g` 0))
25	24	eqeq1d 1892	. . . . . . . 8 |- (t = g -> ((t` 0) = Y <-> (g` 0) = Y))
26		fveq1 4680	. . . . . . . . 9 |- (t = g -> (t` 1) = (g` 1))
27	26	eqeq1d 1892	. . . . . . . 8 |- (t = g -> ((t` 1) = Y <-> (g` 1) = Y))
28	25, 27	anbi12d 690	. . . . . . 7 |- (t = g -> (((t` 0) = Y /\ (t` 1) = Y) <-> ((g` 0) = Y /\ (g` 1) = Y)))
29	28, 22	elrab2 2416	. . . . . 6 |- (g e. S <-> (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y)))
30	16, 23, 29	syl2anb 504	. . . . 5 |- ((f e. S /\ g e. S) -> ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)))
31	30	ssopab2i 3574	. . . 4 |- {<.f, g>. | (f e. S /\ g e. S)} C_ {<.f, g>. | ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0))}
32	12, 31	eqsstri 2647	. . 3 |- (S X. S) C_ {<.f, g>. | ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0))}
33	2, 11, 32	sylancl 525	. 2 |- ((J e. Top /\ Y e. X) -> ((*p` J) |` (S X. S)) Fn (S X. S))
34		oprvres 4963	. . . . . 6 |- ((f e. S /\ g e. S) -> (f((*p` J) |` (S X. S))g) = (f(*p` J)g))
35	34	adantl 424	. . . . 5 |- (((J e. Top /\ Y e. X) /\ (f e. S /\ g e. S)) -> (f((*p` J) |` (S X. S))g) = (f(*p` J)g))
36		pcocn 16076	. . . . . . . . . 10 |- ((J e. Top /\ (f e. (II Cn J) /\ g e. (II Cn J) /\ (f` 1) = (g` 0))) -> (f(*p` J)g) e. (II Cn J))
37		df-3an 860	. . . . . . . . . . 11 |- ((f e. (II Cn J) /\ g e. (II Cn J) /\ (f` 1) = (g` 0)) <-> ((f e. (II Cn J) /\ g e. (II Cn J)) /\ (f` 1) = (g` 0)))
38	16, 37	sylibr 217	. . . . . . . . . 10 |- (((f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)) /\ (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y))) -> (f e. (II Cn J) /\ g e. (II Cn J) /\ (f` 1) = (g` 0)))
39	36, 38	sylan2 500	. . . . . . . . 9 |- ((J e. Top /\ ((f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)) /\ (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y)))) -> (f(*p` J)g) e. (II Cn J))
40		pco0 16077	. . . . . . . . . . 11 |- ((J e. Top /\ (f e. (II Cn J) /\ g e. (II Cn J) /\ (f` 1) = (g` 0))) -> ((f(*p` J)g)` 0) = (f` 0))
41	40, 38	sylan2 500	. . . . . . . . . 10 |- ((J e. Top /\ ((f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)) /\ (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y)))) -> ((f(*p` J)g)` 0) = (f` 0))
42		simplrl 454	. . . . . . . . . . 11 |- (((f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)) /\ (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y))) -> (f` 0) = Y)
43	42	adantl 424	. . . . . . . . . 10 |- ((J e. Top /\ ((f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)) /\ (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y)))) -> (f` 0) = Y)
44	41, 43	eqtrd 1925	. . . . . . . . 9 |- ((J e. Top /\ ((f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)) /\ (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y)))) -> ((f(*p` J)g)` 0) = Y)
45		pco1 16078	. . . . . . . . . . 11 |- ((J e. Top /\ (f e. (II Cn J) /\ g e. (II Cn J) /\ (f` 1) = (g` 0))) -> ((f(*p` J)g)` 1) = (g` 1))
46	45, 38	sylan2 500	. . . . . . . . . 10 |- ((J e. Top /\ ((f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)) /\ (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y)))) -> ((f(*p` J)g)` 1) = (g` 1))
47		simprrr 459	. . . . . . . . . . 11 |- (((f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)) /\ (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y))) -> (g` 1) = Y)
48	47	adantl 424	. . . . . . . . . 10 |- ((J e. Top /\ ((f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)) /\ (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y)))) -> (g` 1) = Y)
49	46, 48	eqtrd 1925	. . . . . . . . 9 |- ((J e. Top /\ ((f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)) /\ (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y)))) -> ((f(*p` J)g)` 1) = Y)
50	39, 44, 49	jca32 312	. . . . . . . 8 |- ((J e. Top /\ ((f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)) /\ (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y)))) -> ((f(*p` J)g) e. (II Cn J) /\ (((f(*p` J)g)` 0) = Y /\ ((f(*p` J)g)` 1) = Y)))
51		fveq1 4680	. . . . . . . . . . 11 |- (t = (f(*p` J)g) -> (t` 0) = ((f(*p` J)g)` 0))
52	51	eqeq1d 1892	. . . . . . . . . 10 |- (t = (f(*p` J)g) -> ((t` 0) = Y <-> ((f(*p` J)g)` 0) = Y))
53		fveq1 4680	. . . . . . . . . . 11 |- (t = (f(*p` J)g) -> (t` 1) = ((f(*p` J)g)` 1))
54	53	eqeq1d 1892	. . . . . . . . . 10 |- (t = (f(*p` J)g) -> ((t` 1) = Y <-> ((f(*p` J)g)` 1) = Y))
55	52, 54	anbi12d 690	. . . . . . . . 9 |- (t = (f(*p` J)g) -> (((t` 0) = Y /\ (t` 1) = Y) <-> (((f(*p` J)g)` 0) = Y /\ ((f(*p` J)g)` 1) = Y)))
56	55, 22	elrab2 2416	. . . . . . . 8 |- ((f(*p` J)g) e. S <-> ((f(*p` J)g) e. (II Cn J) /\ (((f(*p` J)g)` 0) = Y /\ ((f(*p` J)g)` 1) = Y)))
57	50, 56	sylibr 217	. . . . . . 7 |- ((J e. Top /\ ((f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)) /\ (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y)))) -> (f(*p` J)g) e. S)
58	23, 29	anbi12i 540	. . . . . . 7 |- ((f e. S /\ g e. S) <-> ((f e. (II Cn J) /\ ((f` 0) = Y /\ (f` 1) = Y)) /\ (g e. (II Cn J) /\ ((g` 0) = Y /\ (g` 1) = Y))))
59	57, 58	sylan2b 501	. . . . . 6 |- ((J e. Top /\ (f e. S /\ g e. S)) -> (f(*p` J)g) e. S)
60	59	adantlr 429	. . . . 5 |- (((J e. Top /\ Y e. X) /\ (f e. S /\ g e. S)) -> (f(*p` J)g) e. S)
61	35, 60	eqeltrd 1971	. . . 4 |- (((J e. Top /\ Y e. X) /\ (f e. S /\ g e. S)) -> (f((*p` J) |` (S X. S))g) e. S)
62	61	ex 402	. . 3 |- ((J e. Top /\ Y e. X) -> ((f e. S /\ g e. S) -> (f((*p` J) |` (S X. S))g) e. S))
63	62	r19.21aivv 2183	. 2 |- ((J e. Top /\ Y e. X) -> A.f e. S A.g e. S (f((*p` J) |` (S X. S))g) e. S)
64	1, 33, 63	sylanbrc 527	1 |- ((J e. Top /\ Y e. X) -> ((*p` J) |` (S X. S)):(S X. S)-->S)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E!weu 1771  A.wral 2105  {crab 2108   C_ wss 2593  ifcif 2982  U.cuni 3177   class class class wbr 3338  {copab 3395   X. cxp 3984   |` cres 3988   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  0cc0 6386  1c1 6387   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448  2c2 7145  [,]cicc 7527  Topctop 8857   Cn ccn 9028  IIcii 15865  *pcpco 16067

This theorem is referenced by:  pi1f 16093  pi1val 16094

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  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-pco 16069

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