Theorem pi1cof 22859

Description: Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
pi1co.p	|- P = ( J pi1 A )
pi1co.q	|- Q = ( K pi1 B )
pi1co.v	|- V = ( Base ` P )
pi1co.g	|- G = ran ( g e. U. V |-> <. [ g ] ( ~=ph ` J ) , [ ( F o. g ) ] ( ~=ph ` K ) >. )
pi1co.j	|- ( ph -> J e. (TopOn ` X ) )
pi1co.f	|- ( ph -> F e. ( J Cn K ) )
pi1co.a	|- ( ph -> A e. X )
pi1co.b	|- ( ph -> ( F ` A ) = B )

Assertion

Ref	Expression
pi1cof	|- ( ph -> G : V --> ( Base ` Q ) )

Distinct variable groups:   A, g   g, F   g, J   ph, g   g, K   P, g   Q, g   g, V

Allowed substitution hints:   B( g)   G( g)   X( g)

Proof of Theorem pi1cof

Dummy variables s h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pi1co.g	. . . 4 |- G = ran ( g e. U. V |-> <. [ g ] ( ~=ph ` J ) , [ ( F o. g ) ] ( ~=ph ` K ) >. )
2		fvex 6201	. . . . 5 |- ( ~=ph ` J ) e. _V
3		ecexg 7746	. . . . 5 |- ( ( ~=ph ` J ) e. _V -> [ g ] ( ~=ph ` J ) e. _V )
4	2, 3	mp1i 13	. . . 4 |- ( ( ph /\ g e. U. V ) -> [ g ] ( ~=ph ` J ) e. _V )
5		pi1co.q	. . . . 5 |- Q = ( K pi1 B )
6		eqid 2622	. . . . 5 |- ( Base ` Q ) = ( Base ` Q )
7		pi1co.f	. . . . . . . 8 |- ( ph -> F e. ( J Cn K ) )
8		cntop2 21045	. . . . . . . 8 |- ( F e. ( J Cn K ) -> K e. Top )
9	7, 8	syl 17	. . . . . . 7 |- ( ph -> K e. Top )
10		eqid 2622	. . . . . . . 8 |- U. K = U. K
11	10	toptopon 20722	. . . . . . 7 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
12	9, 11	sylib 208	. . . . . 6 |- ( ph -> K e. (TopOn ` U. K ) )
13	12	adantr 481	. . . . 5 |- ( ( ph /\ g e. U. V ) -> K e. (TopOn ` U. K ) )
14		pi1co.b	. . . . . . 7 |- ( ph -> ( F ` A ) = B )
15		pi1co.j	. . . . . . . . 9 |- ( ph -> J e. (TopOn ` X ) )
16		cnf2 21053	. . . . . . . . 9 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` U. K ) /\ F e. ( J Cn K ) ) -> F : X --> U. K )
17	15, 12, 7, 16	syl3anc 1326	. . . . . . . 8 |- ( ph -> F : X --> U. K )
18		pi1co.a	. . . . . . . 8 |- ( ph -> A e. X )
19	17, 18	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( F ` A ) e. U. K )
20	14, 19	eqeltrrd 2702	. . . . . 6 |- ( ph -> B e. U. K )
21	20	adantr 481	. . . . 5 |- ( ( ph /\ g e. U. V ) -> B e. U. K )
22		pi1co.p	. . . . . . . . 9 |- P = ( J pi1 A )
23		pi1co.v	. . . . . . . . . 10 |- V = ( Base ` P )
24	23	a1i 11	. . . . . . . . 9 |- ( ph -> V = ( Base ` P ) )
25	22, 15, 18, 24	pi1eluni 22842	. . . . . . . 8 |- ( ph -> ( g e. U. V <-> ( g e. ( II Cn J ) /\ ( g ` 0 ) = A /\ ( g ` 1 ) = A ) ) )
26	25	biimpa 501	. . . . . . 7 |- ( ( ph /\ g e. U. V ) -> ( g e. ( II Cn J ) /\ ( g ` 0 ) = A /\ ( g ` 1 ) = A ) )
27	26	simp1d 1073	. . . . . 6 |- ( ( ph /\ g e. U. V ) -> g e. ( II Cn J ) )
28	7	adantr 481	. . . . . 6 |- ( ( ph /\ g e. U. V ) -> F e. ( J Cn K ) )
29		cnco 21070	. . . . . 6 |- ( ( g e. ( II Cn J ) /\ F e. ( J Cn K ) ) -> ( F o. g ) e. ( II Cn K ) )
30	27, 28, 29	syl2anc 693	. . . . 5 |- ( ( ph /\ g e. U. V ) -> ( F o. g ) e. ( II Cn K ) )
31		iitopon 22682	. . . . . . . . 9 |- II e. (TopOn ` ( 0 [,] 1 ) )
32	31	a1i 11	. . . . . . . 8 |- ( ( ph /\ g e. U. V ) -> II e. (TopOn ` ( 0 [,] 1 ) ) )
33	15	adantr 481	. . . . . . . 8 |- ( ( ph /\ g e. U. V ) -> J e. (TopOn ` X ) )
34		cnf2 21053	. . . . . . . 8 |- ( ( II e. (TopOn ` ( 0 [,] 1 ) ) /\ J e. (TopOn ` X ) /\ g e. ( II Cn J ) ) -> g : ( 0 [,] 1 ) --> X )
35	32, 33, 27, 34	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ g e. U. V ) -> g : ( 0 [,] 1 ) --> X )
36		0elunit 12290	. . . . . . 7 |- 0 e. ( 0 [,] 1 )
37		fvco3 6275	. . . . . . 7 |- ( ( g : ( 0 [,] 1 ) --> X /\ 0 e. ( 0 [,] 1 ) ) -> ( ( F o. g ) ` 0 ) = ( F ` ( g ` 0 ) ) )
38	35, 36, 37	sylancl 694	. . . . . 6 |- ( ( ph /\ g e. U. V ) -> ( ( F o. g ) ` 0 ) = ( F ` ( g ` 0 ) ) )
39	26	simp2d 1074	. . . . . . 7 |- ( ( ph /\ g e. U. V ) -> ( g ` 0 ) = A )
40	39	fveq2d 6195	. . . . . 6 |- ( ( ph /\ g e. U. V ) -> ( F ` ( g ` 0 ) ) = ( F ` A ) )
41	14	adantr 481	. . . . . 6 |- ( ( ph /\ g e. U. V ) -> ( F ` A ) = B )
42	38, 40, 41	3eqtrd 2660	. . . . 5 |- ( ( ph /\ g e. U. V ) -> ( ( F o. g ) ` 0 ) = B )
43		1elunit 12291	. . . . . . 7 |- 1 e. ( 0 [,] 1 )
44		fvco3 6275	. . . . . . 7 |- ( ( g : ( 0 [,] 1 ) --> X /\ 1 e. ( 0 [,] 1 ) ) -> ( ( F o. g ) ` 1 ) = ( F ` ( g ` 1 ) ) )
45	35, 43, 44	sylancl 694	. . . . . 6 |- ( ( ph /\ g e. U. V ) -> ( ( F o. g ) ` 1 ) = ( F ` ( g ` 1 ) ) )
46	26	simp3d 1075	. . . . . . 7 |- ( ( ph /\ g e. U. V ) -> ( g ` 1 ) = A )
47	46	fveq2d 6195	. . . . . 6 |- ( ( ph /\ g e. U. V ) -> ( F ` ( g ` 1 ) ) = ( F ` A ) )
48	45, 47, 41	3eqtrd 2660	. . . . 5 |- ( ( ph /\ g e. U. V ) -> ( ( F o. g ) ` 1 ) = B )
49	5, 6, 13, 21, 30, 42, 48	elpi1i 22846	. . . 4 |- ( ( ph /\ g e. U. V ) -> [ ( F o. g ) ] ( ~=ph ` K ) e. ( Base ` Q ) )
50		eceq1 7782	. . . 4 |- ( g = h -> [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) )
51		coeq2 5280	. . . . 5 |- ( g = h -> ( F o. g ) = ( F o. h ) )
52	51	eceq1d 7783	. . . 4 |- ( g = h -> [ ( F o. g ) ] ( ~=ph ` K ) = [ ( F o. h ) ] ( ~=ph ` K ) )
53		phtpcer 22794	. . . . . 6 |- ( ~=ph ` K ) Er ( II Cn K )
54	53	a1i 11	. . . . 5 |- ( ( ph /\ ( g e. U. V /\ h e. U. V /\ [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) ) ) -> ( ~=ph ` K ) Er ( II Cn K ) )
55		simpr3 1069	. . . . . . 7 |- ( ( ph /\ ( g e. U. V /\ h e. U. V /\ [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) ) ) -> [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) )
56		phtpcer 22794	. . . . . . . . 9 |- ( ~=ph ` J ) Er ( II Cn J )
57	56	a1i 11	. . . . . . . 8 |- ( ( ph /\ ( g e. U. V /\ h e. U. V /\ [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) ) ) -> ( ~=ph ` J ) Er ( II Cn J ) )
58		simpr1 1067	. . . . . . . . . 10 |- ( ( ph /\ ( g e. U. V /\ h e. U. V /\ [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) ) ) -> g e. U. V )
59	25	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( g e. U. V /\ h e. U. V /\ [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) ) ) -> ( g e. U. V <-> ( g e. ( II Cn J ) /\ ( g ` 0 ) = A /\ ( g ` 1 ) = A ) ) )
60	58, 59	mpbid 222	. . . . . . . . 9 |- ( ( ph /\ ( g e. U. V /\ h e. U. V /\ [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) ) ) -> ( g e. ( II Cn J ) /\ ( g ` 0 ) = A /\ ( g ` 1 ) = A ) )
61	60	simp1d 1073	. . . . . . . 8 |- ( ( ph /\ ( g e. U. V /\ h e. U. V /\ [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) ) ) -> g e. ( II Cn J ) )
62	57, 61	erth 7791	. . . . . . 7 |- ( ( ph /\ ( g e. U. V /\ h e. U. V /\ [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) ) ) -> ( g ( ~=ph ` J ) h <-> [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) ) )
63	55, 62	mpbird 247	. . . . . 6 |- ( ( ph /\ ( g e. U. V /\ h e. U. V /\ [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) ) ) -> g ( ~=ph ` J ) h )
64	7	adantr 481	. . . . . 6 |- ( ( ph /\ ( g e. U. V /\ h e. U. V /\ [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) ) ) -> F e. ( J Cn K ) )
65	63, 64	phtpcco2 22799	. . . . 5 |- ( ( ph /\ ( g e. U. V /\ h e. U. V /\ [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) ) ) -> ( F o. g ) ( ~=ph ` K ) ( F o. h ) )
66	54, 65	erthi 7793	. . . 4 |- ( ( ph /\ ( g e. U. V /\ h e. U. V /\ [ g ] ( ~=ph ` J ) = [ h ] ( ~=ph ` J ) ) ) -> [ ( F o. g ) ] ( ~=ph ` K ) = [ ( F o. h ) ] ( ~=ph ` K ) )
67	1, 4, 49, 50, 52, 66	fliftfund 6563	. . 3 |- ( ph -> Fun G )
68	1, 4, 49	fliftf 6565	. . 3 |- ( ph -> ( Fun G <-> G : ran ( g e. U. V |-> [ g ] ( ~=ph ` J ) ) --> ( Base ` Q ) ) )
69	67, 68	mpbid 222	. 2 |- ( ph -> G : ran ( g e. U. V |-> [ g ] ( ~=ph ` J ) ) --> ( Base ` Q ) )
70	22, 15, 18, 24	pi1bas2 22841	. . . 4 |- ( ph -> V = ( U. V /. ( ~=ph ` J ) ) )
71		df-qs 7748	. . . . 5 |- ( U. V /. ( ~=ph ` J ) ) = { s | E. g e. U. V s = [ g ] ( ~=ph ` J ) }
72		eqid 2622	. . . . . 6 |- ( g e. U. V |-> [ g ] ( ~=ph ` J ) ) = ( g e. U. V |-> [ g ] ( ~=ph ` J ) )
73	72	rnmpt 5371	. . . . 5 |- ran ( g e. U. V |-> [ g ] ( ~=ph ` J ) ) = { s | E. g e. U. V s = [ g ] ( ~=ph ` J ) }
74	71, 73	eqtr4i 2647	. . . 4 |- ( U. V /. ( ~=ph ` J ) ) = ran ( g e. U. V |-> [ g ] ( ~=ph ` J ) )
75	70, 74	syl6eq 2672	. . 3 |- ( ph -> V = ran ( g e. U. V |-> [ g ] ( ~=ph ` J ) ) )
76	75	feq2d 6031	. 2 |- ( ph -> ( G : V --> ( Base ` Q ) <-> G : ran ( g e. U. V |-> [ g ] ( ~=ph ` J ) ) --> ( Base ` Q ) ) )
77	69, 76	mpbird 247	1 |- ( ph -> G : V --> ( Base ` Q ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   <.cop 4183   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   Er wer 7739   [cec 7740   /.cqs 7741   0cc0 9936   1c1 9937   [,]cicc 12178   Basecbs 15857   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   IIcii 22678   ~=ph cphtpc 22768   pi1 cpi1 22803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-qus 16169  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-ii 22680  df-htpy 22769  df-phtpy 22770  df-phtpc 22791  df-om1 22806  df-pi1 22808

This theorem is referenced by:  pi1coval  22860  pi1coghm  22861