Theorem cvmliftlem13 31278

Description: Lemma for cvmlift 31281. The initial value of K is P because Q ( 1 ) is a subset of K which takes value P at 0. (Contributed by Mario Carneiro, 16-Feb-2015.)

Hypotheses

Ref	Expression
cvmliftlem.1	|- S = ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. u e. s ( A. v e. ( s \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C ↾t u ) Homeo ( J ↾t k ) ) ) ) } )
cvmliftlem.b	|- B = U. C
cvmliftlem.x	|- X = U. J
cvmliftlem.f	|- ( ph -> F e. ( C CovMap J ) )
cvmliftlem.g	|- ( ph -> G e. ( II Cn J ) )
cvmliftlem.p	|- ( ph -> P e. B )
cvmliftlem.e	|- ( ph -> ( F ` P ) = ( G ` 0 ) )
cvmliftlem.n	|- ( ph -> N e. NN )
cvmliftlem.t	|- ( ph -> T : ( 1 ... N ) --> U_ j e. J ( { j } X. ( S ` j ) ) )
cvmliftlem.a	|- ( ph -> A. k e. ( 1 ... N ) ( G " ( ( ( k - 1 ) / N ) [,] ( k / N ) ) ) C_ ( 1st ` ( T ` k ) ) )
cvmliftlem.l	|- L = ( topGen ` ran (,) )
cvmliftlem.q	|- Q = seq 0 ( ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) , ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) )
cvmliftlem.k	|- K = U_ k e. ( 1 ... N ) ( Q ` k )

Assertion

Ref	Expression
cvmliftlem13	|- ( ph -> ( K ` 0 ) = P )

Distinct variable groups:   v, b, z, B   j, b, k, m, s, u, x, F, v, z   z, L   P, b, k, m, u, v, x, z   C, b, j, k, s, u, v, z   ph, j, s, x, z   N, b, k, m, u, v, x, z   S, b, j, k, s, u, v, x, z   j, X   G, b, j, k, m, s, u, v, x, z   T, b, j, k, m, s, u, v, x, z   J, b, j, k, s, u, v, x, z   Q, b, k, m, u, v, x, z

Allowed substitution hints:   ph( v, u, k, m, b)   B( x, u, j, k, m, s)   C( x, m)   P( j, s)   Q( j, s)   S( m)   J( m)   K( x, z, v, u, j, k, m, s, b)   L( x, v, u, j, k, m, s, b)   N( j, s)   X( x, z, v, u, k, m, s, b)

Proof of Theorem cvmliftlem13

Step	Hyp	Ref	Expression
1		cvmliftlem.1	. . . . . . 7 |- S = ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. u e. s ( A. v e. ( s \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C ↾t u ) Homeo ( J ↾t k ) ) ) ) } )
2		cvmliftlem.b	. . . . . . 7 |- B = U. C
3		cvmliftlem.x	. . . . . . 7 |- X = U. J
4		cvmliftlem.f	. . . . . . 7 |- ( ph -> F e. ( C CovMap J ) )
5		cvmliftlem.g	. . . . . . 7 |- ( ph -> G e. ( II Cn J ) )
6		cvmliftlem.p	. . . . . . 7 |- ( ph -> P e. B )
7		cvmliftlem.e	. . . . . . 7 |- ( ph -> ( F ` P ) = ( G ` 0 ) )
8		cvmliftlem.n	. . . . . . 7 |- ( ph -> N e. NN )
9		cvmliftlem.t	. . . . . . 7 |- ( ph -> T : ( 1 ... N ) --> U_ j e. J ( { j } X. ( S ` j ) ) )
10		cvmliftlem.a	. . . . . . 7 |- ( ph -> A. k e. ( 1 ... N ) ( G " ( ( ( k - 1 ) / N ) [,] ( k / N ) ) ) C_ ( 1st ` ( T ` k ) ) )
11		cvmliftlem.l	. . . . . . 7 |- L = ( topGen ` ran (,) )
12		cvmliftlem.q	. . . . . . 7 |- Q = seq 0 ( ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) , ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) )
13		cvmliftlem.k	. . . . . . 7 |- K = U_ k e. ( 1 ... N ) ( Q ` k )
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	cvmliftlem11 31277	. . . . . 6 |- ( ph -> ( K e. ( II Cn C ) /\ ( F o. K ) = G ) )
15	14	simpld 475	. . . . 5 |- ( ph -> K e. ( II Cn C ) )
16		iiuni 22684	. . . . . 6 |- ( 0 [,] 1 ) = U. II
17	16, 2	cnf 21050	. . . . 5 |- ( K e. ( II Cn C ) -> K : ( 0 [,] 1 ) --> B )
18	15, 17	syl 17	. . . 4 |- ( ph -> K : ( 0 [,] 1 ) --> B )
19		ffun 6048	. . . 4 |- ( K : ( 0 [,] 1 ) --> B -> Fun K )
20	18, 19	syl 17	. . 3 |- ( ph -> Fun K )
21		nnuz 11723	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
22	8, 21	syl6eleq 2711	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` 1 ) )
23		eluzfz1 12348	. . . . . 6 |- ( N e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... N ) )
24	22, 23	syl 17	. . . . 5 |- ( ph -> 1 e. ( 1 ... N ) )
25		fveq2 6191	. . . . . 6 |- ( k = 1 -> ( Q ` k ) = ( Q ` 1 ) )
26	25	ssiun2s 4564	. . . . 5 |- ( 1 e. ( 1 ... N ) -> ( Q ` 1 ) C_ U_ k e. ( 1 ... N ) ( Q ` k ) )
27	24, 26	syl 17	. . . 4 |- ( ph -> ( Q ` 1 ) C_ U_ k e. ( 1 ... N ) ( Q ` k ) )
28	27, 13	syl6sseqr 3652	. . 3 |- ( ph -> ( Q ` 1 ) C_ K )
29		0xr 10086	. . . . . . 7 |- 0 e. RR*
30	29	a1i 11	. . . . . 6 |- ( ph -> 0 e. RR* )
31	8	nnrecred 11066	. . . . . . 7 |- ( ph -> ( 1 / N ) e. RR )
32	31	rexrd 10089	. . . . . 6 |- ( ph -> ( 1 / N ) e. RR* )
33		1red 10055	. . . . . . 7 |- ( ph -> 1 e. RR )
34		0le1 10551	. . . . . . . 8 |- 0 <_ 1
35	34	a1i 11	. . . . . . 7 |- ( ph -> 0 <_ 1 )
36	8	nnred 11035	. . . . . . 7 |- ( ph -> N e. RR )
37	8	nngt0d 11064	. . . . . . 7 |- ( ph -> 0 < N )
38		divge0 10892	. . . . . . 7 |- ( ( ( 1 e. RR /\ 0 <_ 1 ) /\ ( N e. RR /\ 0 < N ) ) -> 0 <_ ( 1 / N ) )
39	33, 35, 36, 37, 38	syl22anc 1327	. . . . . 6 |- ( ph -> 0 <_ ( 1 / N ) )
40		lbicc2 12288	. . . . . 6 |- ( ( 0 e. RR* /\ ( 1 / N ) e. RR* /\ 0 <_ ( 1 / N ) ) -> 0 e. ( 0 [,] ( 1 / N ) ) )
41	30, 32, 39, 40	syl3anc 1326	. . . . 5 |- ( ph -> 0 e. ( 0 [,] ( 1 / N ) ) )
42		1m1e0 11089	. . . . . . . 8 |- ( 1 - 1 ) = 0
43	42	oveq1i 6660	. . . . . . 7 |- ( ( 1 - 1 ) / N ) = ( 0 / N )
44	8	nncnd 11036	. . . . . . . 8 |- ( ph -> N e. CC )
45	8	nnne0d 11065	. . . . . . . 8 |- ( ph -> N =/= 0 )
46	44, 45	div0d 10800	. . . . . . 7 |- ( ph -> ( 0 / N ) = 0 )
47	43, 46	syl5eq 2668	. . . . . 6 |- ( ph -> ( ( 1 - 1 ) / N ) = 0 )
48	47	oveq1d 6665	. . . . 5 |- ( ph -> ( ( ( 1 - 1 ) / N ) [,] ( 1 / N ) ) = ( 0 [,] ( 1 / N ) ) )
49	41, 48	eleqtrrd 2704	. . . 4 |- ( ph -> 0 e. ( ( ( 1 - 1 ) / N ) [,] ( 1 / N ) ) )
50		eqid 2622	. . . . . . . 8 |- ( ( ( 1 - 1 ) / N ) [,] ( 1 / N ) ) = ( ( ( 1 - 1 ) / N ) [,] ( 1 / N ) )
51		simpr 477	. . . . . . . 8 |- ( ( ph /\ 1 e. ( 1 ... N ) ) -> 1 e. ( 1 ... N ) )
52	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 50	cvmliftlem7 31273	. . . . . . . 8 |- ( ( ph /\ 1 e. ( 1 ... N ) ) -> ( ( Q ` ( 1 - 1 ) ) ` ( ( 1 - 1 ) / N ) ) e. ( `' F " { ( G ` ( ( 1 - 1 ) / N ) ) } ) )
53	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 50, 51, 52	cvmliftlem6 31272	. . . . . . 7 |- ( ( ph /\ 1 e. ( 1 ... N ) ) -> ( ( Q ` 1 ) : ( ( ( 1 - 1 ) / N ) [,] ( 1 / N ) ) --> B /\ ( F o. ( Q ` 1 ) ) = ( G |` ( ( ( 1 - 1 ) / N ) [,] ( 1 / N ) ) ) ) )
54	24, 53	mpdan 702	. . . . . 6 |- ( ph -> ( ( Q ` 1 ) : ( ( ( 1 - 1 ) / N ) [,] ( 1 / N ) ) --> B /\ ( F o. ( Q ` 1 ) ) = ( G |` ( ( ( 1 - 1 ) / N ) [,] ( 1 / N ) ) ) ) )
55	54	simpld 475	. . . . 5 |- ( ph -> ( Q ` 1 ) : ( ( ( 1 - 1 ) / N ) [,] ( 1 / N ) ) --> B )
56		fdm 6051	. . . . 5 |- ( ( Q ` 1 ) : ( ( ( 1 - 1 ) / N ) [,] ( 1 / N ) ) --> B -> dom ( Q ` 1 ) = ( ( ( 1 - 1 ) / N ) [,] ( 1 / N ) ) )
57	55, 56	syl 17	. . . 4 |- ( ph -> dom ( Q ` 1 ) = ( ( ( 1 - 1 ) / N ) [,] ( 1 / N ) ) )
58	49, 57	eleqtrrd 2704	. . 3 |- ( ph -> 0 e. dom ( Q ` 1 ) )
59		funssfv 6209	. . 3 |- ( ( Fun K /\ ( Q ` 1 ) C_ K /\ 0 e. dom ( Q ` 1 ) ) -> ( K ` 0 ) = ( ( Q ` 1 ) ` 0 ) )
60	20, 28, 58, 59	syl3anc 1326	. 2 |- ( ph -> ( K ` 0 ) = ( ( Q ` 1 ) ` 0 ) )
61	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cvmliftlem9 31275	. . . 4 |- ( ( ph /\ 1 e. ( 1 ... N ) ) -> ( ( Q ` 1 ) ` ( ( 1 - 1 ) / N ) ) = ( ( Q ` ( 1 - 1 ) ) ` ( ( 1 - 1 ) / N ) ) )
62	24, 61	mpdan 702	. . 3 |- ( ph -> ( ( Q ` 1 ) ` ( ( 1 - 1 ) / N ) ) = ( ( Q ` ( 1 - 1 ) ) ` ( ( 1 - 1 ) / N ) ) )
63	47	fveq2d 6195	. . 3 |- ( ph -> ( ( Q ` 1 ) ` ( ( 1 - 1 ) / N ) ) = ( ( Q ` 1 ) ` 0 ) )
64	42	fveq2i 6194	. . . . . 6 |- ( Q ` ( 1 - 1 ) ) = ( Q ` 0 )
65	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cvmliftlem4 31270	. . . . . 6 |- ( Q ` 0 ) = { <. 0 , P >. }
66	64, 65	eqtri 2644	. . . . 5 |- ( Q ` ( 1 - 1 ) ) = { <. 0 , P >. }
67	66	a1i 11	. . . 4 |- ( ph -> ( Q ` ( 1 - 1 ) ) = { <. 0 , P >. } )
68	67, 47	fveq12d 6197	. . 3 |- ( ph -> ( ( Q ` ( 1 - 1 ) ) ` ( ( 1 - 1 ) / N ) ) = ( { <. 0 , P >. } ` 0 ) )
69	62, 63, 68	3eqtr3d 2664	. 2 |- ( ph -> ( ( Q ` 1 ) ` 0 ) = ( { <. 0 , P >. } ` 0 ) )
70		0nn0 11307	. . 3 |- 0 e. NN0
71		fvsng 6447	. . 3 |- ( ( 0 e. NN0 /\ P e. B ) -> ( { <. 0 , P >. } ` 0 ) = P )
72	70, 6, 71	sylancr 695	. 2 |- ( ph -> ( { <. 0 , P >. } ` 0 ) = P )
73	60, 69, 72	3eqtrd 2660	1 |- ( ph -> ( K ` 0 ) = P )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   RRcr 9935   0cc0 9936   1c1 9937   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZ>=cuz 11687   (,)cioo 12175   [,]cicc 12178   ...cfz 12326   seqcseq 12801   ↾_t crest 16081   topGenctg 16098   Cn ccn 21028   Homeochmeo 21556   IIcii 22678   CovMap ccvm 31237

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-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

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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  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-n0 11293  df-z 11378  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-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-cn 21031  df-hmeo 21558  df-ii 22680  df-cvm 31238

This theorem is referenced by:  cvmliftlem14  31279