Theorem cvmlift2lem7 27341

Description: Lemma for cvmlift2 27348. (Contributed by Mario Carneiro, 7-May-2015.)

Hypotheses

Ref	Expression
cvmlift2.b	|- B = U. C
cvmlift2.f	|- ( ph -> F e. ( C CovMap J ) )
cvmlift2.g	|- ( ph -> G e. ( ( II tX II ) Cn J ) )
cvmlift2.p	|- ( ph -> P e. B )
cvmlift2.i	|- ( ph -> ( F ` P ) = ( 0 G 0 ) )
cvmlift2.h	|- H = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( z G 0 ) ) /\ ( f ` 0 ) = P ) )
cvmlift2.k	|- K = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) )

Assertion

Ref	Expression
cvmlift2lem7	|- ( ph -> ( F o. K ) = G )

Distinct variable groups:   x, f, y, z, F   ph, f, x, y, z   f, J, x, y, z   f, G, x, y, z   f, H, x, y, z   C, f, x, y, z   P, f, x, y, z   x, B, y, z   f, K, x, y, z

Allowed substitution hint:   B( f)

Proof of Theorem cvmlift2lem7

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmlift2.b	. . . . . . . . 9 |- B = U. C
2		cvmlift2.f	. . . . . . . . 9 |- ( ph -> F e. ( C CovMap J ) )
3		cvmlift2.g	. . . . . . . . 9 |- ( ph -> G e. ( ( II tX II ) Cn J ) )
4		cvmlift2.p	. . . . . . . . 9 |- ( ph -> P e. B )
5		cvmlift2.i	. . . . . . . . 9 |- ( ph -> ( F ` P ) = ( 0 G 0 ) )
6		cvmlift2.h	. . . . . . . . 9 |- H = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( z G 0 ) ) /\ ( f ` 0 ) = P ) )
7		eqid 2454	. . . . . . . . 9 |- ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) )
8	1, 2, 3, 4, 5, 6, 7	cvmlift2lem3 27337	. . . . . . . 8 |- ( ( ph /\ x e. ( 0 [,] 1 ) ) -> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) e. ( II Cn C ) /\ ( F o. ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` 0 ) = ( H ` x ) ) )
9	8	adantrr 716	. . . . . . 7 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) e. ( II Cn C ) /\ ( F o. ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` 0 ) = ( H ` x ) ) )
10	9	simp2d 1001	. . . . . 6 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( F o. ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) )
11	10	fveq1d 5800	. . . . 5 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( ( F o. ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ) ` y ) = ( ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) ` y ) )
12	9	simp1d 1000	. . . . . . 7 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) e. ( II Cn C ) )
13		iiuni 20588	. . . . . . . 8 |- ( 0 [,] 1 ) = U. II
14	13, 1	cnf 18981	. . . . . . 7 |- ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) e. ( II Cn C ) -> ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) : ( 0 [,] 1 ) --> B )
15	12, 14	syl 16	. . . . . 6 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) : ( 0 [,] 1 ) --> B )
16		simprr 756	. . . . . 6 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> y e. ( 0 [,] 1 ) )
17		fvco3 5876	. . . . . 6 |- ( ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) : ( 0 [,] 1 ) --> B /\ y e. ( 0 [,] 1 ) ) -> ( ( F o. ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ) ` y ) = ( F ` ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) ) )
18	15, 16, 17	syl2anc 661	. . . . 5 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( ( F o. ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ) ` y ) = ( F ` ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) ) )
19		oveq2 6207	. . . . . . 7 |- ( z = y -> ( x G z ) = ( x G y ) )
20		eqid 2454	. . . . . . 7 |- ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) )
21		ovex 6224	. . . . . . 7 |- ( x G y ) e. _V
22	19, 20, 21	fvmpt 5882	. . . . . 6 |- ( y e. ( 0 [,] 1 ) -> ( ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) ` y ) = ( x G y ) )
23	16, 22	syl 16	. . . . 5 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) ` y ) = ( x G y ) )
24	11, 18, 23	3eqtr3d 2503	. . . 4 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( F ` ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) ) = ( x G y ) )
25	24	3impb 1184	. . 3 |- ( ( ph /\ x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) -> ( F ` ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) ) = ( x G y ) )
26	25	mpt2eq3dva 6258	. 2 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( F ` ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) ) ) = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( x G y ) ) )
27	15, 16	ffvelrnd 5952	. . 3 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) e. B )
28		cvmlift2.k	. . . 4 |- K = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) )
29	28	a1i 11	. . 3 |- ( ph -> K = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) ) )
30		cvmcn 27294	. . . . 5 |- ( F e. ( C CovMap J ) -> F e. ( C Cn J ) )
31		eqid 2454	. . . . . 6 |- U. J = U. J
32	1, 31	cnf 18981	. . . . 5 |- ( F e. ( C Cn J ) -> F : B --> U. J )
33	2, 30, 32	3syl 20	. . . 4 |- ( ph -> F : B --> U. J )
34	33	feqmptd 5852	. . 3 |- ( ph -> F = ( w e. B |-> ( F ` w ) ) )
35		fveq2 5798	. . 3 |- ( w = ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) -> ( F ` w ) = ( F ` ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) ) )
36	27, 29, 34, 35	fmpt2co 6765	. 2 |- ( ph -> ( F o. K ) = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( F ` ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) ) ) )
37		iitop 20587	. . . . . 6 |- II e. Top
38	37, 37, 13, 13	txunii 19297	. . . . 5 |- ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) = U. ( II tX II )
39	38, 31	cnf 18981	. . . 4 |- ( G e. ( ( II tX II ) Cn J ) -> G : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> U. J )
40		ffn 5666	. . . 4 |- ( G : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> U. J -> G Fn ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
41	3, 39, 40	3syl 20	. . 3 |- ( ph -> G Fn ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
42		fnov 6307	. . 3 |- ( G Fn ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) <-> G = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( x G y ) ) )
43	41, 42	sylib 196	. 2 |- ( ph -> G = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( x G y ) ) )
44	26, 36, 43	3eqtr4d 2505	1 |- ( ph -> ( F o. K ) = G )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   U.cuni 4198   |-> cmpt 4457   X. cxp 4945   o. ccom 4951   Fn wfn 5520   -->wf 5521   ` cfv 5525   iota_crio 6159  (class class class)co 6199   |-> cmpt2 6201   0cc0 9392   1c1 9393   [,]cicc 11413   Cn ccn 18959   tX ctx 19264   IIcii 20582   CovMap ccvm 27287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-ec 7212  df-map 7325  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-clim 13083  df-sum 13281  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-cn 18962  df-cnp 18963  df-cmp 19121  df-con 19147  df-lly 19201  df-nlly 19202  df-tx 19266  df-hmeo 19459  df-xms 20026  df-ms 20027  df-tms 20028  df-ii 20584  df-htpy 20673  df-phtpy 20674  df-phtpc 20695  df-pcon 27253  df-scon 27254  df-cvm 27288

This theorem is referenced by:  cvmlift2lem9  27343  cvmlift2lem13  27347