Theorem cvmlift2lem7 27167

Description: Lemma for cvmlift2 27174. (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 2438	. . . . . . . . 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 27163	. . . . . . . 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 5688	. . . . 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 20437	. . . . . . . 8 |- ( 0 [,] 1 ) = U. II
14	13, 1	cnf 18830	. . . . . . 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 5763	. . . . . 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 6094	. . . . . . 7 |- ( z = y -> ( x G z ) = ( x G y ) )
20		eqid 2438	. . . . . . 7 |- ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) )
21		ovex 6111	. . . . . . 7 |- ( x G y ) e. _V
22	19, 20, 21	fvmpt 5769	. . . . . 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 2478	. . . 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 1183	. . 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 6145	. 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 5839	. . 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 27120	. . . . 5 |- ( F e. ( C CovMap J ) -> F e. ( C Cn J ) )
31		eqid 2438	. . . . . 6 |- U. J = U. J
32	1, 31	cnf 18830	. . . . 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 5739	. . 3 |- ( ph -> F = ( w e. B |-> ( F ` w ) ) )
35		fveq2 5686	. . 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 6651	. 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 20436	. . . . . 6 |- II e. Top
38	37, 37, 13, 13	txunii 19146	. . . . 5 |- ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) = U. ( II tX II )
39	38, 31	cnf 18830	. . . 4 |- ( G e. ( ( II tX II ) Cn J ) -> G : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> U. J )
40		ffn 5554	. . . 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 6193	. . 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 2480	1 |- ( ph -> ( F o. K ) = G )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   U.cuni 4086   e. cmpt 4345   X. cxp 4833   o. ccom 4839   Fn wfn 5408   -->wf 5409   ` cfv 5413   iota_crio 6046  (class class class)co 6086   e. cmpt2 6088   0cc0 9274   1c1 9275   [,]cicc 11295   Cn ccn 18808   tX ctx 19113   IIcii 20431   CovMap ccvm 27113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-ec 7095  df-map 7208  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-nei 18682  df-cn 18811  df-cnp 18812  df-cmp 18970  df-con 18996  df-lly 19050  df-nlly 19051  df-tx 19115  df-hmeo 19308  df-xms 19875  df-ms 19876  df-tms 19877  df-ii 20433  df-htpy 20522  df-phtpy 20523  df-phtpc 20544  df-pcon 27079  df-scon 27080  df-cvm 27114

This theorem is referenced by:  cvmlift2lem9  27169  cvmlift2lem13  27173