Theorem cvmlift2lem7 27046

Description: Lemma for cvmlift2 27053. (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 2433	. . . . . . . . 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 27042	. . . . . . . 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 709	. . . . . . 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 994	. . . . . 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 5681	. . . . 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 993	. . . . . . 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 20299	. . . . . . . 8 |- ( 0 [,] 1 ) = U. II
14	13, 1	cnf 18692	. . . . . . 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 749	. . . . . 6 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> y e. ( 0 [,] 1 ) )
17		fvco3 5756	. . . . . 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 654	. . . . 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 6088	. . . . . . 7 |- ( z = y -> ( x G z ) = ( x G y ) )
20		eqid 2433	. . . . . . 7 |- ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) )
21		ovex 6105	. . . . . . 7 |- ( x G y ) e. _V
22	19, 20, 21	fvmpt 5762	. . . . . 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 2473	. . . 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 1176	. . 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 6139	. 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 5832	. . 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 26999	. . . . 5 |- ( F e. ( C CovMap J ) -> F e. ( C Cn J ) )
31		eqid 2433	. . . . . 6 |- U. J = U. J
32	1, 31	cnf 18692	. . . . 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 5732	. . 3 |- ( ph -> F = ( w e. B |-> ( F ` w ) ) )
35		fveq2 5679	. . 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 6645	. 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 20298	. . . . . 6 |- II e. Top
38	37, 37, 13, 13	txunii 19008	. . . . 5 |- ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) = U. ( II tX II )
39	38, 31	cnf 18692	. . . 4 |- ( G e. ( ( II tX II ) Cn J ) -> G : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> U. J )
40		ffn 5547	. . . 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 6187	. . 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 2475	1 |- ( ph -> ( F o. K ) = G )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 958   = wceq 1362   e. wcel 1755   U.cuni 4079   e. cmpt 4338   X. cxp 4825   o. ccom 4831   Fn wfn 5401   -->wf 5402   ` cfv 5406   iota_crio 6038  (class class class)co 6080   e. cmpt2 6082   0cc0 9270   1c1 9271   [,]cicc 11291   Cn ccn 18670   tX ctx 18975   IIcii 20293   CovMap ccvm 26992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-ec 7091  df-map 7204  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-sum 13148  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-cn 18673  df-cnp 18674  df-cmp 18832  df-con 18858  df-lly 18912  df-nlly 18913  df-tx 18977  df-hmeo 19170  df-xms 19737  df-ms 19738  df-tms 19739  df-ii 20295  df-htpy 20384  df-phtpy 20385  df-phtpc 20406  df-pcon 26958  df-scon 26959  df-cvm 26993

This theorem is referenced by:  cvmlift2lem9  27048  cvmlift2lem13  27052