Theorem cvmlift2lem7 28422

Description: Lemma for cvmlift2 28429. (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 2467	. . . . . . . . 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 28418	. . . . . . . 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 1009	. . . . . 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 5868	. . . . 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 1008	. . . . . . 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 21148	. . . . . . . 8 |- ( 0 [,] 1 ) = U. II
14	13, 1	cnf 19541	. . . . . . 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 5944	. . . . . 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 6292	. . . . . . 7 |- ( z = y -> ( x G z ) = ( x G y ) )
20		eqid 2467	. . . . . . 7 |- ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) )
21		ovex 6309	. . . . . . 7 |- ( x G y ) e. _V
22	19, 20, 21	fvmpt 5950	. . . . . 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 2516	. . . 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 1192	. . 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 6345	. 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 6022	. . 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 28375	. . . . 5 |- ( F e. ( C CovMap J ) -> F e. ( C Cn J ) )
31		eqid 2467	. . . . . 6 |- U. J = U. J
32	1, 31	cnf 19541	. . . . 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 5920	. . 3 |- ( ph -> F = ( w e. B |-> ( F ` w ) ) )
35		fveq2 5866	. . 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 6866	. 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 21147	. . . . . 6 |- II e. Top
38	37, 37, 13, 13	txunii 19857	. . . . 5 |- ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) = U. ( II tX II )
39	38, 31	cnf 19541	. . . 4 |- ( G e. ( ( II tX II ) Cn J ) -> G : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> U. J )
40		ffn 5731	. . . 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 6394	. . 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 2518	1 |- ( ph -> ( F o. K ) = G )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   U.cuni 4245   |-> cmpt 4505   X. cxp 4997   o. ccom 5003   Fn wfn 5583   -->wf 5584   ` cfv 5588   iota_crio 6244  (class class class)co 6284   |-> cmpt2 6286   0cc0 9492   1c1 9493   [,]cicc 11532   Cn ccn 19519   tX ctx 19824   IIcii 21142   CovMap ccvm 28368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-ec 7313  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-cn 19522  df-cnp 19523  df-cmp 19681  df-con 19707  df-lly 19761  df-nlly 19762  df-tx 19826  df-hmeo 20019  df-xms 20586  df-ms 20587  df-tms 20588  df-ii 21144  df-htpy 21233  df-phtpy 21234  df-phtpc 21255  df-pcon 28334  df-scon 28335  df-cvm 28369

This theorem is referenced by:  cvmlift2lem9  28424  cvmlift2lem13  28428