Theorem occllem 28162

Description: Lemma for occl 28163. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
occl.1	|- ( ph -> A C_ ~H )
occl.2	|- ( ph -> F e. Cauchy )
occl.3	|- ( ph -> F : NN --> ( _|_ ` A ) )
occl.4	|- ( ph -> B e. A )

Assertion

Ref	Expression
occllem	|- ( ph -> ( ( ~~>v ` F ) .ih B ) = 0 )

Proof of Theorem occllem

Dummy variables x k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
2	1	cnfldhaus 22588	. . 3 |- ( TopOpen ` ℂfld ) e. Haus
3	2	a1i 11	. 2 |- ( ph -> ( TopOpen ` ℂfld ) e. Haus )
4		occl.2	. . . . . . 7 |- ( ph -> F e. Cauchy )
5		ax-hcompl 28059	. . . . . . 7 |- ( F e. Cauchy -> E. x e. ~H F ~~>v x )
6		hlimf 28094	. . . . . . . . . 10 |- ~~>v : dom ~~>v --> ~H
7		ffn 6045	. . . . . . . . . 10 |- ( ~~>v : dom ~~>v --> ~H -> ~~>v Fn dom ~~>v )
8	6, 7	ax-mp 5	. . . . . . . . 9 |- ~~>v Fn dom ~~>v
9		fnbr 5993	. . . . . . . . 9 |- ( ( ~~>v Fn dom ~~>v /\ F ~~>v x ) -> F e. dom ~~>v )
10	8, 9	mpan 706	. . . . . . . 8 |- ( F ~~>v x -> F e. dom ~~>v )
11	10	rexlimivw 3029	. . . . . . 7 |- ( E. x e. ~H F ~~>v x -> F e. dom ~~>v )
12	4, 5, 11	3syl 18	. . . . . 6 |- ( ph -> F e. dom ~~>v )
13		ffun 6048	. . . . . . 7 |- ( ~~>v : dom ~~>v --> ~H -> Fun ~~>v )
14		funfvbrb 6330	. . . . . . 7 |- ( Fun ~~>v -> ( F e. dom ~~>v <-> F ~~>v ( ~~>v ` F ) ) )
15	6, 13, 14	mp2b 10	. . . . . 6 |- ( F e. dom ~~>v <-> F ~~>v ( ~~>v ` F ) )
16	12, 15	sylib 208	. . . . 5 |- ( ph -> F ~~>v ( ~~>v ` F ) )
17		eqid 2622	. . . . . . . 8 |- <. <. +h , .h >. , normh >. = <. <. +h , .h >. , normh >.
18		eqid 2622	. . . . . . . . 9 |- ( normh o. -h ) = ( normh o. -h )
19	17, 18	hhims 28029	. . . . . . . 8 |- ( normh o. -h ) = ( IndMet ` <. <. +h , .h >. , normh >. )
20		eqid 2622	. . . . . . . 8 |- ( MetOpen ` ( normh o. -h ) ) = ( MetOpen ` ( normh o. -h ) )
21	17, 19, 20	hhlm 28056	. . . . . . 7 |- ~~>v = ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) )
22		resss 5422	. . . . . . 7 |- ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) ) C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
23	21, 22	eqsstri 3635	. . . . . 6 |- ~~>v C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
24	23	ssbri 4697	. . . . 5 |- ( F ~~>v ( ~~>v ` F ) -> F ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) ( ~~>v ` F ) )
25	16, 24	syl 17	. . . 4 |- ( ph -> F ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) ( ~~>v ` F ) )
26	18	hilxmet 28052	. . . . . 6 |- ( normh o. -h ) e. ( *Met ` ~H )
27	20	mopntopon 22244	. . . . . 6 |- ( ( normh o. -h ) e. ( *Met ` ~H ) -> ( MetOpen ` ( normh o. -h ) ) e. (TopOn ` ~H ) )
28	26, 27	mp1i 13	. . . . 5 |- ( ph -> ( MetOpen ` ( normh o. -h ) ) e. (TopOn ` ~H ) )
29	28	cnmptid 21464	. . . . 5 |- ( ph -> ( x e. ~H |-> x ) e. ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) ) )
30		occl.1	. . . . . . 7 |- ( ph -> A C_ ~H )
31		occl.4	. . . . . . 7 |- ( ph -> B e. A )
32	30, 31	sseldd 3604	. . . . . 6 |- ( ph -> B e. ~H )
33	28, 28, 32	cnmptc 21465	. . . . 5 |- ( ph -> ( x e. ~H |-> B ) e. ( ( MetOpen ` ( normh o. -h ) ) Cn ( MetOpen ` ( normh o. -h ) ) ) )
34	17	hhnv 28022	. . . . . 6 |- <. <. +h , .h >. , normh >. e. NrmCVec
35	17	hhip 28034	. . . . . . 7 |- .ih = ( .iOLD ` <. <. +h , .h >. , normh >. )
36	35, 19, 20, 1	dipcn 27575	. . . . . 6 |- ( <. <. +h , .h >. , normh >. e. NrmCVec -> .ih e. ( ( ( MetOpen ` ( normh o. -h ) ) tX ( MetOpen ` ( normh o. -h ) ) ) Cn ( TopOpen ` ℂfld ) ) )
37	34, 36	mp1i 13	. . . . 5 |- ( ph -> .ih e. ( ( ( MetOpen ` ( normh o. -h ) ) tX ( MetOpen ` ( normh o. -h ) ) ) Cn ( TopOpen ` ℂfld ) ) )
38	28, 29, 33, 37	cnmpt12f 21469	. . . 4 |- ( ph -> ( x e. ~H |-> ( x .ih B ) ) e. ( ( MetOpen ` ( normh o. -h ) ) Cn ( TopOpen ` ℂfld ) ) )
39	25, 38	lmcn 21109	. . 3 |- ( ph -> ( ( x e. ~H |-> ( x .ih B ) ) o. F ) ( ~~> t ` ( TopOpen ` ℂfld ) ) ( ( x e. ~H |-> ( x .ih B ) ) ` ( ~~>v ` F ) ) )
40		occl.3	. . . . . . . . . . 11 |- ( ph -> F : NN --> ( _|_ ` A ) )
41	40	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. ( _|_ ` A ) )
42		ocel 28140	. . . . . . . . . . . 12 |- ( A C_ ~H -> ( ( F ` k ) e. ( _|_ ` A ) <-> ( ( F ` k ) e. ~H /\ A. x e. A ( ( F ` k ) .ih x ) = 0 ) ) )
43	30, 42	syl 17	. . . . . . . . . . 11 |- ( ph -> ( ( F ` k ) e. ( _|_ ` A ) <-> ( ( F ` k ) e. ~H /\ A. x e. A ( ( F ` k ) .ih x ) = 0 ) ) )
44	43	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ k e. NN ) -> ( ( F ` k ) e. ( _|_ ` A ) <-> ( ( F ` k ) e. ~H /\ A. x e. A ( ( F ` k ) .ih x ) = 0 ) ) )
45	41, 44	mpbid 222	. . . . . . . . 9 |- ( ( ph /\ k e. NN ) -> ( ( F ` k ) e. ~H /\ A. x e. A ( ( F ` k ) .ih x ) = 0 ) )
46	45	simpld 475	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. ~H )
47		oveq1 6657	. . . . . . . . 9 |- ( x = ( F ` k ) -> ( x .ih B ) = ( ( F ` k ) .ih B ) )
48		eqid 2622	. . . . . . . . 9 |- ( x e. ~H |-> ( x .ih B ) ) = ( x e. ~H |-> ( x .ih B ) )
49		ovex 6678	. . . . . . . . 9 |- ( ( F ` k ) .ih B ) e. _V
50	47, 48, 49	fvmpt 6282	. . . . . . . 8 |- ( ( F ` k ) e. ~H -> ( ( x e. ~H |-> ( x .ih B ) ) ` ( F ` k ) ) = ( ( F ` k ) .ih B ) )
51	46, 50	syl 17	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( ( x e. ~H |-> ( x .ih B ) ) ` ( F ` k ) ) = ( ( F ` k ) .ih B ) )
52	31	adantr 481	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> B e. A )
53	45	simprd 479	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> A. x e. A ( ( F ` k ) .ih x ) = 0 )
54		oveq2 6658	. . . . . . . . . 10 |- ( x = B -> ( ( F ` k ) .ih x ) = ( ( F ` k ) .ih B ) )
55	54	eqeq1d 2624	. . . . . . . . 9 |- ( x = B -> ( ( ( F ` k ) .ih x ) = 0 <-> ( ( F ` k ) .ih B ) = 0 ) )
56	55	rspcv 3305	. . . . . . . 8 |- ( B e. A -> ( A. x e. A ( ( F ` k ) .ih x ) = 0 -> ( ( F ` k ) .ih B ) = 0 ) )
57	52, 53, 56	sylc 65	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( ( F ` k ) .ih B ) = 0 )
58	51, 57	eqtrd 2656	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( ( x e. ~H |-> ( x .ih B ) ) ` ( F ` k ) ) = 0 )
59		ocss 28144	. . . . . . . . 9 |- ( A C_ ~H -> ( _|_ ` A ) C_ ~H )
60	30, 59	syl 17	. . . . . . . 8 |- ( ph -> ( _|_ ` A ) C_ ~H )
61	40, 60	fssd 6057	. . . . . . 7 |- ( ph -> F : NN --> ~H )
62		fvco3 6275	. . . . . . 7 |- ( ( F : NN --> ~H /\ k e. NN ) -> ( ( ( x e. ~H |-> ( x .ih B ) ) o. F ) ` k ) = ( ( x e. ~H |-> ( x .ih B ) ) ` ( F ` k ) ) )
63	61, 62	sylan 488	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( ( ( x e. ~H |-> ( x .ih B ) ) o. F ) ` k ) = ( ( x e. ~H |-> ( x .ih B ) ) ` ( F ` k ) ) )
64		c0ex 10034	. . . . . . . 8 |- 0 e. _V
65	64	fvconst2 6469	. . . . . . 7 |- ( k e. NN -> ( ( NN X. { 0 } ) ` k ) = 0 )
66	65	adantl 482	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( ( NN X. { 0 } ) ` k ) = 0 )
67	58, 63, 66	3eqtr4d 2666	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( ( ( x e. ~H |-> ( x .ih B ) ) o. F ) ` k ) = ( ( NN X. { 0 } ) ` k ) )
68	67	ralrimiva 2966	. . . 4 |- ( ph -> A. k e. NN ( ( ( x e. ~H |-> ( x .ih B ) ) o. F ) ` k ) = ( ( NN X. { 0 } ) ` k ) )
69		ovex 6678	. . . . . . 7 |- ( x .ih B ) e. _V
70	69, 48	fnmpti 6022	. . . . . 6 |- ( x e. ~H |-> ( x .ih B ) ) Fn ~H
71		fnfco 6069	. . . . . 6 |- ( ( ( x e. ~H |-> ( x .ih B ) ) Fn ~H /\ F : NN --> ~H ) -> ( ( x e. ~H |-> ( x .ih B ) ) o. F ) Fn NN )
72	70, 61, 71	sylancr 695	. . . . 5 |- ( ph -> ( ( x e. ~H |-> ( x .ih B ) ) o. F ) Fn NN )
73	64	fconst 6091	. . . . . 6 |- ( NN X. { 0 } ) : NN --> { 0 }
74		ffn 6045	. . . . . 6 |- ( ( NN X. { 0 } ) : NN --> { 0 } -> ( NN X. { 0 } ) Fn NN )
75	73, 74	ax-mp 5	. . . . 5 |- ( NN X. { 0 } ) Fn NN
76		eqfnfv 6311	. . . . 5 |- ( ( ( ( x e. ~H |-> ( x .ih B ) ) o. F ) Fn NN /\ ( NN X. { 0 } ) Fn NN ) -> ( ( ( x e. ~H |-> ( x .ih B ) ) o. F ) = ( NN X. { 0 } ) <-> A. k e. NN ( ( ( x e. ~H |-> ( x .ih B ) ) o. F ) ` k ) = ( ( NN X. { 0 } ) ` k ) ) )
77	72, 75, 76	sylancl 694	. . . 4 |- ( ph -> ( ( ( x e. ~H |-> ( x .ih B ) ) o. F ) = ( NN X. { 0 } ) <-> A. k e. NN ( ( ( x e. ~H |-> ( x .ih B ) ) o. F ) ` k ) = ( ( NN X. { 0 } ) ` k ) ) )
78	68, 77	mpbird 247	. . 3 |- ( ph -> ( ( x e. ~H |-> ( x .ih B ) ) o. F ) = ( NN X. { 0 } ) )
79		fvex 6201	. . . . 5 |- ( ~~>v ` F ) e. _V
80	79	hlimveci 28047	. . . 4 |- ( F ~~>v ( ~~>v ` F ) -> ( ~~>v ` F ) e. ~H )
81		oveq1 6657	. . . . 5 |- ( x = ( ~~>v ` F ) -> ( x .ih B ) = ( ( ~~>v ` F ) .ih B ) )
82		ovex 6678	. . . . 5 |- ( ( ~~>v ` F ) .ih B ) e. _V
83	81, 48, 82	fvmpt 6282	. . . 4 |- ( ( ~~>v ` F ) e. ~H -> ( ( x e. ~H |-> ( x .ih B ) ) ` ( ~~>v ` F ) ) = ( ( ~~>v ` F ) .ih B ) )
84	16, 80, 83	3syl 18	. . 3 |- ( ph -> ( ( x e. ~H |-> ( x .ih B ) ) ` ( ~~>v ` F ) ) = ( ( ~~>v ` F ) .ih B ) )
85	39, 78, 84	3brtr3d 4684	. 2 |- ( ph -> ( NN X. { 0 } ) ( ~~> t ` ( TopOpen ` ℂfld ) ) ( ( ~~>v ` F ) .ih B ) )
86	1	cnfldtopon 22586	. . . 4 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
87	86	a1i 11	. . 3 |- ( ph -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
88		0cnd 10033	. . 3 |- ( ph -> 0 e. CC )
89		1zzd 11408	. . 3 |- ( ph -> 1 e. ZZ )
90		nnuz 11723	. . . 4 |- NN = ( ZZ>= ` 1 )
91	90	lmconst 21065	. . 3 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ 0 e. CC /\ 1 e. ZZ ) -> ( NN X. { 0 } ) ( ~~> t ` ( TopOpen ` ℂfld ) ) 0 )
92	87, 88, 89, 91	syl3anc 1326	. 2 |- ( ph -> ( NN X. { 0 } ) ( ~~> t ` ( TopOpen ` ℂfld ) ) 0 )
93	3, 85, 92	lmmo 21184	1 |- ( ph -> ( ( ~~>v ` F ) .ih B ) = 0 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   {csn 4177   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   |` cres 5116   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   0cc0 9936   1c1 9937   NNcn 11020   ZZcz 11377   TopOpenctopn 16082   *Metcxmt 19731   MetOpencmopn 19736  ℂ_fldccnfld 19746  TopOnctopon 20715   Cn ccn 21028   ~~> tclm 21030   Hauscha 21112   tX ctx 21363   NrmCVeccnv 27439   ~Hchil 27776   +h cva 27777   .h csm 27778   .ih csp 27779   normhcno 27780   -h cmv 27782   Cauchyccau 27783   ~~>v chli 27784   _|_cort 27787

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-inf2 8538  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  ax-addf 10015  ax-mulf 10016  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942  ax-hcompl 28059

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  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-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-lm 21033  df-haus 21119  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455  df-ims 27456  df-dip 27556  df-hnorm 27825  df-hvsub 27828  df-hlim 27829  df-sh 28064  df-oc 28109

This theorem is referenced by:  occl  28163