Theorem limccog 37694

Description: Limit of the composition of two functions. If the limit of F at A is B and the limit of G at B is C, then the limit of G o. F at A is C. With respect to limcco 22841 and limccnp 22839, here we drop continuity assumptions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
limccog.1	|- ( ph -> ran F C_ ( dom G \ { B } ) )
limccog.2	|- ( ph -> B e. ( F lim CC A ) )
limccog.3	|- ( ph -> C e. ( G lim CC B ) )

Assertion

Ref	Expression
limccog	|- ( ph -> C e. ( ( G o. F ) lim CC A ) )

Proof of Theorem limccog

Dummy variables u v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limccl 22823	. . 3 |- ( G lim CC B ) C_ CC
2		limccog.3	. . 3 |- ( ph -> C e. ( G lim CC B ) )
3	1, 2	sseldi 3429	. 2 |- ( ph -> C e. CC )
4		limcrcl 22822	. . . . . . . . . . . 12 |- ( C e. ( G lim CC B ) -> ( G : dom G --> CC /\ dom G C_ CC /\ B e. CC ) )
5	2, 4	syl 17	. . . . . . . . . . 11 |- ( ph -> ( G : dom G --> CC /\ dom G C_ CC /\ B e. CC ) )
6	5	simp1d 1019	. . . . . . . . . 10 |- ( ph -> G : dom G --> CC )
7	5	simp2d 1020	. . . . . . . . . 10 |- ( ph -> dom G C_ CC )
8	5	simp3d 1021	. . . . . . . . . 10 |- ( ph -> B e. CC )
9		eqid 2450	. . . . . . . . . 10 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
10	6, 7, 8, 9	ellimc2 22825	. . . . . . . . 9 |- ( ph -> ( C e. ( G lim CC B ) <-> ( C e. CC /\ A. u e. ( TopOpen ` ℂfld ) ( C e. u -> E. v e. ( TopOpen ` ℂfld ) ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) ) ) )
11	2, 10	mpbid 214	. . . . . . . 8 |- ( ph -> ( C e. CC /\ A. u e. ( TopOpen ` ℂfld ) ( C e. u -> E. v e. ( TopOpen ` ℂfld ) ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) ) )
12	11	simprd 465	. . . . . . 7 |- ( ph -> A. u e. ( TopOpen ` ℂfld ) ( C e. u -> E. v e. ( TopOpen ` ℂfld ) ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) )
13	12	r19.21bi 2756	. . . . . 6 |- ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) -> ( C e. u -> E. v e. ( TopOpen ` ℂfld ) ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) )
14	13	imp 431	. . . . 5 |- ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) -> E. v e. ( TopOpen ` ℂfld ) ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) )
15		simp1ll 1070	. . . . . . . 8 |- ( ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) /\ v e. ( TopOpen ` ℂfld ) /\ ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) -> ph )
16		simp2 1008	. . . . . . . 8 |- ( ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) /\ v e. ( TopOpen ` ℂfld ) /\ ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) -> v e. ( TopOpen ` ℂfld ) )
17		simp3l 1035	. . . . . . . 8 |- ( ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) /\ v e. ( TopOpen ` ℂfld ) /\ ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) -> B e. v )
18		limccog.2	. . . . . . . . . . . 12 |- ( ph -> B e. ( F lim CC A ) )
19		limcrcl 22822	. . . . . . . . . . . . . . 15 |- ( B e. ( F lim CC A ) -> ( F : dom F --> CC /\ dom F C_ CC /\ A e. CC ) )
20	18, 19	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ( F : dom F --> CC /\ dom F C_ CC /\ A e. CC ) )
21	20	simp1d 1019	. . . . . . . . . . . . 13 |- ( ph -> F : dom F --> CC )
22	20	simp2d 1020	. . . . . . . . . . . . 13 |- ( ph -> dom F C_ CC )
23	20	simp3d 1021	. . . . . . . . . . . . 13 |- ( ph -> A e. CC )
24	21, 22, 23, 9	ellimc2 22825	. . . . . . . . . . . 12 |- ( ph -> ( B e. ( F lim CC A ) <-> ( B e. CC /\ A. v e. ( TopOpen ` ℂfld ) ( B e. v -> E. w e. ( TopOpen ` ℂfld ) ( A e. w /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) ) ) ) )
25	18, 24	mpbid 214	. . . . . . . . . . 11 |- ( ph -> ( B e. CC /\ A. v e. ( TopOpen ` ℂfld ) ( B e. v -> E. w e. ( TopOpen ` ℂfld ) ( A e. w /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) ) ) )
26	25	simprd 465	. . . . . . . . . 10 |- ( ph -> A. v e. ( TopOpen ` ℂfld ) ( B e. v -> E. w e. ( TopOpen ` ℂfld ) ( A e. w /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) ) )
27	26	r19.21bi 2756	. . . . . . . . 9 |- ( ( ph /\ v e. ( TopOpen ` ℂfld ) ) -> ( B e. v -> E. w e. ( TopOpen ` ℂfld ) ( A e. w /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) ) )
28	27	imp 431	. . . . . . . 8 |- ( ( ( ph /\ v e. ( TopOpen ` ℂfld ) ) /\ B e. v ) -> E. w e. ( TopOpen ` ℂfld ) ( A e. w /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) )
29	15, 16, 17, 28	syl21anc 1266	. . . . . . 7 |- ( ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) /\ v e. ( TopOpen ` ℂfld ) /\ ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) -> E. w e. ( TopOpen ` ℂfld ) ( A e. w /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) )
30		imaco 5339	. . . . . . . . . . 11 |- ( ( G o. F ) " ( w i^i ( dom F \ { A } ) ) ) = ( G " ( F " ( w i^i ( dom F \ { A } ) ) ) )
31	15	ad2antrr 731	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) /\ v e. ( TopOpen ` ℂfld ) /\ ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) /\ w e. ( TopOpen ` ℂfld ) ) /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ph )
32		simpl3r 1063	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) /\ v e. ( TopOpen ` ℂfld ) /\ ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) /\ w e. ( TopOpen ` ℂfld ) ) -> ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u )
33	32	adantr 467	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) /\ v e. ( TopOpen ` ℂfld ) /\ ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) /\ w e. ( TopOpen ` ℂfld ) ) /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u )
34		simpr 463	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) /\ v e. ( TopOpen ` ℂfld ) /\ ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) /\ w e. ( TopOpen ` ℂfld ) ) /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v )
35		simpr 463	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v )
36		imassrn 5178	. . . . . . . . . . . . . . . . . 18 |- ( F " ( w i^i ( dom F \ { A } ) ) ) C_ ran F
37		limccog.1	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ran F C_ ( dom G \ { B } ) )
38	36, 37	syl5ss 3442	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( F " ( w i^i ( dom F \ { A } ) ) ) C_ ( dom G \ { B } ) )
39	38	adantr 467	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ( F " ( w i^i ( dom F \ { A } ) ) ) C_ ( dom G \ { B } ) )
40	35, 39	ssind 3655	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ( F " ( w i^i ( dom F \ { A } ) ) ) C_ ( v i^i ( dom G \ { B } ) ) )
41		imass2 5203	. . . . . . . . . . . . . . 15 |- ( ( F " ( w i^i ( dom F \ { A } ) ) ) C_ ( v i^i ( dom G \ { B } ) ) -> ( G " ( F " ( w i^i ( dom F \ { A } ) ) ) ) C_ ( G " ( v i^i ( dom G \ { B } ) ) ) )
42	40, 41	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ( G " ( F " ( w i^i ( dom F \ { A } ) ) ) ) C_ ( G " ( v i^i ( dom G \ { B } ) ) ) )
43	42	adantlr 720	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ( G " ( F " ( w i^i ( dom F \ { A } ) ) ) ) C_ ( G " ( v i^i ( dom G \ { B } ) ) ) )
44		simplr 761	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u )
45	43, 44	sstrd 3441	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ( G " ( F " ( w i^i ( dom F \ { A } ) ) ) ) C_ u )
46	31, 33, 34, 45	syl21anc 1266	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) /\ v e. ( TopOpen ` ℂfld ) /\ ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) /\ w e. ( TopOpen ` ℂfld ) ) /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ( G " ( F " ( w i^i ( dom F \ { A } ) ) ) ) C_ u )
47	30, 46	syl5eqss 3475	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) /\ v e. ( TopOpen ` ℂfld ) /\ ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) /\ w e. ( TopOpen ` ℂfld ) ) /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ( ( G o. F ) " ( w i^i ( dom F \ { A } ) ) ) C_ u )
48	47	ex 436	. . . . . . . . 9 |- ( ( ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) /\ v e. ( TopOpen ` ℂfld ) /\ ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) /\ w e. ( TopOpen ` ℂfld ) ) -> ( ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v -> ( ( G o. F ) " ( w i^i ( dom F \ { A } ) ) ) C_ u ) )
49	48	anim2d 568	. . . . . . . 8 |- ( ( ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) /\ v e. ( TopOpen ` ℂfld ) /\ ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) /\ w e. ( TopOpen ` ℂfld ) ) -> ( ( A e. w /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ( A e. w /\ ( ( G o. F ) " ( w i^i ( dom F \ { A } ) ) ) C_ u ) ) )
50	49	reximdva 2861	. . . . . . 7 |- ( ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) /\ v e. ( TopOpen ` ℂfld ) /\ ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) -> ( E. w e. ( TopOpen ` ℂfld ) ( A e. w /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> E. w e. ( TopOpen ` ℂfld ) ( A e. w /\ ( ( G o. F ) " ( w i^i ( dom F \ { A } ) ) ) C_ u ) ) )
51	29, 50	mpd 15	. . . . . 6 |- ( ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) /\ v e. ( TopOpen ` ℂfld ) /\ ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) ) -> E. w e. ( TopOpen ` ℂfld ) ( A e. w /\ ( ( G o. F ) " ( w i^i ( dom F \ { A } ) ) ) C_ u ) )
52	51	rexlimdv3a 2880	. . . . 5 |- ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) -> ( E. v e. ( TopOpen ` ℂfld ) ( B e. v /\ ( G " ( v i^i ( dom G \ { B } ) ) ) C_ u ) -> E. w e. ( TopOpen ` ℂfld ) ( A e. w /\ ( ( G o. F ) " ( w i^i ( dom F \ { A } ) ) ) C_ u ) ) )
53	14, 52	mpd 15	. . . 4 |- ( ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) /\ C e. u ) -> E. w e. ( TopOpen ` ℂfld ) ( A e. w /\ ( ( G o. F ) " ( w i^i ( dom F \ { A } ) ) ) C_ u ) )
54	53	ex 436	. . 3 |- ( ( ph /\ u e. ( TopOpen ` ℂfld ) ) -> ( C e. u -> E. w e. ( TopOpen ` ℂfld ) ( A e. w /\ ( ( G o. F ) " ( w i^i ( dom F \ { A } ) ) ) C_ u ) ) )
55	54	ralrimiva 2801	. 2 |- ( ph -> A. u e. ( TopOpen ` ℂfld ) ( C e. u -> E. w e. ( TopOpen ` ℂfld ) ( A e. w /\ ( ( G o. F ) " ( w i^i ( dom F \ { A } ) ) ) C_ u ) ) )
56		ffun 5729	. . . . . . 7 |- ( F : dom F --> CC -> Fun F )
57	21, 56	syl 17	. . . . . 6 |- ( ph -> Fun F )
58		fdmrn 5742	. . . . . 6 |- ( Fun F <-> F : dom F --> ran F )
59	57, 58	sylib 200	. . . . 5 |- ( ph -> F : dom F --> ran F )
60	37	difss2d 3562	. . . . 5 |- ( ph -> ran F C_ dom G )
61	59, 60	fssd 5736	. . . 4 |- ( ph -> F : dom F --> dom G )
62		fco 5737	. . . 4 |- ( ( G : dom G --> CC /\ F : dom F --> dom G ) -> ( G o. F ) : dom F --> CC )
63	6, 61, 62	syl2anc 666	. . 3 |- ( ph -> ( G o. F ) : dom F --> CC )
64	63, 22, 23, 9	ellimc2 22825	. 2 |- ( ph -> ( C e. ( ( G o. F ) lim CC A ) <-> ( C e. CC /\ A. u e. ( TopOpen ` ℂfld ) ( C e. u -> E. w e. ( TopOpen ` ℂfld ) ( A e. w /\ ( ( G o. F ) " ( w i^i ( dom F \ { A } ) ) ) C_ u ) ) ) ) )
65	3, 55, 64	mpbir2and 932	1 |- ( ph -> C e. ( ( G o. F ) lim CC A ) )

Syntax hints:   -> wi 4   /\ wa 371   /\ w3a 984   e. wcel 1886   A.wral 2736   E.wrex 2737   \ cdif 3400   i^i cin 3402   C_ wss 3403   {csn 3967   dom cdm 4833   ran crn 4834   "cima 4836   o. ccom 4837   Fun wfun 5575   -->wf 5577   ` cfv 5581  (class class class)co 6288   CCcc 9534   TopOpenctopn 15313  ℂ_fldccnfld 18963   lim CC climc 22810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fi 7922  df-sup 7953  df-inf 7954  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-fz 11782  df-seq 12211  df-exp 12270  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-plusg 15196  df-mulr 15197  df-starv 15198  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-rest 15314  df-topn 15315  df-topgen 15335  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cnp 20237  df-xms 21328  df-ms 21329  df-limc 22814

This theorem is referenced by:  dirkercncflem2  37960  fourierdlem53  38017  fourierdlem93  38057  fourierdlem111  38075