Theorem limccog 37797

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 22927 and limccnp 22925, 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 22909	. . 3 |- ( G lim CC B ) C_ CC
2		limccog.3	. . 3 |- ( ph -> C e. ( G lim CC B ) )
3	1, 2	sseldi 3416	. 2 |- ( ph -> C e. CC )
4		limcrcl 22908	. . . . . . . . . . . 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 1042	. . . . . . . . . 10 |- ( ph -> G : dom G --> CC )
7	5	simp2d 1043	. . . . . . . . . 10 |- ( ph -> dom G C_ CC )
8	5	simp3d 1044	. . . . . . . . . 10 |- ( ph -> B e. CC )
9		eqid 2471	. . . . . . . . . 10 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
10	6, 7, 8, 9	ellimc2 22911	. . . . . . . . 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 215	. . . . . . . 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 470	. . . . . . 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 2776	. . . . . 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 436	. . . . 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 1093	. . . . . . . 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 1031	. . . . . . . 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 1058	. . . . . . . 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 22908	. . . . . . . . . . . . . . 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 1042	. . . . . . . . . . . . 13 |- ( ph -> F : dom F --> CC )
22	20	simp2d 1043	. . . . . . . . . . . . 13 |- ( ph -> dom F C_ CC )
23	20	simp3d 1044	. . . . . . . . . . . . 13 |- ( ph -> A e. CC )
24	21, 22, 23, 9	ellimc2 22911	. . . . . . . . . . . 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 215	. . . . . . . . . . 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 470	. . . . . . . . . 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 2776	. . . . . . . . 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 436	. . . . . . . 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 1291	. . . . . . 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 5347	. . . . . . . . . . 11 |- ( ( G o. F ) " ( w i^i ( dom F \ { A } ) ) ) = ( G " ( F " ( w i^i ( dom F \ { A } ) ) ) )
31	15	ad2antrr 740	. . . . . . . . . . . 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 1086	. . . . . . . . . . . . 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 472	. . . . . . . . . . . 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 468	. . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v )
36		imassrn 5185	. . . . . . . . . . . . . . . . . 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 3429	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( F " ( w i^i ( dom F \ { A } ) ) ) C_ ( dom G \ { B } ) )
39	38	adantr 472	. . . . . . . . . . . . . . . 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 3647	. . . . . . . . . . . . . . 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 5210	. . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . 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 770	. . . . . . . . . . . . 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 3428	. . . . . . . . . . . 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 1291	. . . . . . . . . . 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 3462	. . . . . . . . . 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 441	. . . . . . . . 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 575	. . . . . . . 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 2858	. . . . . . 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 2873	. . . . 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 441	. . 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 2809	. 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 5742	. . . . . . 7 |- ( F : dom F --> CC -> Fun F )
57	21, 56	syl 17	. . . . . 6 |- ( ph -> Fun F )
58		fdmrn 5756	. . . . . 6 |- ( Fun F <-> F : dom F --> ran F )
59	57, 58	sylib 201	. . . . 5 |- ( ph -> F : dom F --> ran F )
60	37	difss2d 3552	. . . . 5 |- ( ph -> ran F C_ dom G )
61	59, 60	fssd 5750	. . . 4 |- ( ph -> F : dom F --> dom G )
62		fco 5751	. . . 4 |- ( ( G : dom G --> CC /\ F : dom F --> dom G ) -> ( G o. F ) : dom F --> CC )
63	6, 61, 62	syl2anc 673	. . 3 |- ( ph -> ( G o. F ) : dom F --> CC )
64	63, 22, 23, 9	ellimc2 22911	. 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 936	1 |- ( ph -> C e. ( ( G o. F ) lim CC A ) )

Syntax hints:   -> wi 4   /\ wa 376   /\ w3a 1007   e. wcel 1904   A.wral 2756   E.wrex 2757   \ cdif 3387   i^i cin 3389   C_ wss 3390   {csn 3959   dom cdm 4839   ran crn 4840   "cima 4842   o. ccom 4843   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   TopOpenctopn 15398  ℂ_fldccnfld 19047   lim CC climc 22896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-fz 11811  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cnp 20321  df-xms 21413  df-ms 21414  df-limc 22900

This theorem is referenced by:  dirkercncflem2  38078  fourierdlem53  38135  fourierdlem93  38175  fourierdlem111  38193