Theorem limccog 39852

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 23657 and limccnp 23655, 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 23639	. . 3 |- ( G lim CC B ) C_ CC
2		limccog.3	. . 3 |- ( ph -> C e. ( G lim CC B ) )
3	1, 2	sseldi 3601	. 2 |- ( ph -> C e. CC )
4		limcrcl 23638	. . . . . . . . . . . 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 1073	. . . . . . . . . 10 |- ( ph -> G : dom G --> CC )
7	5	simp2d 1074	. . . . . . . . . 10 |- ( ph -> dom G C_ CC )
8	5	simp3d 1075	. . . . . . . . . 10 |- ( ph -> B e. CC )
9		eqid 2622	. . . . . . . . . 10 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
10	6, 7, 8, 9	ellimc2 23641	. . . . . . . . 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 222	. . . . . . . 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 479	. . . . . . 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 2932	. . . . . 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 445	. . . . 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 1124	. . . . . . . 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 1062	. . . . . . . 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 1089	. . . . . . . 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 23638	. . . . . . . . . . . . . . 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 1073	. . . . . . . . . . . . 13 |- ( ph -> F : dom F --> CC )
22	20	simp2d 1074	. . . . . . . . . . . . 13 |- ( ph -> dom F C_ CC )
23	20	simp3d 1075	. . . . . . . . . . . . 13 |- ( ph -> A e. CC )
24	21, 22, 23, 9	ellimc2 23641	. . . . . . . . . . . 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 222	. . . . . . . . . . 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 479	. . . . . . . . . 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 2932	. . . . . . . . 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 445	. . . . . . . 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 1325	. . . . . . 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 5640	. . . . . . . . . . 11 |- ( ( G o. F ) " ( w i^i ( dom F \ { A } ) ) ) = ( G " ( F " ( w i^i ( dom F \ { A } ) ) ) )
31	15	ad2antrr 762	. . . . . . . . . . . 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 1117	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 477	. . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v ) -> ( F " ( w i^i ( dom F \ { A } ) ) ) C_ v )
36		imassrn 5477	. . . . . . . . . . . . . . . . . 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 3614	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( F " ( w i^i ( dom F \ { A } ) ) ) C_ ( dom G \ { B } ) )
39	38	adantr 481	. . . . . . . . . . . . . . . 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 3837	. . . . . . . . . . . . . . 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 5501	. . . . . . . . . . . . . . 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 751	. . . . . . . . . . . . 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 792	. . . . . . . . . . . . 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 3613	. . . . . . . . . . . 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 1325	. . . . . . . . . . 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 3649	. . . . . . . . . 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 450	. . . . . . . . 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 589	. . . . . . . 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 3017	. . . . . . 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 3033	. . . . 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 450	. . 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 2966	. 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 6048	. . . . . . 7 |- ( F : dom F --> CC -> Fun F )
57	21, 56	syl 17	. . . . . 6 |- ( ph -> Fun F )
58		fdmrn 6064	. . . . . 6 |- ( Fun F <-> F : dom F --> ran F )
59	57, 58	sylib 208	. . . . 5 |- ( ph -> F : dom F --> ran F )
60	37	difss2d 3740	. . . . 5 |- ( ph -> ran F C_ dom G )
61	59, 60	fssd 6057	. . . 4 |- ( ph -> F : dom F --> dom G )
62		fco 6058	. . . 4 |- ( ( G : dom G --> CC /\ F : dom F --> dom G ) -> ( G o. F ) : dom F --> CC )
63	6, 61, 62	syl2anc 693	. . 3 |- ( ph -> ( G o. F ) : dom F --> CC )
64	63, 22, 23, 9	ellimc2 23641	. 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 957	1 |- ( ph -> C e. ( ( G o. F ) lim CC A ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   A.wral 2912   E.wrex 2913   \ cdif 3571   i^i cin 3573   C_ wss 3574   {csn 4177   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   TopOpenctopn 16082  ℂ_fldccnfld 19746   lim CC climc 23626

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  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-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  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-cnp 21032  df-xms 22125  df-ms 22126  df-limc 23630

This theorem is referenced by:  dirkercncflem2  40321  fourierdlem53  40376  fourierdlem93  40416  fourierdlem111  40434