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Theorem limcco 25770

Description: Composition of two limits. (Contributed by Mario Carneiro, 29-Dec-2016.)

**Hypotheses**
Ref	Expression
limcco.r	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑅 ≠ 𝐶)) → 𝑅 ∈ 𝐵)
limcco.s	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ ℂ)
limcco.c	⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝑋))
limcco.d	⊢ (𝜑 → 𝐷 ∈ ((𝑦 ∈ 𝐵 ↦ 𝑆) lim_ℂ 𝐶))
limcco.1	⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
limcco.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑅 = 𝐶)) → 𝑇 = 𝐷)

**Assertion**
Ref	Expression
limcco	⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑇) lim_ℂ 𝑋))

Distinct variable groups: 𝑥,𝐴 𝑥,𝑦,𝐵 𝑥,𝐶,𝑦 𝑥,𝐷,𝑦 𝜑,𝑥,𝑦 𝑦,𝑅 𝑥,𝑆 𝑦,𝑇 Allowed substitution hints: 𝐴(𝑦) 𝑅(𝑥) 𝑆(𝑦) 𝑇(𝑥) 𝑋(𝑥,𝑦)

**Proof of Theorem limcco**
Step	Hyp	Ref	Expression
1		limcco.r	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑅 ≠ 𝐶)) → 𝑅 ∈ 𝐵)
2	1	expr 456	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅 ≠ 𝐶 → 𝑅 ∈ 𝐵))
3	2	necon1bd 2943	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑅 ∈ 𝐵 → 𝑅 = 𝐶))
4		limccl 25752	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝑋) ⊆ ℂ
5		limcco.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝑋))
6	4, 5	sselid 3941	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
7	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
8		elsn2g 4624	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (𝑅 ∈ {𝐶} ↔ 𝑅 = 𝐶))
9	7, 8	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅 ∈ {𝐶} ↔ 𝑅 = 𝐶))
10	3, 9	sylibrd 259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑅 ∈ 𝐵 → 𝑅 ∈ {𝐶}))
11	10	orrd 863	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅 ∈ 𝐵 ∨ 𝑅 ∈ {𝐶}))
12		elun 4112	. . . . 5 ⊢ (𝑅 ∈ (𝐵 ∪ {𝐶}) ↔ (𝑅 ∈ 𝐵 ∨ 𝑅 ∈ {𝐶}))
13	11, 12	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ (𝐵 ∪ {𝐶}))
14	13	fmpttd 7069	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶(𝐵 ∪ {𝐶}))
15		eqid 2729	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑦 ∈ 𝐵 ↦ 𝑆)
16		limcco.s	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ ℂ)
17	15, 16	dmmptd 6645	. . . . 5 ⊢ (𝜑 → dom (𝑦 ∈ 𝐵 ↦ 𝑆) = 𝐵)
18		limcco.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ((𝑦 ∈ 𝐵 ↦ 𝑆) lim_ℂ 𝐶))
19		limcrcl 25751	. . . . . . 7 ⊢ (𝐷 ∈ ((𝑦 ∈ 𝐵 ↦ 𝑆) lim_ℂ 𝐶) → ((𝑦 ∈ 𝐵 ↦ 𝑆):dom (𝑦 ∈ 𝐵 ↦ 𝑆)⟶ℂ ∧ dom (𝑦 ∈ 𝐵 ↦ 𝑆) ⊆ ℂ ∧ 𝐶 ∈ ℂ))
20	18, 19	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆):dom (𝑦 ∈ 𝐵 ↦ 𝑆)⟶ℂ ∧ dom (𝑦 ∈ 𝐵 ↦ 𝑆) ⊆ ℂ ∧ 𝐶 ∈ ℂ))
21	20	simp2d 1143	. . . . 5 ⊢ (𝜑 → dom (𝑦 ∈ 𝐵 ↦ 𝑆) ⊆ ℂ)
22	17, 21	eqsstrrd 3979	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ)
23	6	snssd 4769	. . . 4 ⊢ (𝜑 → {𝐶} ⊆ ℂ)
24	22, 23	unssd 4151	. . 3 ⊢ (𝜑 → (𝐵 ∪ {𝐶}) ⊆ ℂ)
25		eqid 2729	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
26		eqid 2729	. . 3 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐵 ∪ {𝐶})) = ((TopOpen‘ℂ_fld) ↾_t (𝐵 ∪ {𝐶}))
27	22, 6, 16, 26, 25	limcmpt 25760	. . . 4 ⊢ (𝜑 → (𝐷 ∈ ((𝑦 ∈ 𝐵 ↦ 𝑆) lim_ℂ 𝐶) ↔ (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐵 ∪ {𝐶})) CnP (TopOpen‘ℂ_fld))‘𝐶)))
28	18, 27	mpbid 232	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐵 ∪ {𝐶})) CnP (TopOpen‘ℂ_fld))‘𝐶))
29	14, 24, 25, 26, 5, 28	limccnp 25768	. 2 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆))‘𝐶) ∈ (((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) lim_ℂ 𝑋))
30		eqid 2729	. . 3 ⊢ (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) = (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆))
31		iftrue 4490	. . 3 ⊢ (𝑦 = 𝐶 → if(𝑦 = 𝐶, 𝐷, 𝑆) = 𝐷)
32		ssun2 4138	. . . 4 ⊢ {𝐶} ⊆ (𝐵 ∪ {𝐶})
33		snssg 4743	. . . . 5 ⊢ (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝑋) → (𝐶 ∈ (𝐵 ∪ {𝐶}) ↔ {𝐶} ⊆ (𝐵 ∪ {𝐶})))
34	5, 33	syl 17	. . . 4 ⊢ (𝜑 → (𝐶 ∈ (𝐵 ∪ {𝐶}) ↔ {𝐶} ⊆ (𝐵 ∪ {𝐶})))
35	32, 34	mpbiri 258	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵 ∪ {𝐶}))
36	30, 31, 35, 18	fvmptd3 6973	. 2 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆))‘𝐶) = 𝐷)
37		eqidd 2730	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅))
38		eqidd 2730	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) = (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)))
39		eqeq1 2733	. . . . . 6 ⊢ (𝑦 = 𝑅 → (𝑦 = 𝐶 ↔ 𝑅 = 𝐶))
40		limcco.1	. . . . . 6 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
41	39, 40	ifbieq2d 4511	. . . . 5 ⊢ (𝑦 = 𝑅 → if(𝑦 = 𝐶, 𝐷, 𝑆) = if(𝑅 = 𝐶, 𝐷, 𝑇))
42	13, 37, 38, 41	fmptco 7083	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) = (𝑥 ∈ 𝐴 ↦ if(𝑅 = 𝐶, 𝐷, 𝑇)))
43		limcco.2	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑅 = 𝐶)) → 𝑇 = 𝐷)
44	43	anassrs 467	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑅 = 𝐶) → 𝑇 = 𝐷)
45	44	ifeq1da 4516	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑅 = 𝐶, 𝑇, 𝑇) = if(𝑅 = 𝐶, 𝐷, 𝑇))
46		ifid 4525	. . . . . 6 ⊢ if(𝑅 = 𝐶, 𝑇, 𝑇) = 𝑇
47	45, 46	eqtr3di 2779	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑅 = 𝐶, 𝐷, 𝑇) = 𝑇)
48	47	mpteq2dva 5195	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑅 = 𝐶, 𝐷, 𝑇)) = (𝑥 ∈ 𝐴 ↦ 𝑇))
49	42, 48	eqtrd 2764	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) = (𝑥 ∈ 𝐴 ↦ 𝑇))
50	49	oveq1d 7384	. 2 ⊢ (𝜑 → (((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) lim_ℂ 𝑋) = ((𝑥 ∈ 𝐴 ↦ 𝑇) lim_ℂ 𝑋))
51	29, 36, 50	3eltr3d 2842	1 ⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑇) lim_ℂ 𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∪ cun 3909 ⊆ wss 3911 ifcif 4484 {csn 4585 ↦ cmpt 5183 dom cdm 5631 ∘ ccom 5635 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ↾_t crest 17359 TopOpenctopn 17360 ℂ_fldccnfld 21240 CnP ccnp 23088 lim_ℂ climc 25739

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-map 8778 df-pm 8779 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9338 df-sup 9369 df-inf 9370 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-fz 13445 df-seq 13943 df-exp 14003 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-slot 17128 df-ndx 17140 df-base 17156 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-rest 17361 df-topn 17362 df-topgen 17382 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cnp 23091 df-xms 24184 df-ms 24185 df-limc 25743

This theorem is referenced by: dvcobr 25825 dvcobrOLD 25826 dvcnvlem 25856 lhop2 25896 fourierdlem60 46137 fourierdlem61 46138 fourierdlem62 46139 fourierdlem73 46150 fourierdlem76 46153

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