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Theorem limcco 25952

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 460	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅 ≠ 𝐶 → 𝑅 ∈ 𝐵))
3	2	necon1bd 2975	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑅 ∈ 𝐵 → 𝑅 = 𝐶))
4		limccl 25934	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝑋) ⊆ ℂ
5		limcco.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝑋))
6	4, 5	sselid 3934	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
7	6	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
8		elsn2g 4623	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (𝑅 ∈ {𝐶} ↔ 𝑅 = 𝐶))
9	7, 8	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅 ∈ {𝐶} ↔ 𝑅 = 𝐶))
10	3, 9	sylibrd 261	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑅 ∈ 𝐵 → 𝑅 ∈ {𝐶}))
11	10	orrd 874	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅 ∈ 𝐵 ∨ 𝑅 ∈ {𝐶}))
12		elun 4106	. . . . 5 ⊢ (𝑅 ∈ (𝐵 ∪ {𝐶}) ↔ (𝑅 ∈ 𝐵 ∨ 𝑅 ∈ {𝐶}))
13	11, 12	sylibr 236	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ (𝐵 ∪ {𝐶}))
14	13	fmpttd 7096	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶(𝐵 ∪ {𝐶}))
15		eqid 2762	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑦 ∈ 𝐵 ↦ 𝑆)
16		limcco.s	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ ℂ)
17	15, 16	dmmptd 6666	. . . . 5 ⊢ (𝜑 → dom (𝑦 ∈ 𝐵 ↦ 𝑆) = 𝐵)
18		limcco.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ((𝑦 ∈ 𝐵 ↦ 𝑆) lim_ℂ 𝐶))
19		limcrcl 25933	. . . . . . 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 1156	. . . . 5 ⊢ (𝜑 → dom (𝑦 ∈ 𝐵 ↦ 𝑆) ⊆ ℂ)
22	17, 21	eqsstrrd 3971	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ)
23	6	snssd 4745	. . . 4 ⊢ (𝜑 → {𝐶} ⊆ ℂ)
24	22, 23	unssd 4144	. . 3 ⊢ (𝜑 → (𝐵 ∪ {𝐶}) ⊆ ℂ)
25		eqid 2762	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
26		eqid 2762	. . 3 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐵 ∪ {𝐶})) = ((TopOpen‘ℂ_fld) ↾_t (𝐵 ∪ {𝐶}))
27	22, 6, 16, 26, 25	limcmpt 25942	. . . 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 234	. . 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 25950	. 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 2762	. . 3 ⊢ (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) = (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆))
31		iftrue 4486	. . 3 ⊢ (𝑦 = 𝐶 → if(𝑦 = 𝐶, 𝐷, 𝑆) = 𝐷)
32		ssun2 4131	. . . 4 ⊢ {𝐶} ⊆ (𝐵 ∪ {𝐶})
33		snssg 4742	. . . . 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 260	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵 ∪ {𝐶}))
36	30, 31, 35, 18	fvmptd3 6999	. 2 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆))‘𝐶) = 𝐷)
37		eqidd 2763	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅))
38		eqidd 2763	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) = (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)))
39		eqeq1 2766	. . . . . 6 ⊢ (𝑦 = 𝑅 → (𝑦 = 𝐶 ↔ 𝑅 = 𝐶))
40		limcco.1	. . . . . 6 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
41	39, 40	ifbieq2d 4507	. . . . 5 ⊢ (𝑦 = 𝑅 → if(𝑦 = 𝐶, 𝐷, 𝑆) = if(𝑅 = 𝐶, 𝐷, 𝑇))
42	13, 37, 38, 41	fmptco 7111	. . . 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 471	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑅 = 𝐶) → 𝑇 = 𝐷)
45	44	ifeq1da 4512	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑅 = 𝐶, 𝑇, 𝑇) = if(𝑅 = 𝐶, 𝐷, 𝑇))
46		ifid 4521	. . . . . 6 ⊢ if(𝑅 = 𝐶, 𝑇, 𝑇) = 𝑇
47	45, 46	eqtr3di 2812	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑅 = 𝐶, 𝐷, 𝑇) = 𝑇)
48	47	mpteq2dva 5193	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑅 = 𝐶, 𝐷, 𝑇)) = (𝑥 ∈ 𝐴 ↦ 𝑇))
49	42, 48	eqtrd 2797	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) = (𝑥 ∈ 𝐴 ↦ 𝑇))
50	49	oveq1d 7411	. 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 2876	1 ⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑇) lim_ℂ 𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ∪ cun 3902 ⊆ wss 3904 ifcif 4480 {csn 4582 ↦ cmpt 5181 dom cdm 5647 ∘ ccom 5651 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ℂcc 11071 ↾_t crest 17449 TopOpenctopn 17450 ℂ_fldccnfld 21421 CnP ccnp 23282 lim_ℂ climc 25921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-map 8810 df-pm 8811 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fi 9357 df-sup 9388 df-inf 9389 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-fz 13513 df-seq 14015 df-exp 14075 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-struct 17183 df-slot 17218 df-ndx 17230 df-base 17246 df-plusg 17299 df-mulr 17300 df-starv 17301 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-rest 17451 df-topn 17452 df-topgen 17472 df-psmet 21413 df-xmet 21414 df-met 21415 df-bl 21416 df-mopn 21417 df-cnfld 21422 df-top 22951 df-topon 22968 df-topsp 22990 df-bases 23003 df-cnp 23285 df-xms 24377 df-ms 24378 df-limc 25925

This theorem is referenced by: dvcobr 26005 dvcnvlem 26035 lhop2 26074 fourierdlem60 46737 fourierdlem61 46738 fourierdlem62 46739 fourierdlem73 46750 fourierdlem76 46753

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