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Theorem limcco 26206

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 462	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅 ≠ 𝐶 → 𝑅 ∈ 𝐵))
3	2	necon1bd 2974	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑅 ∈ 𝐵 → 𝑅 = 𝐶))
4		limccl 26188	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝑋) ⊆ ℂ
5		limcco.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝑋))
6	4, 5	sselid 3929	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
7	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
8		elsn2g 4625	. . . . . . . 8 ⊢ (𝐶 ∈ ℂ → (𝑅 ∈ {𝐶} ↔ 𝑅 = 𝐶))
9	7, 8	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅 ∈ {𝐶} ↔ 𝑅 = 𝐶))
10	3, 9	sylibrd 262	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑅 ∈ 𝐵 → 𝑅 ∈ {𝐶}))
11	10	orrd 877	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅 ∈ 𝐵 ∨ 𝑅 ∈ {𝐶}))
12		elun 4100	. . . . 5 ⊢ (𝑅 ∈ (𝐵 ∪ {𝐶}) ↔ (𝑅 ∈ 𝐵 ∨ 𝑅 ∈ {𝐶}))
13	11, 12	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ (𝐵 ∪ {𝐶}))
14	13	fmpttd 7113	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅):𝐴⟶(𝐵 ∪ {𝐶}))
15		eqid 2761	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑦 ∈ 𝐵 ↦ 𝑆)
16		limcco.s	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ ℂ)
17	15, 16	dmmptd 6682	. . . . 5 ⊢ (𝜑 → dom (𝑦 ∈ 𝐵 ↦ 𝑆) = 𝐵)
18		limcco.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ((𝑦 ∈ 𝐵 ↦ 𝑆) lim_ℂ 𝐶))
19		limcrcl 26187	. . . . . . 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 18	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆):dom (𝑦 ∈ 𝐵 ↦ 𝑆)⟶ℂ ∧ dom (𝑦 ∈ 𝐵 ↦ 𝑆) ⊆ ℂ ∧ 𝐶 ∈ ℂ))
21	20	simp2d 1161	. . . . 5 ⊢ (𝜑 → dom (𝑦 ∈ 𝐵 ↦ 𝑆) ⊆ ℂ)
22	17, 21	eqsstrrd 3966	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ)
23	6	snssd 4747	. . . 4 ⊢ (𝜑 → {𝐶} ⊆ ℂ)
24	22, 23	unssd 4138	. . 3 ⊢ (𝜑 → (𝐵 ∪ {𝐶}) ⊆ ℂ)
25		eqid 2761	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
26		eqid 2761	. . 3 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐵 ∪ {𝐶})) = ((TopOpen‘ℂ_fld) ↾_t (𝐵 ∪ {𝐶}))
27	22, 6, 16, 26, 25	limcmpt 26196	. . . 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 235	. . 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 26204	. 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 2761	. . 3 ⊢ (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) = (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆))
31		iftrue 4488	. . 3 ⊢ (𝑦 = 𝐶 → if(𝑦 = 𝐶, 𝐷, 𝑆) = 𝐷)
32		ssun2 4125	. . . 4 ⊢ {𝐶} ⊆ (𝐵 ∪ {𝐶})
33		snssg 4744	. . . . 5 ⊢ (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝑋) → (𝐶 ∈ (𝐵 ∪ {𝐶}) ↔ {𝐶} ⊆ (𝐵 ∪ {𝐶})))
34	5, 33	syl 18	. . . 4 ⊢ (𝜑 → (𝐶 ∈ (𝐵 ∪ {𝐶}) ↔ {𝐶} ⊆ (𝐵 ∪ {𝐶})))
35	32, 34	mpbiri 261	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵 ∪ {𝐶}))
36	30, 31, 35, 18	fvmptd3 7015	. 2 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆))‘𝐶) = 𝐷)
37		eqidd 2762	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅))
38		eqidd 2762	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) = (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)))
39		eqeq1 2765	. . . . . 6 ⊢ (𝑦 = 𝑅 → (𝑦 = 𝐶 ↔ 𝑅 = 𝐶))
40		limcco.1	. . . . . 6 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
41	39, 40	ifbieq2d 4509	. . . . 5 ⊢ (𝑦 = 𝑅 → if(𝑦 = 𝐶, 𝐷, 𝑆) = if(𝑅 = 𝐶, 𝐷, 𝑇))
42	13, 37, 38, 41	fmptco 7128	. . . 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 473	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑅 = 𝐶) → 𝑇 = 𝐷)
45	44	ifeq1da 4514	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑅 = 𝐶, 𝑇, 𝑇) = if(𝑅 = 𝐶, 𝐷, 𝑇))
46		ifid 4523	. . . . . 6 ⊢ if(𝑅 = 𝐶, 𝑇, 𝑇) = 𝑇
47	45, 46	eqtr3di 2811	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑅 = 𝐶, 𝐷, 𝑇) = 𝑇)
48	47	mpteq2dva 5198	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑅 = 𝐶, 𝐷, 𝑇)) = (𝑥 ∈ 𝐴 ↦ 𝑇))
49	42, 48	eqtrd 2796	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) = (𝑥 ∈ 𝐴 ↦ 𝑇))
50	49	oveq1d 7433	. 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 2875	1 ⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑇) lim_ℂ 𝑋))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∪ cun 3897 ⊆ wss 3899 ifcif 4482 {csn 4584 ↦ cmpt 5186 dom cdm 5651 ∘ ccom 5655 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 ℂcc 11191 ↾_t crest 17584 TopOpenctopn 17585 ℂ_fldccnfld 21671 CnP ccnp 23536 lim_ℂ climc 26175

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-map 8842 df-pm 8843 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fi 9396 df-sup 9427 df-inf 9428 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-fz 13633 df-seq 14138 df-exp 14198 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-struct 17318 df-slot 17353 df-ndx 17365 df-base 17381 df-plusg 17434 df-mulr 17435 df-starv 17436 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-rest 17586 df-topn 17587 df-topgen 17607 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-cnfld 21672 df-top 23205 df-topon 23222 df-topsp 23244 df-bases 23257 df-cnp 23539 df-xms 24632 df-ms 24633 df-limc 26179

This theorem is used by: dvcobr 26259 dvcnvlem 26289 lhop2 26328 fourierdlem60 47145 fourierdlem61 47146 fourierdlem62 47147 fourierdlem73 47158 fourierdlem76 47161

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