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Theorem limcco 25862

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 2951	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑅 ∈ 𝐵 → 𝑅 = 𝐶))
4		limccl 25844	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝑋) ⊆ ℂ
5		limcco.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝑋))
6	4, 5	sselid 3933	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
7	6	adantr 480	. . . . . . . 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 259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑅 ∈ 𝐵 → 𝑅 ∈ {𝐶}))
11	10	orrd 864	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅 ∈ 𝐵 ∨ 𝑅 ∈ {𝐶}))
12		elun 4107	. . . . 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 2737	. . . . . 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 25843	. . . . . . 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 1144	. . . . 5 ⊢ (𝜑 → dom (𝑦 ∈ 𝐵 ↦ 𝑆) ⊆ ℂ)
22	17, 21	eqsstrrd 3971	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ)
23	6	snssd 4767	. . . 4 ⊢ (𝜑 → {𝐶} ⊆ ℂ)
24	22, 23	unssd 4146	. . 3 ⊢ (𝜑 → (𝐵 ∪ {𝐶}) ⊆ ℂ)
25		eqid 2737	. . 3 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
26		eqid 2737	. . 3 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐵 ∪ {𝐶})) = ((TopOpen‘ℂ_fld) ↾_t (𝐵 ∪ {𝐶}))
27	22, 6, 16, 26, 25	limcmpt 25852	. . . 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 25860	. 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 2737	. . 3 ⊢ (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) = (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆))
31		iftrue 4487	. . 3 ⊢ (𝑦 = 𝐶 → if(𝑦 = 𝐶, 𝐷, 𝑆) = 𝐷)
32		ssun2 4133	. . . 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 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 2738	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅))
38		eqidd 2738	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) = (𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)))
39		eqeq1 2741	. . . . . 6 ⊢ (𝑦 = 𝑅 → (𝑦 = 𝐶 ↔ 𝑅 = 𝐶))
40		limcco.1	. . . . . 6 ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)
41	39, 40	ifbieq2d 4508	. . . . 5 ⊢ (𝑦 = 𝑅 → if(𝑦 = 𝐶, 𝐷, 𝑆) = if(𝑅 = 𝐶, 𝐷, 𝑇))
42	13, 37, 38, 41	fmptco 7084	. . . 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 4513	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑅 = 𝐶, 𝑇, 𝑇) = if(𝑅 = 𝐶, 𝐷, 𝑇))
46		ifid 4522	. . . . . 6 ⊢ if(𝑅 = 𝐶, 𝑇, 𝑇) = 𝑇
47	45, 46	eqtr3di 2787	. . . . 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 2772	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∪ {𝐶}) ↦ if(𝑦 = 𝐶, 𝐷, 𝑆)) ∘ (𝑥 ∈ 𝐴 ↦ 𝑅)) = (𝑥 ∈ 𝐴 ↦ 𝑇))
50	49	oveq1d 7383	. 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 2851	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 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∪ cun 3901 ⊆ wss 3903 ifcif 4481 {csn 4582 ↦ cmpt 5181 dom cdm 5632 ∘ ccom 5636 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ↾_t crest 17352 TopOpenctopn 17353 ℂ_fldccnfld 21321 CnP ccnp 23181 lim_ℂ climc 25831

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9326 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-fz 13436 df-seq 13937 df-exp 13997 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17149 df-plusg 17202 df-mulr 17203 df-starv 17204 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-rest 17354 df-topn 17355 df-topgen 17375 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cnp 23184 df-xms 24276 df-ms 24277 df-limc 25835

This theorem is referenced by: dvcobr 25917 dvcobrOLD 25918 dvcnvlem 25948 lhop2 25988 fourierdlem60 46524 fourierdlem61 46525 fourierdlem62 46526 fourierdlem73 46537 fourierdlem76 46540

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