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Theorem divcncf 25414

Description: The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
divcncf.1	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))
divcncf.2	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→(ℂ ∖ {0})))

**Assertion**
Ref	Expression
divcncf	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵)) ∈ (𝑋–cn→ℂ))

Distinct variable groups: 𝑥,𝑋 𝜑,𝑥 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥)

Proof of Theorem divcncf Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		divcncf.1	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))
2		cncff 24860	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
4	3	fvmptelcdm 7065	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
5		divcncf.2	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→(ℂ ∖ {0})))
6		cncff 24860	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→(ℂ ∖ {0})) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(ℂ ∖ {0}))
7	5, 6	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(ℂ ∖ {0}))
8	7	fvmptelcdm 7065	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (ℂ ∖ {0}))
9	8	eldifad 3901	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
10		eldifsni 4735	. . . . 5 ⊢ (𝐵 ∈ (ℂ ∖ {0}) → 𝐵 ≠ 0)
11	8, 10	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ≠ 0)
12	4, 9, 11	divrecd 11934	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵)))
13	12	mpteq2dva 5178	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 · (1 / 𝐵))))
14	8	ralrimiva 3129	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ (ℂ ∖ {0}))
15		eqidd 2737	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵))
16		eqidd 2737	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) = (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)))
17	14, 15, 16	fmptcos 7084	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) ∘ (𝑥 ∈ 𝑋 ↦ 𝐵)) = (𝑥 ∈ 𝑋 ↦ ⦋𝐵 / 𝑦⦌(1 / 𝑦)))
18		csbov2g 7415	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑦⦌(1 / 𝑦) = (1 / ⦋𝐵 / 𝑦⦌𝑦))
19	9, 18	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋𝐵 / 𝑦⦌(1 / 𝑦) = (1 / ⦋𝐵 / 𝑦⦌𝑦))
20		csbvarg 4374	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑦⦌𝑦 = 𝐵)
21	9, 20	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋𝐵 / 𝑦⦌𝑦 = 𝐵)
22	21	oveq2d 7383	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 / ⦋𝐵 / 𝑦⦌𝑦) = (1 / 𝐵))
23	19, 22	eqtrd 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋𝐵 / 𝑦⦌(1 / 𝑦) = (1 / 𝐵))
24	23	mpteq2dva 5178	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⦋𝐵 / 𝑦⦌(1 / 𝑦)) = (𝑥 ∈ 𝑋 ↦ (1 / 𝐵)))
25	17, 24	eqtr2d 2772	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (1 / 𝐵)) = ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) ∘ (𝑥 ∈ 𝑋 ↦ 𝐵)))
26		ax-1cn 11096	. . . . . 6 ⊢ 1 ∈ ℂ
27		eqid 2736	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) = (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))
28	27	cdivcncf 24888	. . . . . 6 ⊢ (1 ∈ ℂ → (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) ∈ ((ℂ ∖ {0})–cn→ℂ))
29	26, 28	mp1i 13	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) ∈ ((ℂ ∖ {0})–cn→ℂ))
30	5, 29	cncfco 24874	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) ∘ (𝑥 ∈ 𝑋 ↦ 𝐵)) ∈ (𝑋–cn→ℂ))
31	25, 30	eqeltrd 2836	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (1 / 𝐵)) ∈ (𝑋–cn→ℂ))
32	1, 31	mulcncf 25413	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · (1 / 𝐵))) ∈ (𝑋–cn→ℂ))
33	13, 32	eqeltrd 2836	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵)) ∈ (𝑋–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ⦋csb 3837 ∖ cdif 3886 {csn 4567 ↦ cmpt 5166 ∘ ccom 5635 ⟶wf 6494 (class class class)co 7367 ℂcc 11036 0cc0 11038 1c1 11039 · cmul 11043 / cdiv 11807 –cn→ccncf 24843

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 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 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-icc 13305 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cn 23192 df-cnp 23193 df-tx 23527 df-hmeo 23720 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845

This theorem is referenced by: logdivsqrle 34794 divcncff 46319 itgcoscmulx 46397 itgsincmulx 46402 dirkeritg 46530 dirkercncflem2 46532 fourierdlem39 46574 fourierdlem58 46592 fourierdlem62 46596 fourierdlem68 46602 fourierdlem76 46610

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