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Mirrors > Home > MPE Home > Th. List > divcncf | Structured version Visualization version GIF version |
Description: The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
divcncf.1 | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) |
divcncf.2 | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→(ℂ ∖ {0}))) |
Ref | Expression |
---|---|
divcncf | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵)) ∈ (𝑋–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divcncf.1 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) | |
2 | cncff 23493 | . . . . . 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 | fvmptelrn 6870 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) |
5 | divcncf.2 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→(ℂ ∖ {0}))) | |
6 | cncff 23493 | . . . . . . 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 | fvmptelrn 6870 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (ℂ ∖ {0})) |
9 | 8 | eldifad 3946 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) |
10 | eldifsni 4714 | . . . . 5 ⊢ (𝐵 ∈ (ℂ ∖ {0}) → 𝐵 ≠ 0) | |
11 | 8, 10 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ≠ 0) |
12 | 4, 9, 11 | divrecd 11411 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐵) = (𝐴 · (1 / 𝐵))) |
13 | 12 | mpteq2dva 5152 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 · (1 / 𝐵)))) |
14 | 8 | ralrimiva 3180 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ (ℂ ∖ {0})) |
15 | eqidd 2820 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)) | |
16 | eqidd 2820 | . . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) = (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) | |
17 | 14, 15, 16 | fmptcos 6886 | . . . . 5 ⊢ (𝜑 → ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) ∘ (𝑥 ∈ 𝑋 ↦ 𝐵)) = (𝑥 ∈ 𝑋 ↦ ⦋𝐵 / 𝑦⦌(1 / 𝑦))) |
18 | csbov2g 7194 | . . . . . . . 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 4381 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → ⦋𝐵 / 𝑦⦌𝑦 = 𝐵) | |
21 | 9, 20 | syl 17 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋𝐵 / 𝑦⦌𝑦 = 𝐵) |
22 | 21 | oveq2d 7164 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 / ⦋𝐵 / 𝑦⦌𝑦) = (1 / 𝐵)) |
23 | 19, 22 | eqtrd 2854 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋𝐵 / 𝑦⦌(1 / 𝑦) = (1 / 𝐵)) |
24 | 23 | mpteq2dva 5152 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⦋𝐵 / 𝑦⦌(1 / 𝑦)) = (𝑥 ∈ 𝑋 ↦ (1 / 𝐵))) |
25 | 17, 24 | eqtr2d 2855 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (1 / 𝐵)) = ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) ∘ (𝑥 ∈ 𝑋 ↦ 𝐵))) |
26 | ax-1cn 10587 | . . . . . 6 ⊢ 1 ∈ ℂ | |
27 | eqid 2819 | . . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) = (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) | |
28 | 27 | cdivcncf 23517 | . . . . . 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 23507 | . . . 4 ⊢ (𝜑 → ((𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦)) ∘ (𝑥 ∈ 𝑋 ↦ 𝐵)) ∈ (𝑋–cn→ℂ)) |
31 | 25, 30 | eqeltrd 2911 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (1 / 𝐵)) ∈ (𝑋–cn→ℂ)) |
32 | 1, 31 | mulcncf 24039 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · (1 / 𝐵))) ∈ (𝑋–cn→ℂ)) |
33 | 13, 32 | eqeltrd 2911 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵)) ∈ (𝑋–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ⦋csb 3881 ∖ cdif 3931 {csn 4559 ↦ cmpt 5137 ∘ ccom 5552 ⟶wf 6344 (class class class)co 7148 ℂcc 10527 0cc0 10529 1c1 10530 · cmul 10534 / cdiv 11289 –cn→ccncf 23476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-mulf 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-icc 12737 df-fz 12885 df-fzo 13026 df-seq 13362 df-exp 13422 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-cnfld 20538 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cn 21827 df-cnp 21828 df-tx 22162 df-hmeo 22355 df-xms 22922 df-ms 22923 df-tms 22924 df-cncf 23478 |
This theorem is referenced by: logdivsqrle 31914 divcncff 42164 itgcoscmulx 42244 itgsincmulx 42249 dirkeritg 42378 dirkercncflem2 42380 fourierdlem39 42422 fourierdlem58 42440 fourierdlem62 42444 fourierdlem68 42450 fourierdlem76 42458 |
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