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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvmulcncf | Structured version Visualization version GIF version |
Description: A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
dvmulcncf.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvmulcncf.f | ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
dvmulcncf.g | ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) |
dvmulcncf.fdv | ⊢ (𝜑 → (𝑆 D 𝐹) ∈ (𝑋–cn→ℂ)) |
dvmulcncf.gdv | ⊢ (𝜑 → (𝑆 D 𝐺) ∈ (𝑋–cn→ℂ)) |
Ref | Expression |
---|---|
dvmulcncf | ⊢ (𝜑 → (𝑆 D (𝐹 ∘f · 𝐺)) ∈ (𝑋–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvmulcncf.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | dvmulcncf.f | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) | |
3 | dvmulcncf.g | . . 3 ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) | |
4 | dvmulcncf.fdv | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐹) ∈ (𝑋–cn→ℂ)) | |
5 | cncff 24338 | . . . 4 ⊢ ((𝑆 D 𝐹) ∈ (𝑋–cn→ℂ) → (𝑆 D 𝐹):𝑋⟶ℂ) | |
6 | fdm 6713 | . . . 4 ⊢ ((𝑆 D 𝐹):𝑋⟶ℂ → dom (𝑆 D 𝐹) = 𝑋) | |
7 | 4, 5, 6 | 3syl 18 | . . 3 ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) |
8 | dvmulcncf.gdv | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐺) ∈ (𝑋–cn→ℂ)) | |
9 | cncff 24338 | . . . 4 ⊢ ((𝑆 D 𝐺) ∈ (𝑋–cn→ℂ) → (𝑆 D 𝐺):𝑋⟶ℂ) | |
10 | fdm 6713 | . . . 4 ⊢ ((𝑆 D 𝐺):𝑋⟶ℂ → dom (𝑆 D 𝐺) = 𝑋) | |
11 | 8, 9, 10 | 3syl 18 | . . 3 ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋) |
12 | 1, 2, 3, 7, 11 | dvmulf 25389 | . 2 ⊢ (𝜑 → (𝑆 D (𝐹 ∘f · 𝐺)) = (((𝑆 D 𝐹) ∘f · 𝐺) ∘f + ((𝑆 D 𝐺) ∘f · 𝐹))) |
13 | ax-resscn 11149 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
14 | sseq1 4003 | . . . . . . . 8 ⊢ (𝑆 = ℝ → (𝑆 ⊆ ℂ ↔ ℝ ⊆ ℂ)) | |
15 | 13, 14 | mpbiri 257 | . . . . . . 7 ⊢ (𝑆 = ℝ → 𝑆 ⊆ ℂ) |
16 | eqimss 4036 | . . . . . . 7 ⊢ (𝑆 = ℂ → 𝑆 ⊆ ℂ) | |
17 | 15, 16 | pm3.2i 471 | . . . . . 6 ⊢ ((𝑆 = ℝ → 𝑆 ⊆ ℂ) ∧ (𝑆 = ℂ → 𝑆 ⊆ ℂ)) |
18 | elpri 4644 | . . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
19 | 1, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑆 = ℝ ∨ 𝑆 = ℂ)) |
20 | pm3.44 958 | . . . . . 6 ⊢ (((𝑆 = ℝ → 𝑆 ⊆ ℂ) ∧ (𝑆 = ℂ → 𝑆 ⊆ ℂ)) → ((𝑆 = ℝ ∨ 𝑆 = ℂ) → 𝑆 ⊆ ℂ)) | |
21 | 17, 19, 20 | mpsyl 68 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
22 | dvbsss 25348 | . . . . . 6 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆 | |
23 | 7, 22 | eqsstrrdi 4033 | . . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
24 | dvcn 25367 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐺:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆) ∧ dom (𝑆 D 𝐺) = 𝑋) → 𝐺 ∈ (𝑋–cn→ℂ)) | |
25 | 21, 3, 23, 11, 24 | syl31anc 1373 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→ℂ)) |
26 | 4, 25 | mulcncff 44359 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐹) ∘f · 𝐺) ∈ (𝑋–cn→ℂ)) |
27 | dvcn 25367 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆) ∧ dom (𝑆 D 𝐹) = 𝑋) → 𝐹 ∈ (𝑋–cn→ℂ)) | |
28 | 21, 2, 23, 7, 27 | syl31anc 1373 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℂ)) |
29 | 8, 28 | mulcncff 44359 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐺) ∘f · 𝐹) ∈ (𝑋–cn→ℂ)) |
30 | 26, 29 | addcncff 44373 | . 2 ⊢ (𝜑 → (((𝑆 D 𝐹) ∘f · 𝐺) ∘f + ((𝑆 D 𝐺) ∘f · 𝐹)) ∈ (𝑋–cn→ℂ)) |
31 | 12, 30 | eqeltrd 2832 | 1 ⊢ (𝜑 → (𝑆 D (𝐹 ∘f · 𝐺)) ∈ (𝑋–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ⊆ wss 3944 {cpr 4624 dom cdm 5669 ⟶wf 6528 (class class class)co 7393 ∘f cof 7651 ℂcc 11090 ℝcr 11091 + caddc 11095 · cmul 11097 –cn→ccncf 24321 D cdv 25309 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 ax-addf 11171 ax-mulf 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-2o 8449 df-er 8686 df-map 8805 df-pm 8806 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9487 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-q 12915 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-icc 13313 df-fz 13467 df-fzo 13610 df-seq 13949 df-exp 14010 df-hash 14273 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17350 df-topn 17351 df-0g 17369 df-gsum 17370 df-topgen 17371 df-pt 17372 df-prds 17375 df-xrs 17430 df-qtop 17435 df-imas 17436 df-xps 17438 df-mre 17512 df-mrc 17513 df-acs 17515 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-mulg 18923 df-cntz 19147 df-cmn 19614 df-psmet 20870 df-xmet 20871 df-met 20872 df-bl 20873 df-mopn 20874 df-fbas 20875 df-fg 20876 df-cnfld 20879 df-top 22325 df-topon 22342 df-topsp 22364 df-bases 22378 df-cld 22452 df-ntr 22453 df-cls 22454 df-nei 22531 df-lp 22569 df-perf 22570 df-cn 22660 df-cnp 22661 df-haus 22748 df-tx 22995 df-hmeo 23188 df-fil 23279 df-fm 23371 df-flim 23372 df-flf 23373 df-xms 23755 df-ms 23756 df-tms 23757 df-cncf 24323 df-limc 25312 df-dv 25313 |
This theorem is referenced by: fourierdlem72 44667 |
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