Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvdivcncf | Structured version Visualization version GIF version |
Description: A sufficient condition for the derivative of a quotient to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
dvdivcncf.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvdivcncf.f | ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
dvdivcncf.g | ⊢ (𝜑 → 𝐺:𝑋⟶(ℂ ∖ {0})) |
dvdivcncf.fdv | ⊢ (𝜑 → (𝑆 D 𝐹) ∈ (𝑋–cn→ℂ)) |
dvdivcncf.gdv | ⊢ (𝜑 → (𝑆 D 𝐺) ∈ (𝑋–cn→ℂ)) |
Ref | Expression |
---|---|
dvdivcncf | ⊢ (𝜑 → (𝑆 D (𝐹 ∘𝑓 / 𝐺)) ∈ (𝑋–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdivcncf.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | dvdivcncf.f | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) | |
3 | dvdivcncf.g | . . 3 ⊢ (𝜑 → 𝐺:𝑋⟶(ℂ ∖ {0})) | |
4 | dvdivcncf.fdv | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐹) ∈ (𝑋–cn→ℂ)) | |
5 | cncff 22696 | . . . 4 ⊢ ((𝑆 D 𝐹) ∈ (𝑋–cn→ℂ) → (𝑆 D 𝐹):𝑋⟶ℂ) | |
6 | fdm 6051 | . . . 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 | dvdivcncf.gdv | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐺) ∈ (𝑋–cn→ℂ)) | |
9 | cncff 22696 | . . . 4 ⊢ ((𝑆 D 𝐺) ∈ (𝑋–cn→ℂ) → (𝑆 D 𝐺):𝑋⟶ℂ) | |
10 | fdm 6051 | . . . 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 | dvdivf 40137 | . 2 ⊢ (𝜑 → (𝑆 D (𝐹 ∘𝑓 / 𝐺)) = ((((𝑆 D 𝐹) ∘𝑓 · 𝐺) ∘𝑓 − ((𝑆 D 𝐺) ∘𝑓 · 𝐹)) ∘𝑓 / (𝐺 ∘𝑓 · 𝐺))) |
13 | ax-resscn 9993 | . . . . . . . . 9 ⊢ ℝ ⊆ ℂ | |
14 | sseq1 3626 | . . . . . . . . 9 ⊢ (𝑆 = ℝ → (𝑆 ⊆ ℂ ↔ ℝ ⊆ ℂ)) | |
15 | 13, 14 | mpbiri 248 | . . . . . . . 8 ⊢ (𝑆 = ℝ → 𝑆 ⊆ ℂ) |
16 | eqimss 3657 | . . . . . . . 8 ⊢ (𝑆 = ℂ → 𝑆 ⊆ ℂ) | |
17 | 15, 16 | pm3.2i 471 | . . . . . . 7 ⊢ ((𝑆 = ℝ → 𝑆 ⊆ ℂ) ∧ (𝑆 = ℂ → 𝑆 ⊆ ℂ)) |
18 | elpri 4197 | . . . . . . . 8 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
19 | 1, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑆 = ℝ ∨ 𝑆 = ℂ)) |
20 | pm3.44 533 | . . . . . . 7 ⊢ (((𝑆 = ℝ → 𝑆 ⊆ ℂ) ∧ (𝑆 = ℂ → 𝑆 ⊆ ℂ)) → ((𝑆 = ℝ ∨ 𝑆 = ℂ) → 𝑆 ⊆ ℂ)) | |
21 | 17, 19, 20 | mpsyl 68 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
22 | difssd 3738 | . . . . . . 7 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ) | |
23 | 3, 22 | fssd 6057 | . . . . . 6 ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) |
24 | dvbsss 23666 | . . . . . . 7 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆 | |
25 | 7, 24 | syl6eqssr 3656 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
26 | dvcn 23684 | . . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐺:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆) ∧ dom (𝑆 D 𝐺) = 𝑋) → 𝐺 ∈ (𝑋–cn→ℂ)) | |
27 | 21, 23, 25, 11, 26 | syl31anc 1329 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→ℂ)) |
28 | 4, 27 | mulcncff 40081 | . . . 4 ⊢ (𝜑 → ((𝑆 D 𝐹) ∘𝑓 · 𝐺) ∈ (𝑋–cn→ℂ)) |
29 | dvcn 23684 | . . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆) ∧ dom (𝑆 D 𝐹) = 𝑋) → 𝐹 ∈ (𝑋–cn→ℂ)) | |
30 | 21, 2, 25, 7, 29 | syl31anc 1329 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℂ)) |
31 | 8, 30 | mulcncff 40081 | . . . 4 ⊢ (𝜑 → ((𝑆 D 𝐺) ∘𝑓 · 𝐹) ∈ (𝑋–cn→ℂ)) |
32 | 28, 31 | subcncff 40093 | . . 3 ⊢ (𝜑 → (((𝑆 D 𝐹) ∘𝑓 · 𝐺) ∘𝑓 − ((𝑆 D 𝐺) ∘𝑓 · 𝐹)) ∈ (𝑋–cn→ℂ)) |
33 | eldifi 3732 | . . . . . . . . 9 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ∈ ℂ) | |
34 | 33 | adantr 481 | . . . . . . . 8 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ) |
35 | eldifi 3732 | . . . . . . . . 9 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ) | |
36 | 35 | adantl 482 | . . . . . . . 8 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ ℂ) |
37 | 34, 36 | mulcld 10060 | . . . . . . 7 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 · 𝑦) ∈ ℂ) |
38 | eldifsni 4320 | . . . . . . . . 9 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → 𝑥 ≠ 0) | |
39 | 38 | adantr 481 | . . . . . . . 8 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑥 ≠ 0) |
40 | eldifsni 4320 | . . . . . . . . 9 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0) | |
41 | 40 | adantl 482 | . . . . . . . 8 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ≠ 0) |
42 | 34, 36, 39, 41 | mulne0d 10679 | . . . . . . 7 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 · 𝑦) ≠ 0) |
43 | eldifsn 4317 | . . . . . . 7 ⊢ ((𝑥 · 𝑦) ∈ (ℂ ∖ {0}) ↔ ((𝑥 · 𝑦) ∈ ℂ ∧ (𝑥 · 𝑦) ≠ 0)) | |
44 | 37, 42, 43 | sylanbrc 698 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 · 𝑦) ∈ (ℂ ∖ {0})) |
45 | 44 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0}))) → (𝑥 · 𝑦) ∈ (ℂ ∖ {0})) |
46 | 1, 25 | ssexd 4805 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ V) |
47 | inidm 3822 | . . . . 5 ⊢ (𝑋 ∩ 𝑋) = 𝑋 | |
48 | 45, 3, 3, 46, 46, 47 | off 6912 | . . . 4 ⊢ (𝜑 → (𝐺 ∘𝑓 · 𝐺):𝑋⟶(ℂ ∖ {0})) |
49 | 27, 27 | mulcncff 40081 | . . . . 5 ⊢ (𝜑 → (𝐺 ∘𝑓 · 𝐺) ∈ (𝑋–cn→ℂ)) |
50 | cncffvrn 22701 | . . . . 5 ⊢ (((ℂ ∖ {0}) ⊆ ℂ ∧ (𝐺 ∘𝑓 · 𝐺) ∈ (𝑋–cn→ℂ)) → ((𝐺 ∘𝑓 · 𝐺) ∈ (𝑋–cn→(ℂ ∖ {0})) ↔ (𝐺 ∘𝑓 · 𝐺):𝑋⟶(ℂ ∖ {0}))) | |
51 | 22, 49, 50 | syl2anc 693 | . . . 4 ⊢ (𝜑 → ((𝐺 ∘𝑓 · 𝐺) ∈ (𝑋–cn→(ℂ ∖ {0})) ↔ (𝐺 ∘𝑓 · 𝐺):𝑋⟶(ℂ ∖ {0}))) |
52 | 48, 51 | mpbird 247 | . . 3 ⊢ (𝜑 → (𝐺 ∘𝑓 · 𝐺) ∈ (𝑋–cn→(ℂ ∖ {0}))) |
53 | 32, 52 | divcncff 40104 | . 2 ⊢ (𝜑 → ((((𝑆 D 𝐹) ∘𝑓 · 𝐺) ∘𝑓 − ((𝑆 D 𝐺) ∘𝑓 · 𝐹)) ∘𝑓 / (𝐺 ∘𝑓 · 𝐺)) ∈ (𝑋–cn→ℂ)) |
54 | 12, 53 | eqeltrd 2701 | 1 ⊢ (𝜑 → (𝑆 D (𝐹 ∘𝑓 / 𝐺)) ∈ (𝑋–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 {cpr 4179 dom cdm 5114 ⟶wf 5884 (class class class)co 6650 ∘𝑓 cof 6895 ℂcc 9934 ℝcr 9935 0cc0 9936 · cmul 9941 − cmin 10266 / cdiv 10684 –cn→ccncf 22679 D cdv 23627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-t1 21118 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 |
This theorem is referenced by: fourierdlem58 40381 fourierdlem59 40382 |
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