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Mirrors > Home > MPE Home > Th. List > dvmptdiv | Structured version Visualization version GIF version |
Description: Function-builder for derivative, quotient rule. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
dvmptdiv.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvmptdiv.a | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) |
dvmptdiv.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) |
dvmptdiv.da | ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) |
dvmptdiv.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ (ℂ ∖ {0})) |
dvmptdiv.d | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ℂ) |
dvmptdiv.dc | ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷)) |
Ref | Expression |
---|---|
dvmptdiv | ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (((𝐵 · 𝐶) − (𝐷 · 𝐴)) / (𝐶↑2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvmptdiv.a | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) | |
2 | dvmptdiv.c | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ (ℂ ∖ {0})) | |
3 | 2 | eldifad 3892 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ) |
4 | eldifsn 4714 | . . . . . . 7 ⊢ (𝐶 ∈ (ℂ ∖ {0}) ↔ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) | |
5 | 2, 4 | sylib 221 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) |
6 | 5 | simprd 499 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ≠ 0) |
7 | 1, 3, 6 | divrecd 11635 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶))) |
8 | 7 | mpteq2dva 5164 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (𝐴 · (1 / 𝐶)))) |
9 | 8 | oveq2d 7247 | . 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) = (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 · (1 / 𝐶))))) |
10 | dvmptdiv.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
11 | dvmptdiv.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) | |
12 | dvmptdiv.da | . . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) | |
13 | 3, 6 | reccld 11625 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 / 𝐶) ∈ ℂ) |
14 | 1cnd 10852 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 1 ∈ ℂ) | |
15 | dvmptdiv.d | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ℂ) | |
16 | 14, 15 | mulcld 10877 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 · 𝐷) ∈ ℂ) |
17 | 3 | sqcld 13738 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶↑2) ∈ ℂ) |
18 | 6 | neneqd 2946 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝐶 = 0) |
19 | sqeq0 13716 | . . . . . . . 8 ⊢ (𝐶 ∈ ℂ → ((𝐶↑2) = 0 ↔ 𝐶 = 0)) | |
20 | 3, 19 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐶↑2) = 0 ↔ 𝐶 = 0)) |
21 | 18, 20 | mtbird 328 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ (𝐶↑2) = 0) |
22 | 21 | neqned 2948 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶↑2) ≠ 0) |
23 | 16, 17, 22 | divcld 11632 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((1 · 𝐷) / (𝐶↑2)) ∈ ℂ) |
24 | 23 | negcld 11200 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → -((1 · 𝐷) / (𝐶↑2)) ∈ ℂ) |
25 | 1cnd 10852 | . . . 4 ⊢ (𝜑 → 1 ∈ ℂ) | |
26 | dvmptdiv.dc | . . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷)) | |
27 | 10, 25, 2, 15, 26 | dvrecg 24894 | . . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (1 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ -((1 · 𝐷) / (𝐶↑2)))) |
28 | 10, 1, 11, 12, 13, 24, 27 | dvmptmul 24882 | . 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 · (1 / 𝐶)))) = (𝑥 ∈ 𝑋 ↦ ((𝐵 · (1 / 𝐶)) + (-((1 · 𝐷) / (𝐶↑2)) · 𝐴)))) |
29 | 10, 1, 11, 12 | dvmptcl 24880 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) |
30 | 29, 3 | mulcld 10877 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 · 𝐶) ∈ ℂ) |
31 | 30, 17, 22 | divcld 11632 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐵 · 𝐶) / (𝐶↑2)) ∈ ℂ) |
32 | 15, 1 | mulcld 10877 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐷 · 𝐴) ∈ ℂ) |
33 | 32, 17, 22 | divcld 11632 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐷 · 𝐴) / (𝐶↑2)) ∈ ℂ) |
34 | 31, 33 | negsubd 11219 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐵 · 𝐶) / (𝐶↑2)) + -((𝐷 · 𝐴) / (𝐶↑2))) = (((𝐵 · 𝐶) / (𝐶↑2)) − ((𝐷 · 𝐴) / (𝐶↑2)))) |
35 | 29, 14, 3, 6 | div12d 11668 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 · (1 / 𝐶)) = (1 · (𝐵 / 𝐶))) |
36 | 29, 3, 6 | divcld 11632 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 / 𝐶) ∈ ℂ) |
37 | 36 | mulid2d 10875 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 · (𝐵 / 𝐶)) = (𝐵 / 𝐶)) |
38 | 3 | sqvald 13737 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶↑2) = (𝐶 · 𝐶)) |
39 | 38 | oveq2d 7247 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐵 · 𝐶) / (𝐶↑2)) = ((𝐵 · 𝐶) / (𝐶 · 𝐶))) |
40 | 29, 3, 3, 6, 6 | divcan5rd 11659 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐵 · 𝐶) / (𝐶 · 𝐶)) = (𝐵 / 𝐶)) |
41 | 39, 40 | eqtr2d 2779 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 / 𝐶) = ((𝐵 · 𝐶) / (𝐶↑2))) |
42 | 35, 37, 41 | 3eqtrd 2782 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 · (1 / 𝐶)) = ((𝐵 · 𝐶) / (𝐶↑2))) |
43 | 15 | mulid2d 10875 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (1 · 𝐷) = 𝐷) |
44 | 43 | oveq1d 7246 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((1 · 𝐷) / (𝐶↑2)) = (𝐷 / (𝐶↑2))) |
45 | 44 | negeqd 11096 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → -((1 · 𝐷) / (𝐶↑2)) = -(𝐷 / (𝐶↑2))) |
46 | 45 | oveq1d 7246 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-((1 · 𝐷) / (𝐶↑2)) · 𝐴) = (-(𝐷 / (𝐶↑2)) · 𝐴)) |
47 | 15, 17, 22 | divcld 11632 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐷 / (𝐶↑2)) ∈ ℂ) |
48 | 47, 1 | mulneg1d 11309 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-(𝐷 / (𝐶↑2)) · 𝐴) = -((𝐷 / (𝐶↑2)) · 𝐴)) |
49 | 15, 1, 17, 22 | div23d 11669 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐷 · 𝐴) / (𝐶↑2)) = ((𝐷 / (𝐶↑2)) · 𝐴)) |
50 | 49 | eqcomd 2744 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐷 / (𝐶↑2)) · 𝐴) = ((𝐷 · 𝐴) / (𝐶↑2))) |
51 | 50 | negeqd 11096 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → -((𝐷 / (𝐶↑2)) · 𝐴) = -((𝐷 · 𝐴) / (𝐶↑2))) |
52 | 46, 48, 51 | 3eqtrd 2782 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (-((1 · 𝐷) / (𝐶↑2)) · 𝐴) = -((𝐷 · 𝐴) / (𝐶↑2))) |
53 | 42, 52 | oveq12d 7249 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐵 · (1 / 𝐶)) + (-((1 · 𝐷) / (𝐶↑2)) · 𝐴)) = (((𝐵 · 𝐶) / (𝐶↑2)) + -((𝐷 · 𝐴) / (𝐶↑2)))) |
54 | 30, 32, 17, 22 | divsubdird 11671 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐵 · 𝐶) − (𝐷 · 𝐴)) / (𝐶↑2)) = (((𝐵 · 𝐶) / (𝐶↑2)) − ((𝐷 · 𝐴) / (𝐶↑2)))) |
55 | 34, 53, 54 | 3eqtr4d 2788 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐵 · (1 / 𝐶)) + (-((1 · 𝐷) / (𝐶↑2)) · 𝐴)) = (((𝐵 · 𝐶) − (𝐷 · 𝐴)) / (𝐶↑2))) |
56 | 55 | mpteq2dva 5164 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐵 · (1 / 𝐶)) + (-((1 · 𝐷) / (𝐶↑2)) · 𝐴))) = (𝑥 ∈ 𝑋 ↦ (((𝐵 · 𝐶) − (𝐷 · 𝐴)) / (𝐶↑2)))) |
57 | 9, 28, 56 | 3eqtrd 2782 | 1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (((𝐵 · 𝐶) − (𝐷 · 𝐴)) / (𝐶↑2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ≠ wne 2941 ∖ cdif 3877 {csn 4555 {cpr 4557 ↦ cmpt 5149 (class class class)co 7231 ℂcc 10751 ℝcr 10752 0cc0 10753 1c1 10754 + caddc 10756 · cmul 10758 − cmin 11086 -cneg 11087 / cdiv 11513 2c2 11909 ↑cexp 13659 D cdv 24784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 ax-pre-sup 10831 ax-addf 10832 ax-mulf 10833 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-int 4874 df-iun 4920 df-iin 4921 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-se 5524 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-isom 6406 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-of 7487 df-om 7663 df-1st 7779 df-2nd 7780 df-supp 7924 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-2o 8223 df-er 8411 df-map 8530 df-pm 8531 df-ixp 8599 df-en 8647 df-dom 8648 df-sdom 8649 df-fin 8650 df-fsupp 9010 df-fi 9051 df-sup 9082 df-inf 9083 df-oi 9150 df-card 9579 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-div 11514 df-nn 11855 df-2 11917 df-3 11918 df-4 11919 df-5 11920 df-6 11921 df-7 11922 df-8 11923 df-9 11924 df-n0 12115 df-z 12201 df-dec 12318 df-uz 12463 df-q 12569 df-rp 12611 df-xneg 12728 df-xadd 12729 df-xmul 12730 df-icc 12966 df-fz 13120 df-fzo 13263 df-seq 13599 df-exp 13660 df-hash 13921 df-cj 14686 df-re 14687 df-im 14688 df-sqrt 14822 df-abs 14823 df-struct 16724 df-sets 16741 df-slot 16759 df-ndx 16769 df-base 16785 df-ress 16809 df-plusg 16839 df-mulr 16840 df-starv 16841 df-sca 16842 df-vsca 16843 df-ip 16844 df-tset 16845 df-ple 16846 df-ds 16848 df-unif 16849 df-hom 16850 df-cco 16851 df-rest 16951 df-topn 16952 df-0g 16970 df-gsum 16971 df-topgen 16972 df-pt 16973 df-prds 16976 df-xrs 17031 df-qtop 17036 df-imas 17037 df-xps 17039 df-mre 17113 df-mrc 17114 df-acs 17116 df-mgm 18138 df-sgrp 18187 df-mnd 18198 df-submnd 18243 df-mulg 18513 df-cntz 18735 df-cmn 19196 df-psmet 20379 df-xmet 20380 df-met 20381 df-bl 20382 df-mopn 20383 df-fbas 20384 df-fg 20385 df-cnfld 20388 df-top 21815 df-topon 21832 df-topsp 21854 df-bases 21867 df-cld 21940 df-ntr 21941 df-cls 21942 df-nei 22019 df-lp 22057 df-perf 22058 df-cn 22148 df-cnp 22149 df-t1 22235 df-haus 22236 df-tx 22483 df-hmeo 22676 df-fil 22767 df-fm 22859 df-flim 22860 df-flf 22861 df-xms 23242 df-ms 23243 df-tms 23244 df-cncf 23799 df-limc 24787 df-dv 24788 |
This theorem is referenced by: dvdivf 43166 dvdivbd 43167 fourierdlem56 43406 fourierdlem57 43407 |
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