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Mirrors > Home > MPE Home > Th. List > dvmptdivc | Structured version Visualization version GIF version |
Description: Function-builder for derivative, division rule for constant divisor. (Contributed by Mario Carneiro, 18-May-2016.) |
Ref | Expression |
---|---|
dvmptadd.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvmptadd.a | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) |
dvmptadd.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) |
dvmptadd.da | ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) |
dvmptcmul.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
dvmptdivc.0 | ⊢ (𝜑 → 𝐶 ≠ 0) |
Ref | Expression |
---|---|
dvmptdivc | ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvmptadd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | dvmptadd.a | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) | |
3 | dvmptadd.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) | |
4 | dvmptadd.da | . . 3 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) | |
5 | dvmptcmul.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
6 | dvmptdivc.0 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 0) | |
7 | 5, 6 | reccld 12028 | . . 3 ⊢ (𝜑 → (1 / 𝐶) ∈ ℂ) |
8 | 1, 2, 3, 4, 7 | dvmptcmul 25984 | . 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((1 / 𝐶) · 𝐴))) = (𝑥 ∈ 𝑋 ↦ ((1 / 𝐶) · 𝐵))) |
9 | 5 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ) |
10 | 6 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ≠ 0) |
11 | 2, 9, 10 | divrec2d 12039 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐶) = ((1 / 𝐶) · 𝐴)) |
12 | 11 | mpteq2dva 5245 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ ((1 / 𝐶) · 𝐴))) |
13 | 12 | oveq2d 7432 | . 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ((1 / 𝐶) · 𝐴)))) |
14 | 1, 2, 3, 4 | dvmptcl 25979 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) |
15 | 14, 9, 10 | divrec2d 12039 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 / 𝐶) = ((1 / 𝐶) · 𝐵)) |
16 | 15 | mpteq2dva 5245 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ ((1 / 𝐶) · 𝐵))) |
17 | 8, 13, 16 | 3eqtr4d 2776 | 1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 {cpr 4625 ↦ cmpt 5228 (class class class)co 7416 ℂcc 11147 ℝcr 11148 0cc0 11149 1c1 11150 · cmul 11154 / cdiv 11912 D cdv 25880 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-pm 8850 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-fi 9447 df-sup 9478 df-inf 9479 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-icc 13379 df-fz 13533 df-fzo 13676 df-seq 14016 df-exp 14076 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-xrs 17512 df-qtop 17517 df-imas 17518 df-xps 17520 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-mulg 19058 df-cntz 19307 df-cmn 19776 df-psmet 21331 df-xmet 21332 df-met 21333 df-bl 21334 df-mopn 21335 df-fbas 21336 df-fg 21337 df-cnfld 21340 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22937 df-cld 23011 df-ntr 23012 df-cls 23013 df-nei 23090 df-lp 23128 df-perf 23129 df-cn 23219 df-cnp 23220 df-haus 23307 df-tx 23554 df-hmeo 23747 df-fil 23838 df-fm 23930 df-flim 23931 df-flf 23932 df-xms 24314 df-ms 24315 df-tms 24316 df-cncf 24886 df-limc 25883 df-dv 25884 |
This theorem is referenced by: dvsincos 26001 dvlip 26014 advlogexp 26679 logdivsum 27559 itgexpif 34465 areacirclem1 37422 dvrelog2b 41778 dvnmptdivc 45595 itgcoscmulx 45626 itgsincmulx 45631 dirkeritg 45759 fourierdlem39 45803 fourierdlem56 45819 |
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