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Mirrors > Home > MPE Home > Th. List > dvconst | Structured version Visualization version GIF version |
Description: Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
dvconst | ⊢ (𝐴 ∈ ℂ → (ℂ D (ℂ × {𝐴})) = (ℂ × {0})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconst6g 6777 | . 2 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}):ℂ⟶ℂ) | |
2 | simpr2 1196 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → 𝑧 ∈ ℂ) | |
3 | fvconst2g 7198 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((ℂ × {𝐴})‘𝑧) = 𝐴) | |
4 | 2, 3 | syldan 592 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → ((ℂ × {𝐴})‘𝑧) = 𝐴) |
5 | fvconst2g 7198 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((ℂ × {𝐴})‘𝑥) = 𝐴) | |
6 | 5 | 3ad2antr1 1189 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → ((ℂ × {𝐴})‘𝑥) = 𝐴) |
7 | 4, 6 | oveq12d 7422 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → (((ℂ × {𝐴})‘𝑧) − ((ℂ × {𝐴})‘𝑥)) = (𝐴 − 𝐴)) |
8 | subid 11475 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 − 𝐴) = 0) | |
9 | 8 | adantr 482 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → (𝐴 − 𝐴) = 0) |
10 | 7, 9 | eqtrd 2773 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → (((ℂ × {𝐴})‘𝑧) − ((ℂ × {𝐴})‘𝑥)) = 0) |
11 | 10 | oveq1d 7419 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → ((((ℂ × {𝐴})‘𝑧) − ((ℂ × {𝐴})‘𝑥)) / (𝑧 − 𝑥)) = (0 / (𝑧 − 𝑥))) |
12 | simpr1 1195 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → 𝑥 ∈ ℂ) | |
13 | 2, 12 | subcld 11567 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → (𝑧 − 𝑥) ∈ ℂ) |
14 | simpr3 1197 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → 𝑧 ≠ 𝑥) | |
15 | 2, 12, 14 | subne0d 11576 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → (𝑧 − 𝑥) ≠ 0) |
16 | 13, 15 | div0d 11985 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → (0 / (𝑧 − 𝑥)) = 0) |
17 | 11, 16 | eqtrd 2773 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → ((((ℂ × {𝐴})‘𝑧) − ((ℂ × {𝐴})‘𝑥)) / (𝑧 − 𝑥)) = 0) |
18 | 0cn 11202 | . 2 ⊢ 0 ∈ ℂ | |
19 | 1, 17, 18 | dvidlem 25414 | 1 ⊢ (𝐴 ∈ ℂ → (ℂ D (ℂ × {𝐴})) = (ℂ × {0})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 {csn 4627 × cxp 5673 ‘cfv 6540 (class class class)co 7404 ℂcc 11104 0cc0 11106 − cmin 11440 / cdiv 11867 D cdv 25362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-icc 13327 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-rest 17364 df-topn 17365 df-topgen 17385 df-psmet 20921 df-xmet 20922 df-met 20923 df-bl 20924 df-mopn 20925 df-fbas 20926 df-fg 20927 df-cnfld 20930 df-top 22378 df-topon 22395 df-topsp 22417 df-bases 22431 df-cld 22505 df-ntr 22506 df-cls 22507 df-nei 22584 df-lp 22622 df-perf 22623 df-cn 22713 df-cnp 22714 df-haus 22801 df-fil 23332 df-fm 23424 df-flim 23425 df-flf 23426 df-xms 23808 df-ms 23809 df-cncf 24376 df-limc 25365 df-dv 25366 |
This theorem is referenced by: dvcmul 25443 dvcmulf 25444 dvexp2 25453 dvmptc 25457 dvef 25479 dvsconst 43022 binomcxplemnotnn0 43048 |
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