dvexp2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvexp2		Structured version Visualization version GIF version

Theorem dvexp2 23916

Description: Derivative of an exponential, possibly zero power. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

**Assertion**
Ref	Expression
dvexp2	⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))

Distinct variable group: 𝑥,𝑁

**Proof of Theorem dvexp2**
Step	Hyp	Ref	Expression
1		elnn0 11486	. 2 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		dvexp 23915	. . . 4 ⊢ (𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
3		nnne0 11245	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
4	3	neneqd 2937	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ¬ 𝑁 = 0)
5	4	iffalsed 4241	. . . . 5 ⊢ (𝑁 ∈ ℕ → if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1))))
6	5	mpteq2dv 4897	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
7	2, 6	eqtr4d 2797	. . 3 ⊢ (𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
8		oveq2 6821	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝑥↑𝑁) = (𝑥↑0))
9		exp0 13058	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
10	8, 9	sylan9eq 2814	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑁) = 1)
11	10	mpteq2dva 4896	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ 1))
12		fconstmpt 5320	. . . . . . . 8 ⊢ (ℂ × {1}) = (𝑥 ∈ ℂ ↦ 1)
13	11, 12	syl6eqr 2812	. . . . . . 7 ⊢ (𝑁 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (ℂ × {1}))
14	13	oveq2d 6829	. . . . . 6 ⊢ (𝑁 = 0 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (ℂ D (ℂ × {1})))
15		ax-1cn 10186	. . . . . . 7 ⊢ 1 ∈ ℂ
16		dvconst 23879	. . . . . . 7 ⊢ (1 ∈ ℂ → (ℂ D (ℂ × {1})) = (ℂ × {0}))
17	15, 16	ax-mp 5	. . . . . 6 ⊢ (ℂ D (ℂ × {1})) = (ℂ × {0})
18	14, 17	syl6eq 2810	. . . . 5 ⊢ (𝑁 = 0 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (ℂ × {0}))
19		fconstmpt 5320	. . . . 5 ⊢ (ℂ × {0}) = (𝑥 ∈ ℂ ↦ 0)
20	18, 19	syl6eq 2810	. . . 4 ⊢ (𝑁 = 0 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ 0))
21		iftrue 4236	. . . . 5 ⊢ (𝑁 = 0 → if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))) = 0)
22	21	mpteq2dv 4897	. . . 4 ⊢ (𝑁 = 0 → (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))) = (𝑥 ∈ ℂ ↦ 0))
23	20, 22	eqtr4d 2797	. . 3 ⊢ (𝑁 = 0 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
24	7, 23	jaoi 393	. 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
25	1, 24	sylbi 207	1 ⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 382 = wceq 1632 ∈ wcel 2139 ifcif 4230 {csn 4321 ↦ cmpt 4881 × cxp 5264 (class class class)co 6813 ℂcc 10126 0cc0 10128 1c1 10129 · cmul 10133 − cmin 10458 ℕcn 11212 ℕ₀cn0 11484 ↑cexp 13054 D cdv 23826

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 ax-addf 10207 ax-mulf 10208

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-er 7911 df-map 8025 df-pm 8026 df-ixp 8075 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-fi 8482 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-cda 9182 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-q 11982 df-rp 12026 df-xneg 12139 df-xadd 12140 df-xmul 12141 df-icc 12375 df-fz 12520 df-fzo 12660 df-seq 12996 df-exp 13055 df-hash 13312 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-starv 16158 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-unif 16167 df-hom 16168 df-cco 16169 df-rest 16285 df-topn 16286 df-0g 16304 df-gsum 16305 df-topgen 16306 df-pt 16307 df-prds 16310 df-xrs 16364 df-qtop 16369 df-imas 16370 df-xps 16372 df-mre 16448 df-mrc 16449 df-acs 16451 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-mulg 17742 df-cntz 17950 df-cmn 18395 df-psmet 19940 df-xmet 19941 df-met 19942 df-bl 19943 df-mopn 19944 df-fbas 19945 df-fg 19946 df-cnfld 19949 df-top 20901 df-topon 20918 df-topsp 20939 df-bases 20952 df-cld 21025 df-ntr 21026 df-cls 21027 df-nei 21104 df-lp 21142 df-perf 21143 df-cn 21233 df-cnp 21234 df-haus 21321 df-tx 21567 df-hmeo 21760 df-fil 21851 df-fm 21943 df-flim 21944 df-flf 21945 df-xms 22326 df-ms 22327 df-tms 22328 df-cncf 22882 df-limc 23829 df-dv 23830

This theorem is referenced by: dvexp3 23940 dvply1 24238 dvtaylp 24323 pserdvlem2 24381

W3C validator