dvcxp2 - Metamath Proof Explorer

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Theorem dvcxp2 26672

Description: The derivative of a complex power with respect to the second argument. (Contributed by Mario Carneiro, 24-Feb-2015.)

**Assertion**
Ref	Expression
dvcxp2	⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴↑_𝑐𝑥))) = (𝑥 ∈ ℂ ↦ ((log‘𝐴) · (𝐴↑_𝑐𝑥))))

Distinct variable group: 𝑥,𝐴

Proof of Theorem dvcxp2 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnelprrecn 11094	. . . 4 ⊢ ℂ ∈ {ℝ, ℂ}
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ℂ ∈ {ℝ, ℂ})
3		simpr 484	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
4		relogcl 26506	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
5	4	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (log‘𝐴) ∈ ℝ)
6	5	recnd 11135	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (log‘𝐴) ∈ ℂ)
7	3, 6	mulcld 11127	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝑥 · (log‘𝐴)) ∈ ℂ)
8		efcl 15984	. . . 4 ⊢ (𝑦 ∈ ℂ → (exp‘𝑦) ∈ ℂ)
9	8	adantl 481	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑦 ∈ ℂ) → (exp‘𝑦) ∈ ℂ)
10	3, 6	mulcomd 11128	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝑥 · (log‘𝐴)) = ((log‘𝐴) · 𝑥))
11	10	mpteq2dva 5179	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ (𝑥 · (log‘𝐴))) = (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 𝑥)))
12	11	oveq2d 7357	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥 · (log‘𝐴)))) = (ℂ D (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 𝑥))))
13		1cnd 11102	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
14	2	dvmptid 25883	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1))
15	4	recnd 11135	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
16	2, 3, 13, 14, 15	dvmptcmul 25890	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 𝑥))) = (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 1)))
17	6	mulridd 11124	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((log‘𝐴) · 1) = (log‘𝐴))
18	17	mpteq2dva 5179	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 1)) = (𝑥 ∈ ℂ ↦ (log‘𝐴)))
19	12, 16, 18	3eqtrd 2770	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥 · (log‘𝐴)))) = (𝑥 ∈ ℂ ↦ (log‘𝐴)))
20		dvef 25906	. . . 4 ⊢ (ℂ D exp) = exp
21		eff 15983	. . . . . . . 8 ⊢ exp:ℂ⟶ℂ
22	21	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → exp:ℂ⟶ℂ)
23	22	feqmptd 6885	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
24	23	eqcomd 2737	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (𝑦 ∈ ℂ ↦ (exp‘𝑦)) = exp)
25	24	oveq2d 7357	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑦 ∈ ℂ ↦ (exp‘𝑦))) = (ℂ D exp))
26	20, 25, 24	3eqtr4a 2792	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑦 ∈ ℂ ↦ (exp‘𝑦))) = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
27		fveq2 6817	. . 3 ⊢ (𝑦 = (𝑥 · (log‘𝐴)) → (exp‘𝑦) = (exp‘(𝑥 · (log‘𝐴))))
28	2, 2, 7, 5, 9, 9, 19, 26, 27, 27	dvmptco 25898	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (exp‘(𝑥 · (log‘𝐴))))) = (𝑥 ∈ ℂ ↦ ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴))))
29		rpcn 12896	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
30	29	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)
31		rpne0 12902	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
32	31	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → 𝐴 ≠ 0)
33	30, 32, 3	cxpefd 26643	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝐴↑_𝑐𝑥) = (exp‘(𝑥 · (log‘𝐴))))
34	33	mpteq2dva 5179	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ (𝐴↑_𝑐𝑥)) = (𝑥 ∈ ℂ ↦ (exp‘(𝑥 · (log‘𝐴)))))
35	34	oveq2d 7357	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴↑_𝑐𝑥))) = (ℂ D (𝑥 ∈ ℂ ↦ (exp‘(𝑥 · (log‘𝐴))))))
36	30, 3	cxpcld 26639	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝐴↑_𝑐𝑥) ∈ ℂ)
37	6, 36	mulcomd 11128	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((log‘𝐴) · (𝐴↑_𝑐𝑥)) = ((𝐴↑_𝑐𝑥) · (log‘𝐴)))
38	33	oveq1d 7356	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((𝐴↑_𝑐𝑥) · (log‘𝐴)) = ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴)))
39	37, 38	eqtrd 2766	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((log‘𝐴) · (𝐴↑_𝑐𝑥)) = ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴)))
40	39	mpteq2dva 5179	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ ((log‘𝐴) · (𝐴↑_𝑐𝑥))) = (𝑥 ∈ ℂ ↦ ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴))))
41	28, 35, 40	3eqtr4d 2776	1 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴↑_𝑐𝑥))) = (𝑥 ∈ ℂ ↦ ((log‘𝐴) · (𝐴↑_𝑐𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 {cpr 4573 ↦ cmpt 5167 ⟶wf 6472 ‘cfv 6476 (class class class)co 7341 ℂcc 10999 ℝcr 11000 0cc0 11001 1c1 11002 · cmul 11006 ℝ⁺crp 12885 expce 15963 D cdv 25786 logclog 26485 ↑_𝑐ccxp 26486

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079 ax-addf 11080

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9827 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-q 12842 df-rp 12886 df-xneg 13006 df-xadd 13007 df-xmul 13008 df-ioo 13244 df-ioc 13245 df-ico 13246 df-icc 13247 df-fz 13403 df-fzo 13550 df-fl 13691 df-mod 13769 df-seq 13904 df-exp 13964 df-fac 14176 df-bc 14205 df-hash 14233 df-shft 14969 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-limsup 15373 df-clim 15390 df-rlim 15391 df-sum 15589 df-ef 15969 df-sin 15971 df-cos 15972 df-pi 15974 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-starv 17171 df-sca 17172 df-vsca 17173 df-ip 17174 df-tset 17175 df-ple 17176 df-ds 17178 df-unif 17179 df-hom 17180 df-cco 17181 df-rest 17321 df-topn 17322 df-0g 17340 df-gsum 17341 df-topgen 17342 df-pt 17343 df-prds 17346 df-xrs 17401 df-qtop 17406 df-imas 17407 df-xps 17409 df-mre 17483 df-mrc 17484 df-acs 17486 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19224 df-cmn 19689 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22804 df-topon 22821 df-topsp 22843 df-bases 22856 df-cld 22929 df-ntr 22930 df-cls 22931 df-nei 23008 df-lp 23046 df-perf 23047 df-cn 23137 df-cnp 23138 df-haus 23225 df-tx 23472 df-hmeo 23665 df-fil 23756 df-fm 23848 df-flim 23849 df-flf 23850 df-xms 24230 df-ms 24231 df-tms 24232 df-cncf 24793 df-limc 25789 df-dv 25790 df-log 26487 df-cxp 26488

This theorem is referenced by: etransclem46 46318

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