dvcxp2 - Metamath Proof Explorer

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Theorem dvcxp2 25429

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 10668	. . . 4 ⊢ ℂ ∈ {ℝ, ℂ}
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ℂ ∈ {ℝ, ℂ})
3		simpr 488	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
4		relogcl 25266	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
5	4	adantr 484	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (log‘𝐴) ∈ ℝ)
6	5	recnd 10707	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (log‘𝐴) ∈ ℂ)
7	3, 6	mulcld 10699	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝑥 · (log‘𝐴)) ∈ ℂ)
8		efcl 15484	. . . 4 ⊢ (𝑦 ∈ ℂ → (exp‘𝑦) ∈ ℂ)
9	8	adantl 485	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑦 ∈ ℂ) → (exp‘𝑦) ∈ ℂ)
10	3, 6	mulcomd 10700	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝑥 · (log‘𝐴)) = ((log‘𝐴) · 𝑥))
11	10	mpteq2dva 5127	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ (𝑥 · (log‘𝐴))) = (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 𝑥)))
12	11	oveq2d 7166	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥 · (log‘𝐴)))) = (ℂ D (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 𝑥))))
13		1cnd 10674	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
14	2	dvmptid 24656	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1))
15	4	recnd 10707	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
16	2, 3, 13, 14, 15	dvmptcmul 24663	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 𝑥))) = (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 1)))
17	6	mulid1d 10696	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((log‘𝐴) · 1) = (log‘𝐴))
18	17	mpteq2dva 5127	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 1)) = (𝑥 ∈ ℂ ↦ (log‘𝐴)))
19	12, 16, 18	3eqtrd 2797	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥 · (log‘𝐴)))) = (𝑥 ∈ ℂ ↦ (log‘𝐴)))
20		dvef 24679	. . . 4 ⊢ (ℂ D exp) = exp
21		eff 15483	. . . . . . . 8 ⊢ exp:ℂ⟶ℂ
22	21	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → exp:ℂ⟶ℂ)
23	22	feqmptd 6721	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
24	23	eqcomd 2764	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (𝑦 ∈ ℂ ↦ (exp‘𝑦)) = exp)
25	24	oveq2d 7166	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑦 ∈ ℂ ↦ (exp‘𝑦))) = (ℂ D exp))
26	20, 25, 24	3eqtr4a 2819	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑦 ∈ ℂ ↦ (exp‘𝑦))) = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
27		fveq2 6658	. . 3 ⊢ (𝑦 = (𝑥 · (log‘𝐴)) → (exp‘𝑦) = (exp‘(𝑥 · (log‘𝐴))))
28	2, 2, 7, 5, 9, 9, 19, 26, 27, 27	dvmptco 24671	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (exp‘(𝑥 · (log‘𝐴))))) = (𝑥 ∈ ℂ ↦ ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴))))
29		rpcn 12440	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
30	29	adantr 484	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)
31		rpne0 12446	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
32	31	adantr 484	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → 𝐴 ≠ 0)
33	30, 32, 3	cxpefd 25402	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝐴↑_𝑐𝑥) = (exp‘(𝑥 · (log‘𝐴))))
34	33	mpteq2dva 5127	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ (𝐴↑_𝑐𝑥)) = (𝑥 ∈ ℂ ↦ (exp‘(𝑥 · (log‘𝐴)))))
35	34	oveq2d 7166	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴↑_𝑐𝑥))) = (ℂ D (𝑥 ∈ ℂ ↦ (exp‘(𝑥 · (log‘𝐴))))))
36	30, 3	cxpcld 25398	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝐴↑_𝑐𝑥) ∈ ℂ)
37	6, 36	mulcomd 10700	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((log‘𝐴) · (𝐴↑_𝑐𝑥)) = ((𝐴↑_𝑐𝑥) · (log‘𝐴)))
38	33	oveq1d 7165	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((𝐴↑_𝑐𝑥) · (log‘𝐴)) = ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴)))
39	37, 38	eqtrd 2793	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((log‘𝐴) · (𝐴↑_𝑐𝑥)) = ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴)))
40	39	mpteq2dva 5127	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ ((log‘𝐴) · (𝐴↑_𝑐𝑥))) = (𝑥 ∈ ℂ ↦ ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴))))
41	28, 35, 40	3eqtr4d 2803	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 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 {cpr 4524 ↦ cmpt 5112 ⟶wf 6331 ‘cfv 6335 (class class class)co 7150 ℂcc 10573 ℝcr 10574 0cc0 10575 1c1 10576 · cmul 10580 ℝ⁺crp 12430 expce 15463 D cdv 24562 logclog 25245 ↑_𝑐ccxp 25246

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5156 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-inf2 9137 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652 ax-pre-sup 10653 ax-addf 10654 ax-mulf 10655

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-int 4839 df-iun 4885 df-iin 4886 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-se 5484 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-isom 6344 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7405 df-om 7580 df-1st 7693 df-2nd 7694 df-supp 7836 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-1o 8112 df-2o 8113 df-er 8299 df-map 8418 df-pm 8419 df-ixp 8480 df-en 8528 df-dom 8529 df-sdom 8530 df-fin 8531 df-fsupp 8867 df-fi 8908 df-sup 8939 df-inf 8940 df-oi 9007 df-card 9401 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911 df-div 11336 df-nn 11675 df-2 11737 df-3 11738 df-4 11739 df-5 11740 df-6 11741 df-7 11742 df-8 11743 df-9 11744 df-n0 11935 df-z 12021 df-dec 12138 df-uz 12283 df-q 12389 df-rp 12431 df-xneg 12548 df-xadd 12549 df-xmul 12550 df-ioo 12783 df-ioc 12784 df-ico 12785 df-icc 12786 df-fz 12940 df-fzo 13083 df-fl 13211 df-mod 13287 df-seq 13419 df-exp 13480 df-fac 13684 df-bc 13713 df-hash 13741 df-shft 14474 df-cj 14506 df-re 14507 df-im 14508 df-sqrt 14642 df-abs 14643 df-limsup 14876 df-clim 14893 df-rlim 14894 df-sum 15091 df-ef 15469 df-sin 15471 df-cos 15472 df-pi 15474 df-struct 16543 df-ndx 16544 df-slot 16545 df-base 16547 df-sets 16548 df-ress 16549 df-plusg 16636 df-mulr 16637 df-starv 16638 df-sca 16639 df-vsca 16640 df-ip 16641 df-tset 16642 df-ple 16643 df-ds 16645 df-unif 16646 df-hom 16647 df-cco 16648 df-rest 16754 df-topn 16755 df-0g 16773 df-gsum 16774 df-topgen 16775 df-pt 16776 df-prds 16779 df-xrs 16833 df-qtop 16838 df-imas 16839 df-xps 16841 df-mre 16915 df-mrc 16916 df-acs 16918 df-mgm 17918 df-sgrp 17967 df-mnd 17978 df-submnd 18023 df-mulg 18292 df-cntz 18514 df-cmn 18975 df-psmet 20158 df-xmet 20159 df-met 20160 df-bl 20161 df-mopn 20162 df-fbas 20163 df-fg 20164 df-cnfld 20167 df-top 21594 df-topon 21611 df-topsp 21633 df-bases 21646 df-cld 21719 df-ntr 21720 df-cls 21721 df-nei 21798 df-lp 21836 df-perf 21837 df-cn 21927 df-cnp 21928 df-haus 22015 df-tx 22262 df-hmeo 22455 df-fil 22546 df-fm 22638 df-flim 22639 df-flf 22640 df-xms 23022 df-ms 23023 df-tms 23024 df-cncf 23579 df-limc 24565 df-dv 24566 df-log 25247 df-cxp 25248

This theorem is referenced by: etransclem46 43310

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