dvcxp2 - Metamath Proof Explorer

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Theorem dvcxp2 26667

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 11121	. . . 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 26501	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
5	4	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (log‘𝐴) ∈ ℝ)
6	5	recnd 11162	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (log‘𝐴) ∈ ℂ)
7	3, 6	mulcld 11154	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝑥 · (log‘𝐴)) ∈ ℂ)
8		efcl 16008	. . . 4 ⊢ (𝑦 ∈ ℂ → (exp‘𝑦) ∈ ℂ)
9	8	adantl 481	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑦 ∈ ℂ) → (exp‘𝑦) ∈ ℂ)
10	3, 6	mulcomd 11155	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝑥 · (log‘𝐴)) = ((log‘𝐴) · 𝑥))
11	10	mpteq2dva 5188	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ (𝑥 · (log‘𝐴))) = (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 𝑥)))
12	11	oveq2d 7369	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥 · (log‘𝐴)))) = (ℂ D (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 𝑥))))
13		1cnd 11129	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
14	2	dvmptid 25878	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1))
15	4	recnd 11162	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
16	2, 3, 13, 14, 15	dvmptcmul 25885	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 𝑥))) = (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 1)))
17	6	mulridd 11151	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((log‘𝐴) · 1) = (log‘𝐴))
18	17	mpteq2dva 5188	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 1)) = (𝑥 ∈ ℂ ↦ (log‘𝐴)))
19	12, 16, 18	3eqtrd 2768	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥 · (log‘𝐴)))) = (𝑥 ∈ ℂ ↦ (log‘𝐴)))
20		dvef 25901	. . . 4 ⊢ (ℂ D exp) = exp
21		eff 16007	. . . . . . . 8 ⊢ exp:ℂ⟶ℂ
22	21	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → exp:ℂ⟶ℂ)
23	22	feqmptd 6895	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
24	23	eqcomd 2735	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (𝑦 ∈ ℂ ↦ (exp‘𝑦)) = exp)
25	24	oveq2d 7369	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑦 ∈ ℂ ↦ (exp‘𝑦))) = (ℂ D exp))
26	20, 25, 24	3eqtr4a 2790	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑦 ∈ ℂ ↦ (exp‘𝑦))) = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
27		fveq2 6826	. . 3 ⊢ (𝑦 = (𝑥 · (log‘𝐴)) → (exp‘𝑦) = (exp‘(𝑥 · (log‘𝐴))))
28	2, 2, 7, 5, 9, 9, 19, 26, 27, 27	dvmptco 25893	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (exp‘(𝑥 · (log‘𝐴))))) = (𝑥 ∈ ℂ ↦ ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴))))
29		rpcn 12923	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
30	29	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)
31		rpne0 12929	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
32	31	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → 𝐴 ≠ 0)
33	30, 32, 3	cxpefd 26638	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝐴↑_𝑐𝑥) = (exp‘(𝑥 · (log‘𝐴))))
34	33	mpteq2dva 5188	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ (𝐴↑_𝑐𝑥)) = (𝑥 ∈ ℂ ↦ (exp‘(𝑥 · (log‘𝐴)))))
35	34	oveq2d 7369	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴↑_𝑐𝑥))) = (ℂ D (𝑥 ∈ ℂ ↦ (exp‘(𝑥 · (log‘𝐴))))))
36	30, 3	cxpcld 26634	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝐴↑_𝑐𝑥) ∈ ℂ)
37	6, 36	mulcomd 11155	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((log‘𝐴) · (𝐴↑_𝑐𝑥)) = ((𝐴↑_𝑐𝑥) · (log‘𝐴)))
38	33	oveq1d 7368	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((𝐴↑_𝑐𝑥) · (log‘𝐴)) = ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴)))
39	37, 38	eqtrd 2764	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((log‘𝐴) · (𝐴↑_𝑐𝑥)) = ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴)))
40	39	mpteq2dva 5188	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ ((log‘𝐴) · (𝐴↑_𝑐𝑥))) = (𝑥 ∈ ℂ ↦ ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴))))
41	28, 35, 40	3eqtr4d 2774	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 1540 ∈ wcel 2109 ≠ wne 2925 {cpr 4581 ↦ cmpt 5176 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 · cmul 11033 ℝ⁺crp 12912 expce 15987 D cdv 25781 logclog 26480 ↑_𝑐ccxp 26481

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12611 df-uz 12755 df-q 12869 df-rp 12913 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13271 df-ioc 13272 df-ico 13273 df-icc 13274 df-fz 13430 df-fzo 13577 df-fl 13715 df-mod 13793 df-seq 13928 df-exp 13988 df-fac 14200 df-bc 14229 df-hash 14257 df-shft 14993 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-limsup 15397 df-clim 15414 df-rlim 15415 df-sum 15613 df-ef 15993 df-sin 15995 df-cos 15996 df-pi 15998 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-ress 17161 df-plusg 17193 df-mulr 17194 df-starv 17195 df-sca 17196 df-vsca 17197 df-ip 17198 df-tset 17199 df-ple 17200 df-ds 17202 df-unif 17203 df-hom 17204 df-cco 17205 df-rest 17345 df-topn 17346 df-0g 17364 df-gsum 17365 df-topgen 17366 df-pt 17367 df-prds 17370 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18533 df-sgrp 18612 df-mnd 18628 df-submnd 18677 df-mulg 18966 df-cntz 19215 df-cmn 19680 df-psmet 21272 df-xmet 21273 df-met 21274 df-bl 21275 df-mopn 21276 df-fbas 21277 df-fg 21278 df-cnfld 21281 df-top 22798 df-topon 22815 df-topsp 22837 df-bases 22850 df-cld 22923 df-ntr 22924 df-cls 22925 df-nei 23002 df-lp 23040 df-perf 23041 df-cn 23131 df-cnp 23132 df-haus 23219 df-tx 23466 df-hmeo 23659 df-fil 23750 df-fm 23842 df-flim 23843 df-flf 23844 df-xms 24225 df-ms 24226 df-tms 24227 df-cncf 24788 df-limc 25784 df-dv 25785 df-log 26482 df-cxp 26483

This theorem is referenced by: etransclem46 46281

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