dvcxp2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvcxp2		Structured version Visualization version GIF version

Theorem dvcxp2 26781

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 11161	. . . 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 26615	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
5	4	adantr 484	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (log‘𝐴) ∈ ℝ)
6	5	recnd 11205	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (log‘𝐴) ∈ ℂ)
7	3, 6	mulcld 11197	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝑥 · (log‘𝐴)) ∈ ℂ)
8		efcl 16093	. . . 4 ⊢ (𝑦 ∈ ℂ → (exp‘𝑦) ∈ ℂ)
9	8	adantl 485	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑦 ∈ ℂ) → (exp‘𝑦) ∈ ℂ)
10	3, 6	mulcomd 11198	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝑥 · (log‘𝐴)) = ((log‘𝐴) · 𝑥))
11	10	mpteq2dva 5192	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ (𝑥 · (log‘𝐴))) = (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 𝑥)))
12	11	oveq2d 7406	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥 · (log‘𝐴)))) = (ℂ D (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 𝑥))))
13		1cnd 11170	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
14	2	dvmptid 25997	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1))
15	4	recnd 11205	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
16	2, 3, 13, 14, 15	dvmptcmul 26004	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 𝑥))) = (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 1)))
17	6	mulridd 11194	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((log‘𝐴) · 1) = (log‘𝐴))
18	17	mpteq2dva 5192	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ ((log‘𝐴) · 1)) = (𝑥 ∈ ℂ ↦ (log‘𝐴)))
19	12, 16, 18	3eqtrd 2800	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥 · (log‘𝐴)))) = (𝑥 ∈ ℂ ↦ (log‘𝐴)))
20		dvef 26020	. . . 4 ⊢ (ℂ D exp) = exp
21		eff 16092	. . . . . . . 8 ⊢ exp:ℂ⟶ℂ
22	21	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → exp:ℂ⟶ℂ)
23	22	feqmptd 6929	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
24	23	eqcomd 2767	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (𝑦 ∈ ℂ ↦ (exp‘𝑦)) = exp)
25	24	oveq2d 7406	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑦 ∈ ℂ ↦ (exp‘𝑦))) = (ℂ D exp))
26	20, 25, 24	3eqtr4a 2822	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑦 ∈ ℂ ↦ (exp‘𝑦))) = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
27		fveq2 6861	. . 3 ⊢ (𝑦 = (𝑥 · (log‘𝐴)) → (exp‘𝑦) = (exp‘(𝑥 · (log‘𝐴))))
28	2, 2, 7, 5, 9, 9, 19, 26, 27, 27	dvmptco 26012	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (exp‘(𝑥 · (log‘𝐴))))) = (𝑥 ∈ ℂ ↦ ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴))))
29		rpcn 12999	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
30	29	adantr 484	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)
31		rpne0 13005	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
32	31	adantr 484	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → 𝐴 ≠ 0)
33	30, 32, 3	cxpefd 26752	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝐴↑_𝑐𝑥) = (exp‘(𝑥 · (log‘𝐴))))
34	33	mpteq2dva 5192	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ (𝐴↑_𝑐𝑥)) = (𝑥 ∈ ℂ ↦ (exp‘(𝑥 · (log‘𝐴)))))
35	34	oveq2d 7406	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴↑_𝑐𝑥))) = (ℂ D (𝑥 ∈ ℂ ↦ (exp‘(𝑥 · (log‘𝐴))))))
36	30, 3	cxpcld 26748	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → (𝐴↑_𝑐𝑥) ∈ ℂ)
37	6, 36	mulcomd 11198	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((log‘𝐴) · (𝐴↑_𝑐𝑥)) = ((𝐴↑_𝑐𝑥) · (log‘𝐴)))
38	33	oveq1d 7405	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((𝐴↑_𝑐𝑥) · (log‘𝐴)) = ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴)))
39	37, 38	eqtrd 2796	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℂ) → ((log‘𝐴) · (𝐴↑_𝑐𝑥)) = ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴)))
40	39	mpteq2dva 5192	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (𝑥 ∈ ℂ ↦ ((log‘𝐴) · (𝐴↑_𝑐𝑥))) = (𝑥 ∈ ℂ ↦ ((exp‘(𝑥 · (log‘𝐴))) · (log‘𝐴))))
41	28, 35, 40	3eqtr4d 2806	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 1559 ∈ wcel 2141 ≠ wne 2956 {cpr 4583 ↦ cmpt 5180 ⟶wf 6511 ‘cfv 6515 (class class class)co 7390 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 · cmul 11073 ℝ⁺crp 12988 expce 16072 D cdv 25903 logclog 26594 ↑_𝑐ccxp 26595

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7712 ax-inf2 9591 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7654 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-er 8671 df-map 8803 df-pm 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9303 df-fi 9352 df-sup 9383 df-inf 9384 df-oi 9453 df-card 9892 df-pnf 11213 df-mnf 11214 df-xr 11215 df-ltxr 11216 df-le 11217 df-sub 11411 df-neg 11412 df-div 11840 df-nn 12206 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12477 df-z 12564 df-dec 12684 df-uz 12835 df-q 12945 df-rp 12989 df-xneg 13109 df-xadd 13110 df-xmul 13111 df-ioo 13348 df-ioc 13349 df-ico 13350 df-icc 13351 df-fz 13508 df-fzo 13655 df-fl 13797 df-mod 13875 df-seq 14010 df-exp 14070 df-fac 14282 df-bc 14311 df-hash 14339 df-shft 15075 df-cj 15107 df-re 15108 df-im 15109 df-sqrt 15243 df-abs 15244 df-limsup 15479 df-clim 15496 df-rlim 15497 df-sum 15695 df-ef 16078 df-sin 16080 df-cos 16081 df-pi 16083 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17248 df-plusg 17280 df-mulr 17281 df-starv 17282 df-sca 17283 df-vsca 17284 df-ip 17285 df-tset 17286 df-ple 17287 df-ds 17289 df-unif 17290 df-hom 17291 df-cco 17292 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-xrs 17513 df-qtop 17518 df-imas 17519 df-xps 17521 df-mre 17595 df-mrc 17596 df-acs 17598 df-mgm 18655 df-sgrp 18734 df-mnd 18750 df-submnd 18799 df-mulg 19091 df-cntz 19338 df-cmn 19803 df-psmet 21394 df-xmet 21395 df-met 21396 df-bl 21397 df-mopn 21398 df-fbas 21399 df-fg 21400 df-cnfld 21403 df-top 22932 df-topon 22949 df-topsp 22971 df-bases 22984 df-cld 23057 df-ntr 23058 df-cls 23059 df-nei 23136 df-lp 23174 df-perf 23175 df-cn 23265 df-cnp 23266 df-haus 23353 df-tx 23600 df-hmeo 23793 df-fil 23884 df-fm 23976 df-flim 23977 df-flf 23978 df-xms 24358 df-ms 24359 df-tms 24360 df-cncf 24918 df-limc 25906 df-dv 25907 df-log 26596 df-cxp 26597

This theorem is referenced by: etransclem46 46807

W3C validator