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Theorem rpcxpef 15533

Description: Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)

**Assertion**
Ref	Expression
rpcxpef	⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))

Proof of Theorem rpcxpef Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
2		simpl 109	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℝ⁺)
3	2	relogcld 15521	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (log‘𝐴) ∈ ℝ)
4	3	recnd 8143	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (log‘𝐴) ∈ ℂ)
5	1, 4	mulcld 8135	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐵 · (log‘𝐴)) ∈ ℂ)
6		efcl 12141	. . 3 ⊢ ((𝐵 · (log‘𝐴)) ∈ ℂ → (exp‘(𝐵 · (log‘𝐴))) ∈ ℂ)
7	5, 6	syl 14	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (exp‘(𝐵 · (log‘𝐴))) ∈ ℂ)
8		fveq2 5603	. . . . 5 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
9	8	oveq2d 5990	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 · (log‘𝑥)) = (𝑦 · (log‘𝐴)))
10	9	fveq2d 5607	. . 3 ⊢ (𝑥 = 𝐴 → (exp‘(𝑦 · (log‘𝑥))) = (exp‘(𝑦 · (log‘𝐴))))
11		fvoveq1 5997	. . 3 ⊢ (𝑦 = 𝐵 → (exp‘(𝑦 · (log‘𝐴))) = (exp‘(𝐵 · (log‘𝐴))))
12		df-rpcxp 15498	. . 3 ⊢ ↑_𝑐 = (𝑥 ∈ ℝ⁺, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥))))
13	10, 11, 12	ovmpog 6110	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ (exp‘(𝐵 · (log‘𝐴))) ∈ ℂ) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
14	7, 13	mpd3an3 1353	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1375 ∈ wcel 2180 ‘cfv 5294 (class class class)co 5974 ℂcc 7965 · cmul 7972 ℝ⁺crp 9817 expce 12119 logclog 15495 ↑_𝑐ccxp 15496

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-nul 4189 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-iinf 4657 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-mulrcl 8066 ax-addcom 8067 ax-mulcom 8068 ax-addass 8069 ax-mulass 8070 ax-distr 8071 ax-i2m1 8072 ax-0lt1 8073 ax-1rid 8074 ax-0id 8075 ax-rnegex 8076 ax-precex 8077 ax-cnre 8078 ax-pre-ltirr 8079 ax-pre-ltwlin 8080 ax-pre-lttrn 8081 ax-pre-apti 8082 ax-pre-ltadd 8083 ax-pre-mulgt0 8084 ax-pre-mulext 8085 ax-arch 8086 ax-caucvg 8087 ax-pre-suploc 8088 ax-addf 8089 ax-mulf 8090

This theorem depends on definitions: df-bi 117 df-stab 835 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-reu 2495 df-rmo 2496 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-if 3583 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-disj 4039 df-br 4063 df-opab 4125 df-mpt 4126 df-tr 4162 df-id 4361 df-po 4364 df-iso 4365 df-iord 4434 df-on 4436 df-ilim 4437 df-suc 4439 df-iom 4660 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-isom 5303 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-of 6188 df-1st 6256 df-2nd 6257 df-recs 6421 df-irdg 6486 df-frec 6507 df-1o 6532 df-oadd 6536 df-er 6650 df-map 6767 df-pm 6768 df-en 6858 df-dom 6859 df-fin 6860 df-sup 7119 df-inf 7120 df-pnf 8151 df-mnf 8152 df-xr 8153 df-ltxr 8154 df-le 8155 df-sub 8287 df-neg 8288 df-reap 8690 df-ap 8697 df-div 8788 df-inn 9079 df-2 9137 df-3 9138 df-4 9139 df-n0 9338 df-z 9415 df-uz 9691 df-q 9783 df-rp 9818 df-xneg 9936 df-xadd 9937 df-ioo 10056 df-ico 10058 df-icc 10059 df-fz 10173 df-fzo 10307 df-seqfrec 10637 df-exp 10728 df-fac 10915 df-bc 10937 df-ihash 10965 df-shft 11292 df-cj 11319 df-re 11320 df-im 11321 df-rsqrt 11475 df-abs 11476 df-clim 11756 df-sumdc 11831 df-ef 12125 df-e 12126 df-rest 13240 df-topgen 13259 df-psmet 14472 df-xmet 14473 df-met 14474 df-bl 14475 df-mopn 14476 df-top 14637 df-topon 14650 df-bases 14682 df-ntr 14735 df-cn 14827 df-cnp 14828 df-tx 14892 df-cncf 15210 df-limced 15295 df-dvap 15296 df-relog 15497 df-rpcxp 15498

This theorem is referenced by: cxpexprp 15534 logcxp 15536 1cxp 15539 ecxp 15540 rpcncxpcl 15541 rpcxpcl 15542 cxpap0 15543 rpcxpadd 15544 rpmulcxp 15548 cxpmul 15551 abscxp 15554 cxplt 15555 rpcxple2 15557 rpcxplt2 15558 apcxp2 15578 rpabscxpbnd 15579 rpcxplogb 15603

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