rpcxpef - Intuitionistic Logic Explorer

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Theorem rpcxpef 15214

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 15202	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (log‘𝐴) ∈ ℝ)
4	3	recnd 8072	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (log‘𝐴) ∈ ℂ)
5	1, 4	mulcld 8064	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐵 · (log‘𝐴)) ∈ ℂ)
6		efcl 11846	. . 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 5561	. . . . 5 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
9	8	oveq2d 5941	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 · (log‘𝑥)) = (𝑦 · (log‘𝐴)))
10	9	fveq2d 5565	. . 3 ⊢ (𝑥 = 𝐴 → (exp‘(𝑦 · (log‘𝑥))) = (exp‘(𝑦 · (log‘𝐴))))
11		fvoveq1 5948	. . 3 ⊢ (𝑦 = 𝐵 → (exp‘(𝑦 · (log‘𝐴))) = (exp‘(𝐵 · (log‘𝐴))))
12		df-rpcxp 15179	. . 3 ⊢ ↑_𝑐 = (𝑥 ∈ ℝ⁺, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥))))
13	10, 11, 12	ovmpog 6061	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ (exp‘(𝐵 · (log‘𝐴))) ∈ ℂ) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
14	7, 13	mpd3an3 1349	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2167 ‘cfv 5259 (class class class)co 5925 ℂcc 7894 · cmul 7901 ℝ⁺crp 9745 expce 11824 logclog 15176 ↑_𝑐ccxp 15177

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016 ax-pre-suploc 8017 ax-addf 8018 ax-mulf 8019

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-disj 4012 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-of 6139 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-frec 6458 df-1o 6483 df-oadd 6487 df-er 6601 df-map 6718 df-pm 6719 df-en 6809 df-dom 6810 df-fin 6811 df-sup 7059 df-inf 7060 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-n0 9267 df-z 9344 df-uz 9619 df-q 9711 df-rp 9746 df-xneg 9864 df-xadd 9865 df-ioo 9984 df-ico 9986 df-icc 9987 df-fz 10101 df-fzo 10235 df-seqfrec 10557 df-exp 10648 df-fac 10835 df-bc 10857 df-ihash 10885 df-shft 10997 df-cj 11024 df-re 11025 df-im 11026 df-rsqrt 11180 df-abs 11181 df-clim 11461 df-sumdc 11536 df-ef 11830 df-e 11831 df-rest 12943 df-topgen 12962 df-psmet 14175 df-xmet 14176 df-met 14177 df-bl 14178 df-mopn 14179 df-top 14318 df-topon 14331 df-bases 14363 df-ntr 14416 df-cn 14508 df-cnp 14509 df-tx 14573 df-cncf 14891 df-limced 14976 df-dvap 14977 df-relog 15178 df-rpcxp 15179

This theorem is referenced by: cxpexprp 15215 logcxp 15217 1cxp 15220 ecxp 15221 rpcncxpcl 15222 rpcxpcl 15223 cxpap0 15224 rpcxpadd 15225 rpmulcxp 15229 cxpmul 15232 abscxp 15235 cxplt 15236 rpcxple2 15238 rpcxplt2 15239 apcxp2 15259 rpabscxpbnd 15260 rpcxplogb 15284

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