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Theorem rpcxpef 15410

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 15398	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (log‘𝐴) ∈ ℝ)
4	3	recnd 8108	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (log‘𝐴) ∈ ℂ)
5	1, 4	mulcld 8100	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐵 · (log‘𝐴)) ∈ ℂ)
6		efcl 12019	. . 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 5583	. . . . 5 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
9	8	oveq2d 5967	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 · (log‘𝑥)) = (𝑦 · (log‘𝐴)))
10	9	fveq2d 5587	. . 3 ⊢ (𝑥 = 𝐴 → (exp‘(𝑦 · (log‘𝑥))) = (exp‘(𝑦 · (log‘𝐴))))
11		fvoveq1 5974	. . 3 ⊢ (𝑦 = 𝐵 → (exp‘(𝑦 · (log‘𝐴))) = (exp‘(𝐵 · (log‘𝐴))))
12		df-rpcxp 15375	. . 3 ⊢ ↑_𝑐 = (𝑥 ∈ ℝ⁺, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥))))
13	10, 11, 12	ovmpog 6087	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ (exp‘(𝐵 · (log‘𝐴))) ∈ ℂ) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
14	7, 13	mpd3an3 1351	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1373 ∈ wcel 2177 ‘cfv 5276 (class class class)co 5951 ℂcc 7930 · cmul 7937 ℝ⁺crp 9782 expce 11997 logclog 15372 ↑_𝑐ccxp 15373

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4163 ax-sep 4166 ax-nul 4174 ax-pow 4222 ax-pr 4257 ax-un 4484 ax-setind 4589 ax-iinf 4640 ax-cnex 8023 ax-resscn 8024 ax-1cn 8025 ax-1re 8026 ax-icn 8027 ax-addcl 8028 ax-addrcl 8029 ax-mulcl 8030 ax-mulrcl 8031 ax-addcom 8032 ax-mulcom 8033 ax-addass 8034 ax-mulass 8035 ax-distr 8036 ax-i2m1 8037 ax-0lt1 8038 ax-1rid 8039 ax-0id 8040 ax-rnegex 8041 ax-precex 8042 ax-cnre 8043 ax-pre-ltirr 8044 ax-pre-ltwlin 8045 ax-pre-lttrn 8046 ax-pre-apti 8047 ax-pre-ltadd 8048 ax-pre-mulgt0 8049 ax-pre-mulext 8050 ax-arch 8051 ax-caucvg 8052 ax-pre-suploc 8053 ax-addf 8054 ax-mulf 8055

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3000 df-csb 3095 df-dif 3169 df-un 3171 df-in 3173 df-ss 3180 df-nul 3462 df-if 3573 df-pw 3619 df-sn 3640 df-pr 3641 df-op 3643 df-uni 3853 df-int 3888 df-iun 3931 df-disj 4024 df-br 4048 df-opab 4110 df-mpt 4111 df-tr 4147 df-id 4344 df-po 4347 df-iso 4348 df-iord 4417 df-on 4419 df-ilim 4420 df-suc 4422 df-iom 4643 df-xp 4685 df-rel 4686 df-cnv 4687 df-co 4688 df-dm 4689 df-rn 4690 df-res 4691 df-ima 4692 df-iota 5237 df-fun 5278 df-fn 5279 df-f 5280 df-f1 5281 df-fo 5282 df-f1o 5283 df-fv 5284 df-isom 5285 df-riota 5906 df-ov 5954 df-oprab 5955 df-mpo 5956 df-of 6165 df-1st 6233 df-2nd 6234 df-recs 6398 df-irdg 6463 df-frec 6484 df-1o 6509 df-oadd 6513 df-er 6627 df-map 6744 df-pm 6745 df-en 6835 df-dom 6836 df-fin 6837 df-sup 7093 df-inf 7094 df-pnf 8116 df-mnf 8117 df-xr 8118 df-ltxr 8119 df-le 8120 df-sub 8252 df-neg 8253 df-reap 8655 df-ap 8662 df-div 8753 df-inn 9044 df-2 9102 df-3 9103 df-4 9104 df-n0 9303 df-z 9380 df-uz 9656 df-q 9748 df-rp 9783 df-xneg 9901 df-xadd 9902 df-ioo 10021 df-ico 10023 df-icc 10024 df-fz 10138 df-fzo 10272 df-seqfrec 10600 df-exp 10691 df-fac 10878 df-bc 10900 df-ihash 10928 df-shft 11170 df-cj 11197 df-re 11198 df-im 11199 df-rsqrt 11353 df-abs 11354 df-clim 11634 df-sumdc 11709 df-ef 12003 df-e 12004 df-rest 13117 df-topgen 13136 df-psmet 14349 df-xmet 14350 df-met 14351 df-bl 14352 df-mopn 14353 df-top 14514 df-topon 14527 df-bases 14559 df-ntr 14612 df-cn 14704 df-cnp 14705 df-tx 14769 df-cncf 15087 df-limced 15172 df-dvap 15173 df-relog 15374 df-rpcxp 15375

This theorem is referenced by: cxpexprp 15411 logcxp 15413 1cxp 15416 ecxp 15417 rpcncxpcl 15418 rpcxpcl 15419 cxpap0 15420 rpcxpadd 15421 rpmulcxp 15425 cxpmul 15428 abscxp 15431 cxplt 15432 rpcxple2 15434 rpcxplt2 15435 apcxp2 15455 rpabscxpbnd 15456 rpcxplogb 15480

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