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Theorem rpcxpef 15746

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 15734	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (log‘𝐴) ∈ ℝ)
4	3	recnd 8298	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (log‘𝐴) ∈ ℂ)
5	1, 4	mulcld 8290	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐵 · (log‘𝐴)) ∈ ℂ)
6		efcl 12343	. . 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 5669	. . . . 5 ⊢ (𝑥 = 𝐴 → (log‘𝑥) = (log‘𝐴))
9	8	oveq2d 6065	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 · (log‘𝑥)) = (𝑦 · (log‘𝐴)))
10	9	fveq2d 5673	. . 3 ⊢ (𝑥 = 𝐴 → (exp‘(𝑦 · (log‘𝑥))) = (exp‘(𝑦 · (log‘𝐴))))
11		fvoveq1 6072	. . 3 ⊢ (𝑦 = 𝐵 → (exp‘(𝑦 · (log‘𝐴))) = (exp‘(𝐵 · (log‘𝐴))))
12		df-rpcxp 15711	. . 3 ⊢ ↑_𝑐 = (𝑥 ∈ ℝ⁺, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥))))
13	10, 11, 12	ovmpog 6187	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ ∧ (exp‘(𝐵 · (log‘𝐴))) ∈ ℂ) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
14	7, 13	mpd3an3 1375	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℂ) → (𝐴↑_𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1398 ∈ wcel 2203 ‘cfv 5351 (class class class)co 6049 ℂcc 8121 · cmul 8128 ℝ⁺crp 9982 expce 12321 logclog 15708 ↑_𝑐ccxp 15709

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8214 ax-resscn 8215 ax-1cn 8216 ax-1re 8217 ax-icn 8218 ax-addcl 8219 ax-addrcl 8220 ax-mulcl 8221 ax-mulrcl 8222 ax-addcom 8223 ax-mulcom 8224 ax-addass 8225 ax-mulass 8226 ax-distr 8227 ax-i2m1 8228 ax-0lt1 8229 ax-1rid 8230 ax-0id 8231 ax-rnegex 8232 ax-precex 8233 ax-cnre 8234 ax-pre-ltirr 8235 ax-pre-ltwlin 8236 ax-pre-lttrn 8237 ax-pre-apti 8238 ax-pre-ltadd 8239 ax-pre-mulgt0 8240 ax-pre-mulext 8241 ax-arch 8242 ax-caucvg 8243 ax-pre-suploc 8244 ax-addf 8245 ax-mulf 8246

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-disj 4085 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-isom 5360 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-of 6265 df-1st 6333 df-2nd 6334 df-recs 6535 df-irdg 6600 df-frec 6621 df-1o 6646 df-oadd 6650 df-er 6766 df-map 6883 df-pm 6884 df-en 6975 df-dom 6976 df-fin 6977 df-sup 7274 df-inf 7275 df-pnf 8306 df-mnf 8307 df-xr 8308 df-ltxr 8309 df-le 8310 df-sub 8442 df-neg 8443 df-reap 8845 df-ap 8852 df-div 8943 df-inn 9234 df-2 9292 df-3 9293 df-4 9294 df-n0 9493 df-z 9574 df-uz 9850 df-q 9948 df-rp 9983 df-xneg 10101 df-xadd 10102 df-ioo 10221 df-ico 10223 df-icc 10224 df-fz 10339 df-fzo 10473 df-seqfrec 10806 df-exp 10897 df-fac 11084 df-bc 11106 df-ihash 11134 df-shft 11493 df-cj 11520 df-re 11521 df-im 11522 df-rsqrt 11676 df-abs 11677 df-clim 11957 df-sumdc 12032 df-ef 12327 df-e 12328 df-rest 13443 df-topgen 13462 df-psmet 14678 df-xmet 14679 df-met 14680 df-bl 14681 df-mopn 14682 df-top 14850 df-topon 14863 df-bases 14895 df-ntr 14948 df-cn 15040 df-cnp 15041 df-tx 15105 df-cncf 15423 df-limced 15508 df-dvap 15509 df-relog 15710 df-rpcxp 15711

This theorem is referenced by: cxpexprp 15747 logcxp 15749 1cxp 15752 ecxp 15753 rpcncxpcl 15754 rpcxpcl 15755 cxpap0 15756 rpcxpadd 15757 rpmulcxp 15761 cxpmul 15764 abscxp 15767 cxplt 15768 rpcxple2 15770 rpcxplt2 15771 apcxp2 15791 rpabscxpbnd 15792 rpcxplogb 15816

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