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Mirrors > Home > MPE Home > Th. List > cxpefd | Structured version Visualization version GIF version |
Description: Value of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.) |
Ref | Expression |
---|---|
cxp0d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
cxpefd.2 | ⊢ (𝜑 → 𝐴 ≠ 0) |
cxpefd.3 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
cxpefd | ⊢ (𝜑 → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cxp0d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | cxpefd.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) | |
3 | cxpefd.3 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | cxpef 26142 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) | |
5 | 1, 2, 3, 4 | syl3anc 1372 | 1 ⊢ (𝜑 → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ‘cfv 6535 (class class class)co 7396 ℂcc 11095 0cc0 11097 · cmul 11102 expce 15992 logclog 26032 ↑𝑐ccxp 26033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-mulcl 11159 ax-i2m1 11165 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6487 df-fun 6537 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-cxp 26035 |
This theorem is referenced by: dvcxp1 26215 dvcxp2 26216 dvcncxp1 26218 cxpcn 26220 abscxpbnd 26228 root1eq1 26230 cxpeq 26232 cxplogb 26258 efiatan 26384 efiatan2 26389 efrlim 26441 cxp2limlem 26447 cxploglim 26449 amgmlem 26461 zetacvg 26486 gamcvg2lem 26530 bposlem9 26762 chtppilimlem1 26943 ostth2lem4 27106 ostth2 27107 ostth3 27108 iprodgam 34643 aks4d1p1p1 40834 proot1ex 41814 logcxp0 47061 |
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