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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 25400 | . 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 2113 ≠ wne 2934 ‘cfv 6333 (class class class)co 7164 ℂcc 10606 0cc0 10608 · cmul 10613 expce 15500 logclog 25290 ↑𝑐ccxp 25291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-mulcl 10670 ax-i2m1 10676 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6291 df-fun 6335 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-cxp 25293 |
This theorem is referenced by: dvcxp1 25473 dvcxp2 25474 dvcncxp1 25476 cxpcn 25478 abscxpbnd 25486 root1eq1 25488 cxpeq 25490 cxplogb 25516 efiatan 25642 efiatan2 25647 efrlim 25699 cxp2limlem 25705 cxploglim 25707 amgmlem 25719 zetacvg 25744 gamcvg2lem 25788 bposlem9 26020 chtppilimlem1 26201 ostth2lem4 26364 ostth2 26365 ostth3 26366 iprodgam 33271 aks4d1p1p1 39679 cxpgt0d 39905 proot1ex 40582 logcxp0 45399 |
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