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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 26707 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) | |
| 5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 · cmul 11160 expce 16097 logclog 26596 ↑𝑐ccxp 26597 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-i2m1 11223 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-cxp 26599 |
| This theorem is referenced by: dvcxp1 26782 dvcxp2 26783 dvcncxp1 26785 cxpcn 26787 cxpcnOLD 26788 abscxpbnd 26796 root1eq1 26798 cxpeq 26800 cxplogb 26829 efiatan 26955 efiatan2 26960 efrlim 27012 efrlimOLD 27013 cxp2limlem 27019 cxploglim 27021 amgmlem 27033 zetacvg 27058 gamcvg2lem 27102 bposlem9 27336 chtppilimlem1 27517 ostth2lem4 27680 ostth2 27681 ostth3 27682 iprodgam 35742 aks4d1p1p1 42064 cxp112d 42377 cxp111d 42378 proot1ex 43208 logcxp0 48456 |
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