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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 26722 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) | |
5 | 1, 2, 3, 4 | syl3anc 1370 | 1 ⊢ (𝜑 → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 · cmul 11158 expce 16094 logclog 26611 ↑𝑐ccxp 26612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-i2m1 11221 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-cxp 26614 |
This theorem is referenced by: dvcxp1 26797 dvcxp2 26798 dvcncxp1 26800 cxpcn 26802 cxpcnOLD 26803 abscxpbnd 26811 root1eq1 26813 cxpeq 26815 cxplogb 26844 efiatan 26970 efiatan2 26975 efrlim 27027 efrlimOLD 27028 cxp2limlem 27034 cxploglim 27036 amgmlem 27048 zetacvg 27073 gamcvg2lem 27117 bposlem9 27351 chtppilimlem1 27532 ostth2lem4 27695 ostth2 27696 ostth3 27697 iprodgam 35722 aks4d1p1p1 42045 cxp112d 42356 cxp111d 42357 proot1ex 43185 logcxp0 48385 |
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