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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 25250 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) | |
5 | 1, 2, 3, 4 | syl3anc 1367 | 1 ⊢ (𝜑 → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 · cmul 10544 expce 15417 logclog 25140 ↑𝑐ccxp 25141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-mulcl 10601 ax-i2m1 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-cxp 25143 |
This theorem is referenced by: dvcxp1 25323 dvcxp2 25324 dvcncxp1 25326 cxpcn 25328 abscxpbnd 25336 root1eq1 25338 cxpeq 25340 cxplogb 25366 efiatan 25492 efiatan2 25497 efrlim 25549 cxp2limlem 25555 cxploglim 25557 amgmlem 25569 zetacvg 25594 gamcvg2lem 25638 bposlem9 25870 chtppilimlem1 26051 ostth2lem4 26214 ostth2 26215 ostth3 26216 iprodgam 32976 cxpgt0d 39187 proot1ex 39808 logcxp0 44602 |
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