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Mirrors > Home > MPE Home > Th. List > cxpef | Structured version Visualization version GIF version |
Description: Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) |
Ref | Expression |
---|---|
cxpef | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cxpval 26691 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴))))) | |
2 | 1 | 3adant2 1128 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴))))) |
3 | simp2 1134 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → 𝐴 ≠ 0) | |
4 | 3 | neneqd 2935 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → ¬ 𝐴 = 0) |
5 | 4 | iffalsed 4544 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))) = (exp‘(𝐵 · (log‘𝐴)))) |
6 | 2, 5 | eqtrd 2766 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ifcif 4533 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 0cc0 11158 1c1 11159 · cmul 11163 expce 16063 logclog 26581 ↑𝑐ccxp 26582 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-mulcl 11220 ax-i2m1 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6506 df-fun 6556 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-cxp 26584 |
This theorem is referenced by: cxpexpz 26694 logcxp 26696 1cxp 26699 ecxp 26700 rpcxpcl 26703 cxpne0 26704 cxpadd 26706 mulcxp 26712 cxpmul 26715 abscxp 26719 abscxp2 26720 cxplt 26721 cxple2 26724 cxpsqrtlem 26729 cxpsqrt 26730 cxpefd 26739 1cubrlem 26869 bposlem9 27321 iexpire 35557 aks4d1p1p1 41762 |
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