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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 25249 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴))))) | |
2 | 1 | 3adant2 1127 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴))))) |
3 | simp2 1133 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → 𝐴 ≠ 0) | |
4 | 3 | neneqd 3023 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → ¬ 𝐴 = 0) |
5 | 4 | iffalsed 4480 | . 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 2858 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴↑𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ifcif 4469 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 · 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: cxpexpz 25252 logcxp 25254 1cxp 25257 ecxp 25258 rpcxpcl 25261 cxpne0 25262 cxpadd 25264 mulcxp 25270 cxpmul 25273 abscxp 25277 abscxp2 25278 cxplt 25279 cxple2 25282 cxpsqrtlem 25287 cxpsqrt 25288 cxpefd 25297 1cubrlem 25421 bposlem9 25870 iexpire 32969 |
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