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| Mirrors > Home > MPE Home > Th. List > 0cxp | Structured version Visualization version GIF version | ||
| Description: Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.) |
| Ref | Expression |
|---|---|
| 0cxp | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑𝑐𝐴) = 0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0cn 11253 | . . . 4 ⊢ 0 ∈ ℂ | |
| 2 | cxpval 26706 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0↑𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0))))) | |
| 3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐴 ∈ ℂ → (0↑𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0))))) |
| 4 | eqid 2737 | . . . 4 ⊢ 0 = 0 | |
| 5 | 4 | iftruei 4532 | . . 3 ⊢ if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))) = if(𝐴 = 0, 1, 0) |
| 6 | 3, 5 | eqtrdi 2793 | . 2 ⊢ (𝐴 ∈ ℂ → (0↑𝑐𝐴) = if(𝐴 = 0, 1, 0)) |
| 7 | ifnefalse 4537 | . 2 ⊢ (𝐴 ≠ 0 → if(𝐴 = 0, 1, 0) = 0) | |
| 8 | 6, 7 | sylan9eq 2797 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑𝑐𝐴) = 0) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ifcif 4525 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 · 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: cxpexp 26710 cxpeq0 26720 cxpge0 26725 mulcxplem 26726 cxpmul2 26731 cxple2 26739 cxpsqrt 26745 0cxpd 26752 cxpsqrtth 26772 abscxpbnd 26796 |
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