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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 9911 | . . . 4 ⊢ 0 ∈ ℂ | |
2 | cxpval 24210 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0↑𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0))))) | |
3 | 1, 2 | mpan 702 | . . 3 ⊢ (𝐴 ∈ ℂ → (0↑𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0))))) |
4 | eqid 2610 | . . . 4 ⊢ 0 = 0 | |
5 | 4 | iftruei 4043 | . . 3 ⊢ if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))) = if(𝐴 = 0, 1, 0) |
6 | 3, 5 | syl6eq 2660 | . 2 ⊢ (𝐴 ∈ ℂ → (0↑𝑐𝐴) = if(𝐴 = 0, 1, 0)) |
7 | ifnefalse 4048 | . 2 ⊢ (𝐴 ≠ 0 → if(𝐴 = 0, 1, 0) = 0) | |
8 | 6, 7 | sylan9eq 2664 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑𝑐𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ifcif 4036 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 · cmul 9820 expce 14631 logclog 24105 ↑𝑐ccxp 24106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-mulcl 9877 ax-i2m1 9883 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-cxp 24108 |
This theorem is referenced by: cxpexp 24214 cxpeq0 24224 cxpge0 24229 mulcxplem 24230 cxpmul2 24235 cxple2 24243 cxpsqrt 24249 0cxpd 24256 abscxpbnd 24294 |
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