![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11202 | . . . 4 ⊢ 0 ∈ ℂ | |
2 | cxpval 26163 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0↑𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0))))) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐴 ∈ ℂ → (0↑𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0))))) |
4 | eqid 2732 | . . . 4 ⊢ 0 = 0 | |
5 | 4 | iftruei 4534 | . . 3 ⊢ if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))) = if(𝐴 = 0, 1, 0) |
6 | 3, 5 | eqtrdi 2788 | . 2 ⊢ (𝐴 ∈ ℂ → (0↑𝑐𝐴) = if(𝐴 = 0, 1, 0)) |
7 | ifnefalse 4539 | . 2 ⊢ (𝐴 ≠ 0 → if(𝐴 = 0, 1, 0) = 0) | |
8 | 6, 7 | sylan9eq 2792 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑𝑐𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ifcif 4527 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 0cc0 11106 1c1 11107 · cmul 11111 expce 16001 logclog 26054 ↑𝑐ccxp 26055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-mulcl 11168 ax-i2m1 11174 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-cxp 26057 |
This theorem is referenced by: cxpexp 26167 cxpeq0 26177 cxpge0 26182 mulcxplem 26183 cxpmul2 26188 cxple2 26196 cxpsqrt 26202 0cxpd 26209 cxpsqrtth 26228 abscxpbnd 26250 |
Copyright terms: Public domain | W3C validator |