![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0cxpd | Structured version Visualization version GIF version |
Description: Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 30-May-2016.) |
Ref | Expression |
---|---|
cxp0d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
cxpefd.2 | ⊢ (𝜑 → 𝐴 ≠ 0) |
Ref | Expression |
---|---|
0cxpd | ⊢ (𝜑 → (0↑𝑐𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cxp0d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | cxpefd.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) | |
3 | 0cxp 26100 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑𝑐𝐴) = 0) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (0↑𝑐𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 (class class class)co 7392 ℂcc 11089 0cc0 11091 ↑𝑐ccxp 25990 |
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 2702 ax-sep 5291 ax-nul 5298 ax-pr 5419 ax-1cn 11149 ax-icn 11150 ax-addcl 11151 ax-mulcl 11153 ax-i2m1 11159 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3474 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5141 df-opab 5203 df-id 5566 df-xp 5674 df-rel 5675 df-cnv 5676 df-co 5677 df-dm 5678 df-iota 6483 df-fun 6533 df-fv 6539 df-ov 7395 df-oprab 7396 df-mpo 7397 df-cxp 25992 |
This theorem is referenced by: cxpcn3lem 26179 cxpcn3 26180 cxpaddle 26184 cxpeq 26189 amgm 26419 abvcxp 27042 padicabvcxp 27059 |
Copyright terms: Public domain | W3C validator |