0cxp - Metamath Proof Explorer

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Theorem 0cxp 25257

Description: Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.)

**Assertion**
Ref	Expression
0cxp	⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑_𝑐𝐴) = 0)

**Proof of Theorem 0cxp**
Step	Hyp	Ref	Expression
1		0cn 10622	. . . 4 ⊢ 0 ∈ ℂ
2		cxpval 25255	. . . 4 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0↑_𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))))
3	1, 2	mpan 689	. . 3 ⊢ (𝐴 ∈ ℂ → (0↑_𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))))
4		eqid 2798	. . . 4 ⊢ 0 = 0
5	4	iftruei 4432	. . 3 ⊢ if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))) = if(𝐴 = 0, 1, 0)
6	3, 5	eqtrdi 2849	. 2 ⊢ (𝐴 ∈ ℂ → (0↑_𝑐𝐴) = if(𝐴 = 0, 1, 0))
7		ifnefalse 4437	. 2 ⊢ (𝐴 ≠ 0 → if(𝐴 = 0, 1, 0) = 0)
8	6, 7	sylan9eq 2853	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑_𝑐𝐴) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ifcif 4425 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 · cmul 10531 expce 15407 logclog 25146 ↑_𝑐ccxp 25147

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-mulcl 10588 ax-i2m1 10594

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-cxp 25149

This theorem is referenced by: cxpexp 25259 cxpeq0 25269 cxpge0 25274 mulcxplem 25275 cxpmul2 25280 cxple2 25288 cxpsqrt 25294 0cxpd 25301 cxpsqrtth 25320 abscxpbnd 25342

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