0cxp - Metamath Proof Explorer

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Theorem 0cxp 25356

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 10671	. . . 4 ⊢ 0 ∈ ℂ
2		cxpval 25354	. . . 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 2758	. . . 4 ⊢ 0 = 0
5	4	iftruei 4427	. . 3 ⊢ if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))) = if(𝐴 = 0, 1, 0)
6	3, 5	eqtrdi 2809	. 2 ⊢ (𝐴 ∈ ℂ → (0↑_𝑐𝐴) = if(𝐴 = 0, 1, 0))
7		ifnefalse 4432	. 2 ⊢ (𝐴 ≠ 0 → if(𝐴 = 0, 1, 0) = 0)
8	6, 7	sylan9eq 2813	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 2951 ifcif 4420 ‘cfv 6335 (class class class)co 7150 ℂcc 10573 0cc0 10575 1c1 10576 · cmul 10580 expce 15463 logclog 25245 ↑_𝑐ccxp 25246

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 2729 ax-sep 5169 ax-nul 5176 ax-pr 5298 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-mulcl 10637 ax-i2m1 10643

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3697 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-nul 4226 df-if 4421 df-sn 4523 df-pr 4525 df-op 4529 df-uni 4799 df-br 5033 df-opab 5095 df-id 5430 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-iota 6294 df-fun 6337 df-fv 6343 df-ov 7153 df-oprab 7154 df-mpo 7155 df-cxp 25248

This theorem is referenced by: cxpexp 25358 cxpeq0 25368 cxpge0 25373 mulcxplem 25374 cxpmul2 25379 cxple2 25387 cxpsqrt 25393 0cxpd 25400 cxpsqrtth 25419 abscxpbnd 25441

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