0cxp - Metamath Proof Explorer

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Theorem 0cxp 26575

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 11166	. . . 4 ⊢ 0 ∈ ℂ
2		cxpval 26573	. . . 4 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0↑_𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))))
3	1, 2	mpan 690	. . 3 ⊢ (𝐴 ∈ ℂ → (0↑_𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))))
4		eqid 2729	. . . 4 ⊢ 0 = 0
5	4	iftruei 4495	. . 3 ⊢ if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))) = if(𝐴 = 0, 1, 0)
6	3, 5	eqtrdi 2780	. 2 ⊢ (𝐴 ∈ ℂ → (0↑_𝑐𝐴) = if(𝐴 = 0, 1, 0))
7		ifnefalse 4500	. 2 ⊢ (𝐴 ≠ 0 → if(𝐴 = 0, 1, 0) = 0)
8	6, 7	sylan9eq 2784	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑_𝑐𝐴) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ifcif 4488 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 0cc0 11068 1c1 11069 · cmul 11073 expce 16027 logclog 26463 ↑_𝑐ccxp 26464

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-mulcl 11130 ax-i2m1 11136

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-dif 3917 df-un 3919 df-ss 3931 df-nul 4297 df-if 4489 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6464 df-fun 6513 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-cxp 26466

This theorem is referenced by: cxpexp 26577 cxpeq0 26587 cxpge0 26592 mulcxplem 26593 cxpmul2 26598 cxple2 26606 cxpsqrt 26612 0cxpd 26619 cxpsqrtth 26639 abscxpbnd 26663

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