0cxp - Metamath Proof Explorer

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Theorem 0cxp 26911

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 11226	. . . 4 ⊢ 0 ∈ ℂ
2		cxpval 26909	. . . 4 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0↑_𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))))
3	1, 2	mpan 703	. . 3 ⊢ (𝐴 ∈ ℂ → (0↑_𝑐𝐴) = if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))))
4		eqid 2762	. . . 4 ⊢ 0 = 0
5	4	iftruei 4492	. . 3 ⊢ if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))) = if(𝐴 = 0, 1, 0)
6	3, 5	eqtrdi 2813	. 2 ⊢ (𝐴 ∈ ℂ → (0↑_𝑐𝐴) = if(𝐴 = 0, 1, 0))
7		ifnefalse 4497	. 2 ⊢ (𝐴 ≠ 0 → if(𝐴 = 0, 1, 0) = 0)
8	6, 7	sylan9eq 2817	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑_𝑐𝐴) = 0)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ifcif 4485 ‘cfv 6537 (class class class)co 7417 ℂcc 11126 0cc0 11128 1c1 11129 · cmul 11133 expce 16153 logclog 26799 ↑_𝑐ccxp 26800

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pr 5402 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-mulcl 11190 ax-i2m1 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-sbc 3743 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-iota 6493 df-fun 6539 df-fv 6545 df-ov 7420 df-oprab 7421 df-mpo 7422 df-cxp 26802

This theorem is used by: cxpexp 26913 cxpeq0 26923 cxpge0 26928 mulcxplem 26929 cxpmul2 26934 cxple2 26942 cxpsqrt 26948 0cxpd 26955 cxpsqrtth 26975 abscxpbnd 26998

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