0cxp - Metamath Proof Explorer

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Theorem 0cxp 26603

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 11111	. . . 4 ⊢ 0 ∈ ℂ
2		cxpval 26601	. . . 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 2733	. . . 4 ⊢ 0 = 0
5	4	iftruei 4481	. . 3 ⊢ if(0 = 0, if(𝐴 = 0, 1, 0), (exp‘(𝐴 · (log‘0)))) = if(𝐴 = 0, 1, 0)
6	3, 5	eqtrdi 2784	. 2 ⊢ (𝐴 ∈ ℂ → (0↑_𝑐𝐴) = if(𝐴 = 0, 1, 0))
7		ifnefalse 4486	. 2 ⊢ (𝐴 ≠ 0 → if(𝐴 = 0, 1, 0) = 0)
8	6, 7	sylan9eq 2788	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑_𝑐𝐴) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ifcif 4474 ‘cfv 6486 (class class class)co 7352 ℂcc 11011 0cc0 11013 1c1 11014 · cmul 11018 expce 15970 logclog 26491 ↑_𝑐ccxp 26492

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pr 5372 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-mulcl 11075 ax-i2m1 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-sbc 3738 df-dif 3901 df-un 3903 df-ss 3915 df-nul 4283 df-if 4475 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-br 5094 df-opab 5156 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-iota 6442 df-fun 6488 df-fv 6494 df-ov 7355 df-oprab 7356 df-mpo 7357 df-cxp 26494

This theorem is referenced by: cxpexp 26605 cxpeq0 26615 cxpge0 26620 mulcxplem 26621 cxpmul2 26626 cxple2 26634 cxpsqrt 26640 0cxpd 26647 cxpsqrtth 26667 abscxpbnd 26691

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