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Theorem sgnn 15040

Description: The signum of a negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.)

**Assertion**
Ref	Expression
sgnn	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1)

**Proof of Theorem sgnn**
Step	Hyp	Ref	Expression
1		sgnval 15034	. . 3 ⊢ (𝐴 ∈ ℝ^* → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))
2	1	adantr 481	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))
3		0xr 11260	. . . . 5 ⊢ 0 ∈ ℝ^*
4		xrltne 13141	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝐴 < 0) → 0 ≠ 𝐴)
5	3, 4	mp3an2 1449	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → 0 ≠ 𝐴)
6		nesym 2997	. . . 4 ⊢ (0 ≠ 𝐴 ↔ ¬ 𝐴 = 0)
7	5, 6	sylib 217	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → ¬ 𝐴 = 0)
8	7	iffalsed 4539	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) = if(𝐴 < 0, -1, 1))
9		iftrue 4534	. . 3 ⊢ (𝐴 < 0 → if(𝐴 < 0, -1, 1) = -1)
10	9	adantl 482	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → if(𝐴 < 0, -1, 1) = -1)
11	2, 8, 10	3eqtrd 2776	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ifcif 4528 class class class wbr 5148 ‘cfv 6543 0cc0 11109 1c1 11110 ℝ^*cxr 11246 < clt 11247 -cneg 11444 sgncsgn 15032

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-i2m1 11177 ax-rnegex 11180 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-neg 11446 df-sgn 15033

This theorem is referenced by: sgnmnf 15041 sgncl 33532 sgnmul 33536 sgnsub 33538 sgnnbi 33539 sgnsgn 33542

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