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Theorem sgnn 15143

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 15137	. . 3 ⊢ (𝐴 ∈ ℝ^* → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))
2	1	adantr 480	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))
3		0xr 11337	. . . . 5 ⊢ 0 ∈ ℝ^*
4		xrltne 13225	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝐴 < 0) → 0 ≠ 𝐴)
5	3, 4	mp3an2 1449	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → 0 ≠ 𝐴)
6		nesym 3003	. . . 4 ⊢ (0 ≠ 𝐴 ↔ ¬ 𝐴 = 0)
7	5, 6	sylib 218	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → ¬ 𝐴 = 0)
8	7	iffalsed 4559	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) = if(𝐴 < 0, -1, 1))
9		iftrue 4554	. . 3 ⊢ (𝐴 < 0 → if(𝐴 < 0, -1, 1) = -1)
10	9	adantl 481	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → if(𝐴 < 0, -1, 1) = -1)
11	2, 8, 10	3eqtrd 2784	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ifcif 4548 class class class wbr 5166 ‘cfv 6573 0cc0 11184 1c1 11185 ℝ^*cxr 11323 < clt 11324 -cneg 11521 sgncsgn 15135

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-i2m1 11252 ax-rnegex 11255 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-neg 11523 df-sgn 15136

This theorem is referenced by: sgnmnf 15144 sgncl 34503 sgnmul 34507 sgnsub 34509 sgnnbi 34510 sgnsgn 34513

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