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Theorem sgnn 14904

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 14898	. . 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 11123	. . . . 5 ⊢ 0 ∈ ℝ^*
4		xrltne 12998	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝐴 < 0) → 0 ≠ 𝐴)
5	3, 4	mp3an2 1448	. . . 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 4484	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) = if(𝐴 < 0, -1, 1))
9		iftrue 4479	. . 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 2780	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 ifcif 4473 class class class wbr 5092 ‘cfv 6479 0cc0 10972 1c1 10973 ℝ^*cxr 11109 < clt 11110 -cneg 11307 sgncsgn 14896

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-i2m1 11040 ax-rnegex 11043 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-po 5532 df-so 5533 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-ov 7340 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-neg 11309 df-sgn 14897

This theorem is referenced by: sgnmnf 14905 sgncl 32805 sgnmul 32809 sgnsub 32811 sgnnbi 32812 sgnsgn 32815

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