sgnn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > sgnn		Structured version Visualization version GIF version

Theorem sgnn 15094

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 15088	. . 3 ⊢ (𝐴 ∈ ℝ^* → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))
2	1	adantr 479	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)))
3		0xr 11307	. . . . 5 ⊢ 0 ∈ ℝ^*
4		xrltne 13191	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝐴 < 0) → 0 ≠ 𝐴)
5	3, 4	mp3an2 1445	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → 0 ≠ 𝐴)
6		nesym 2986	. . . 4 ⊢ (0 ≠ 𝐴 ↔ ¬ 𝐴 = 0)
7	5, 6	sylib 217	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → ¬ 𝐴 = 0)
8	7	iffalsed 4543	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) = if(𝐴 < 0, -1, 1))
9		iftrue 4538	. . 3 ⊢ (𝐴 < 0 → if(𝐴 < 0, -1, 1) = -1)
10	9	adantl 480	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → if(𝐴 < 0, -1, 1) = -1)
11	2, 8, 10	3eqtrd 2769	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ifcif 4532 class class class wbr 5152 ‘cfv 6553 0cc0 11154 1c1 11155 ℝ^*cxr 11293 < clt 11294 -cneg 11491 sgncsgn 15086

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-i2m1 11222 ax-rnegex 11225 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-po 5593 df-so 5594 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7426 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-neg 11493 df-sgn 15087

This theorem is referenced by: sgnmnf 15095 sgncl 34328 sgnmul 34332 sgnsub 34334 sgnnbi 34335 sgnsgn 34338

W3C validator