Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sgnn | Structured version Visualization version GIF version |
Description: The signum of a negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.) |
Ref | Expression |
---|---|
sgnn | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sgnval 14441 | . . 3 ⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) | |
2 | 1 | adantr 483 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) |
3 | 0xr 10682 | . . . . 5 ⊢ 0 ∈ ℝ* | |
4 | xrltne 12550 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ* ∧ 𝐴 < 0) → 0 ≠ 𝐴) | |
5 | 3, 4 | mp3an2 1445 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 0 ≠ 𝐴) |
6 | nesym 3072 | . . . 4 ⊢ (0 ≠ 𝐴 ↔ ¬ 𝐴 = 0) | |
7 | 5, 6 | sylib 220 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → ¬ 𝐴 = 0) |
8 | 7 | iffalsed 4478 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) = if(𝐴 < 0, -1, 1)) |
9 | iftrue 4473 | . . 3 ⊢ (𝐴 < 0 → if(𝐴 < 0, -1, 1) = -1) | |
10 | 9 | adantl 484 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → if(𝐴 < 0, -1, 1) = -1) |
11 | 2, 8, 10 | 3eqtrd 2860 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ifcif 4467 class class class wbr 5059 ‘cfv 6350 0cc0 10531 1c1 10532 ℝ*cxr 10668 < clt 10669 -cneg 10865 sgncsgn 14439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-i2m1 10599 ax-rnegex 10602 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-neg 10867 df-sgn 14440 |
This theorem is referenced by: sgnmnf 14448 sgncl 31791 sgnmul 31795 sgnsub 31797 sgnnbi 31798 sgnsgn 31801 |
Copyright terms: Public domain | W3C validator |