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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 14447 | . . 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 10688 | . . . . 5 ⊢ 0 ∈ ℝ* | |
4 | xrltne 12557 | . . . . 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 1537 ∈ wcel 2114 ≠ wne 3016 ifcif 4467 class class class wbr 5066 ‘cfv 6355 0cc0 10537 1c1 10538 ℝ*cxr 10674 < clt 10675 -cneg 10871 sgncsgn 14445 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-i2m1 10605 ax-rnegex 10608 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-neg 10873 df-sgn 14446 |
This theorem is referenced by: sgnmnf 14454 sgncl 31796 sgnmul 31800 sgnsub 31802 sgnnbi 31803 sgnsgn 31806 |
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