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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 15061 | . . 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 11285 | . . . . 5 ⊢ 0 ∈ ℝ* | |
4 | xrltne 13168 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ* ∧ 𝐴 < 0) → 0 ≠ 𝐴) | |
5 | 3, 4 | mp3an2 1446 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → 0 ≠ 𝐴) |
6 | nesym 2993 | . . . 4 ⊢ (0 ≠ 𝐴 ↔ ¬ 𝐴 = 0) | |
7 | 5, 6 | sylib 217 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → ¬ 𝐴 = 0) |
8 | 7 | iffalsed 4535 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) = if(𝐴 < 0, -1, 1)) |
9 | iftrue 4530 | . . 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 2772 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ifcif 4524 class class class wbr 5142 ‘cfv 6542 0cc0 11132 1c1 11133 ℝ*cxr 11271 < clt 11272 -cneg 11469 sgncsgn 15059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-i2m1 11200 ax-rnegex 11203 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-neg 11471 df-sgn 15060 |
This theorem is referenced by: sgnmnf 15068 sgncl 34152 sgnmul 34156 sgnsub 34158 sgnnbi 34159 sgnsgn 34162 |
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