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Mirrors > Home > MPE Home > Th. List > sgnp | Structured version Visualization version GIF version |
Description: The signum of a positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.) |
Ref | Expression |
---|---|
sgnp | ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sgnval 14435 | . . 3 ⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) | |
2 | 1 | adantr 481 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) = if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1))) |
3 | 0xr 10676 | . . . . 5 ⊢ 0 ∈ ℝ* | |
4 | xrltne 12544 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 0 < 𝐴) → 𝐴 ≠ 0) | |
5 | 3, 4 | mp3an1 1439 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → 𝐴 ≠ 0) |
6 | 5 | neneqd 3018 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → ¬ 𝐴 = 0) |
7 | 6 | iffalsed 4474 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → if(𝐴 = 0, 0, if(𝐴 < 0, -1, 1)) = if(𝐴 < 0, -1, 1)) |
8 | xrltnsym 12518 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (0 < 𝐴 → ¬ 𝐴 < 0)) | |
9 | 3, 8 | mpan 686 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (0 < 𝐴 → ¬ 𝐴 < 0)) |
10 | 9 | imp 407 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → ¬ 𝐴 < 0) |
11 | 10 | iffalsed 4474 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → if(𝐴 < 0, -1, 1) = 1) |
12 | 2, 7, 11 | 3eqtrd 2857 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ifcif 4463 class class class wbr 5057 ‘cfv 6348 0cc0 10525 1c1 10526 ℝ*cxr 10662 < clt 10663 -cneg 10859 sgncsgn 14433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-i2m1 10593 ax-rnegex 10596 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-neg 10861 df-sgn 14434 |
This theorem is referenced by: sgnrrp 14438 sgn1 14439 sgnpnf 14440 sgncl 31695 sgnmul 31699 sgnmulrp2 31700 sgnsub 31701 sgnpbi 31703 |
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