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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sgncl | Structured version Visualization version GIF version |
Description: Closure of the signum. (Contributed by Thierry Arnoux, 28-Sep-2018.) |
Ref | Expression |
---|---|
sgncl | ⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) ∈ {-1, 0, 1}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → 𝐴 = 0) | |
2 | 1 | fveq2d 6649 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) = (sgn‘0)) |
3 | sgn0 14440 | . . . 4 ⊢ (sgn‘0) = 0 | |
4 | 2, 3 | eqtrdi 2849 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) = 0) |
5 | c0ex 10624 | . . . 4 ⊢ 0 ∈ V | |
6 | 5 | tpid2 4666 | . . 3 ⊢ 0 ∈ {-1, 0, 1} |
7 | 4, 6 | eqeltrdi 2898 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
8 | sgnn 14445 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1) | |
9 | negex 10873 | . . . . . 6 ⊢ -1 ∈ V | |
10 | 9 | tpid1 4664 | . . . . 5 ⊢ -1 ∈ {-1, 0, 1} |
11 | 8, 10 | eqeltrdi 2898 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
12 | 11 | adantlr 714 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) ∧ 𝐴 < 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
13 | sgnp 14441 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) = 1) | |
14 | 1ex 10626 | . . . . . 6 ⊢ 1 ∈ V | |
15 | 14 | tpid3 4669 | . . . . 5 ⊢ 1 ∈ {-1, 0, 1} |
16 | 13, 15 | eqeltrdi 2898 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
17 | 16 | adantlr 714 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) ∧ 0 < 𝐴) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
18 | 0xr 10677 | . . . 4 ⊢ 0 ∈ ℝ* | |
19 | xrlttri2 12523 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴))) | |
20 | 19 | biimpa 480 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) ∧ 𝐴 ≠ 0) → (𝐴 < 0 ∨ 0 < 𝐴)) |
21 | 18, 20 | mpanl2 700 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) → (𝐴 < 0 ∨ 0 < 𝐴)) |
22 | 12, 17, 21 | mpjaodan 956 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
23 | 7, 22 | pm2.61dane 3074 | 1 ⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) ∈ {-1, 0, 1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {ctp 4529 class class class wbr 5030 ‘cfv 6324 0cc0 10526 1c1 10527 ℝ*cxr 10663 < clt 10664 -cneg 10860 sgncsgn 14437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-i2m1 10594 ax-rnegex 10597 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-neg 10862 df-sgn 14438 |
This theorem is referenced by: sgnclre 31907 sgnmulsgn 31917 sgnmulsgp 31918 signstcl 31945 signstf 31946 signstf0 31948 signstfvn 31949 signsvtn0 31950 signstfvneq0 31952 signsvfn 31962 |
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