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 487 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → 𝐴 = 0) | |
2 | 1 | fveq2d 6669 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) = (sgn‘0)) |
3 | sgn0 14442 | . . . 4 ⊢ (sgn‘0) = 0 | |
4 | 2, 3 | syl6eq 2872 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) = 0) |
5 | c0ex 10629 | . . . 4 ⊢ 0 ∈ V | |
6 | 5 | tpid2 4700 | . . 3 ⊢ 0 ∈ {-1, 0, 1} |
7 | 4, 6 | eqeltrdi 2921 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
8 | sgnn 14447 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1) | |
9 | negex 10878 | . . . . . 6 ⊢ -1 ∈ V | |
10 | 9 | tpid1 4698 | . . . . 5 ⊢ -1 ∈ {-1, 0, 1} |
11 | 8, 10 | eqeltrdi 2921 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
12 | 11 | adantlr 713 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) ∧ 𝐴 < 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
13 | sgnp 14443 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) = 1) | |
14 | 1ex 10631 | . . . . . 6 ⊢ 1 ∈ V | |
15 | 14 | tpid3 4703 | . . . . 5 ⊢ 1 ∈ {-1, 0, 1} |
16 | 13, 15 | eqeltrdi 2921 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
17 | 16 | adantlr 713 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) ∧ 0 < 𝐴) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
18 | 0xr 10682 | . . . 4 ⊢ 0 ∈ ℝ* | |
19 | xrlttri2 12529 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴))) | |
20 | 19 | biimpa 479 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) ∧ 𝐴 ≠ 0) → (𝐴 < 0 ∨ 0 < 𝐴)) |
21 | 18, 20 | mpanl2 699 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) → (𝐴 < 0 ∨ 0 < 𝐴)) |
22 | 12, 17, 21 | mpjaodan 955 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
23 | 7, 22 | pm2.61dane 3104 | 1 ⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) ∈ {-1, 0, 1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {ctp 4565 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-tp 4566 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: sgnclre 31792 sgnmulsgn 31802 sgnmulsgp 31803 signstcl 31830 signstf 31831 signstf0 31833 signstfvn 31834 signsvtn0 31835 signstfvneq0 31837 signsvfn 31847 |
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