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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 485 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → 𝐴 = 0) | |
2 | 1 | fveq2d 6882 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) = (sgn‘0)) |
3 | sgn0 15018 | . . . 4 ⊢ (sgn‘0) = 0 | |
4 | 2, 3 | eqtrdi 2787 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) = 0) |
5 | c0ex 11190 | . . . 4 ⊢ 0 ∈ V | |
6 | 5 | tpid2 4767 | . . 3 ⊢ 0 ∈ {-1, 0, 1} |
7 | 4, 6 | eqeltrdi 2840 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
8 | sgnn 15023 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1) | |
9 | negex 11440 | . . . . . 6 ⊢ -1 ∈ V | |
10 | 9 | tpid1 4765 | . . . . 5 ⊢ -1 ∈ {-1, 0, 1} |
11 | 8, 10 | eqeltrdi 2840 | . . . 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 15019 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) = 1) | |
14 | 1ex 11192 | . . . . . 6 ⊢ 1 ∈ V | |
15 | 14 | tpid3 4770 | . . . . 5 ⊢ 1 ∈ {-1, 0, 1} |
16 | 13, 15 | eqeltrdi 2840 | . . . 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 11243 | . . . 4 ⊢ 0 ∈ ℝ* | |
19 | xrlttri2 13103 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴))) | |
20 | 19 | biimpa 477 | . . . 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 957 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
23 | 7, 22 | pm2.61dane 3028 | 1 ⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) ∈ {-1, 0, 1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 {ctp 4626 class class class wbr 5141 ‘cfv 6532 0cc0 11092 1c1 11093 ℝ*cxr 11229 < clt 11230 -cneg 11427 sgncsgn 15015 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-i2m1 11160 ax-rnegex 11163 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-neg 11429 df-sgn 15016 |
This theorem is referenced by: sgnclre 33369 sgnmulsgn 33379 sgnmulsgp 33380 signstcl 33407 signstf 33408 signstf0 33410 signstfvn 33411 signsvtn0 33412 signstfvneq0 33414 signsvfn 33424 |
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