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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 479 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → 𝐴 = 0) | |
2 | 1 | fveq2d 6357 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) = (sgn‘0)) |
3 | sgn0 14048 | . . . 4 ⊢ (sgn‘0) = 0 | |
4 | 2, 3 | syl6eq 2810 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) = 0) |
5 | c0ex 10246 | . . . 4 ⊢ 0 ∈ V | |
6 | 5 | tpid2 4448 | . . 3 ⊢ 0 ∈ {-1, 0, 1} |
7 | 4, 6 | syl6eqel 2847 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
8 | sgnn 14053 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) = -1) | |
9 | negex 10491 | . . . . . 6 ⊢ -1 ∈ V | |
10 | 9 | tpid1 4447 | . . . . 5 ⊢ -1 ∈ {-1, 0, 1} |
11 | 8, 10 | syl6eqel 2847 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 < 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
12 | 11 | adantlr 753 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) ∧ 𝐴 < 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
13 | sgnp 14049 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) = 1) | |
14 | 1ex 10247 | . . . . . 6 ⊢ 1 ∈ V | |
15 | 14 | tpid3 4450 | . . . . 5 ⊢ 1 ∈ {-1, 0, 1} |
16 | 13, 15 | syl6eqel 2847 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
17 | 16 | adantlr 753 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) ∧ 0 < 𝐴) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
18 | 0xr 10298 | . . . 4 ⊢ 0 ∈ ℝ* | |
19 | xrlttri2 12188 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 ≠ 0 ↔ (𝐴 < 0 ∨ 0 < 𝐴))) | |
20 | 19 | biimpa 502 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) ∧ 𝐴 ≠ 0) → (𝐴 < 0 ∨ 0 < 𝐴)) |
21 | 18, 20 | mpanl2 719 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) → (𝐴 < 0 ∨ 0 < 𝐴)) |
22 | 12, 17, 21 | mpjaodan 862 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ 0) → (sgn‘𝐴) ∈ {-1, 0, 1}) |
23 | 7, 22 | pm2.61dane 3019 | 1 ⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) ∈ {-1, 0, 1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 {ctp 4325 class class class wbr 4804 ‘cfv 6049 0cc0 10148 1c1 10149 ℝ*cxr 10285 < clt 10286 -cneg 10479 sgncsgn 14045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-i2m1 10216 ax-1ne0 10217 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6817 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-neg 10481 df-sgn 14046 |
This theorem is referenced by: sgnclre 30931 sgnmulsgn 30941 sgnmulsgp 30942 signstcl 30972 signstf 30973 signstf0 30975 signstfvn 30976 signsvtn0 30977 signstfvneq0 30979 signsvfn 30989 |
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