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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sgnclre | Structured version Visualization version GIF version |
Description: Closure of the signum. (Contributed by Thierry Arnoux, 28-Sep-2018.) |
Ref | Expression |
---|---|
sgnclre | ⊢ (𝐴 ∈ ℝ → (sgn‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1rr 12202 | . . 3 ⊢ -1 ∈ ℝ | |
2 | 0re 11091 | . . 3 ⊢ 0 ∈ ℝ | |
3 | 1re 11089 | . . 3 ⊢ 1 ∈ ℝ | |
4 | tpssi 4795 | . . 3 ⊢ ((-1 ∈ ℝ ∧ 0 ∈ ℝ ∧ 1 ∈ ℝ) → {-1, 0, 1} ⊆ ℝ) | |
5 | 1, 2, 3, 4 | mp3an 1462 | . 2 ⊢ {-1, 0, 1} ⊆ ℝ |
6 | rexr 11135 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
7 | sgncl 32918 | . . 3 ⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) ∈ {-1, 0, 1}) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → (sgn‘𝐴) ∈ {-1, 0, 1}) |
9 | 5, 8 | sselid 3941 | 1 ⊢ (𝐴 ∈ ℝ → (sgn‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ⊆ wss 3909 {ctp 4589 ‘cfv 6492 ℝcr 10984 0cc0 10985 1c1 10986 ℝ*cxr 11122 -cneg 11320 sgncsgn 14906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-sub 11321 df-neg 11322 df-sgn 14907 |
This theorem is referenced by: sgnmul 32922 sgnmulrp2 32923 signstf0 32960 signstfvneq0 32964 signsvfn 32974 signsvfpn 32977 signsvfnn 32978 |
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