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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sgnsf | Structured version Visualization version GIF version |
Description: The sign function. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
Ref | Expression |
---|---|
sgnsval.b | ⊢ 𝐵 = (Base‘𝑅) |
sgnsval.0 | ⊢ 0 = (0g‘𝑅) |
sgnsval.l | ⊢ < = (lt‘𝑅) |
sgnsval.s | ⊢ 𝑆 = (sgns‘𝑅) |
Ref | Expression |
---|---|
sgnsf | ⊢ (𝑅 ∈ 𝑉 → 𝑆:𝐵⟶{-1, 0, 1}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sgnsval.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | sgnsval.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
3 | sgnsval.l | . . 3 ⊢ < = (lt‘𝑅) | |
4 | sgnsval.s | . . 3 ⊢ 𝑆 = (sgns‘𝑅) | |
5 | 1, 2, 3, 4 | sgnsv 30057 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) |
6 | c0ex 10246 | . . . . 5 ⊢ 0 ∈ V | |
7 | 6 | tpid2 4448 | . . . 4 ⊢ 0 ∈ {-1, 0, 1} |
8 | 1ex 10247 | . . . . . 6 ⊢ 1 ∈ V | |
9 | 8 | tpid3 4450 | . . . . 5 ⊢ 1 ∈ {-1, 0, 1} |
10 | negex 10491 | . . . . . 6 ⊢ -1 ∈ V | |
11 | 10 | tpid1 4447 | . . . . 5 ⊢ -1 ∈ {-1, 0, 1} |
12 | 9, 11 | keepel 4299 | . . . 4 ⊢ if( 0 < 𝑥, 1, -1) ∈ {-1, 0, 1} |
13 | 7, 12 | keepel 4299 | . . 3 ⊢ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)) ∈ {-1, 0, 1} |
14 | 13 | a1i 11 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)) ∈ {-1, 0, 1}) |
15 | 5, 14 | fmpt3d 6550 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑆:𝐵⟶{-1, 0, 1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ifcif 4230 {ctp 4325 class class class wbr 4804 ⟶wf 6045 ‘cfv 6049 0cc0 10148 1c1 10149 -cneg 10479 Basecbs 16079 0gc0g 16322 ltcplt 17162 sgnscsgns 30055 |
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-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pr 5055 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-mulcl 10210 ax-i2m1 10216 |
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-ral 3055 df-rex 3056 df-reu 3057 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-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 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-neg 10481 df-sgns 30056 |
This theorem is referenced by: (None) |
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