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 29058 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) |
6 | c0ex 9913 | . . . . 5 ⊢ 0 ∈ V | |
7 | 6 | tpid2 4247 | . . . 4 ⊢ 0 ∈ {-1, 0, 1} |
8 | 1ex 9914 | . . . . . 6 ⊢ 1 ∈ V | |
9 | 8 | tpid3 4250 | . . . . 5 ⊢ 1 ∈ {-1, 0, 1} |
10 | negex 10158 | . . . . . 6 ⊢ -1 ∈ V | |
11 | 10 | tpid1 4246 | . . . . 5 ⊢ -1 ∈ {-1, 0, 1} |
12 | 9, 11 | keepel 4105 | . . . 4 ⊢ if( 0 < 𝑥, 1, -1) ∈ {-1, 0, 1} |
13 | 7, 12 | keepel 4105 | . . 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 6293 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑆:𝐵⟶{-1, 0, 1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ifcif 4036 {ctp 4129 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 0cc0 9815 1c1 9816 -cneg 10146 Basecbs 15695 0gc0g 15923 ltcplt 16764 sgnscsgns 29056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-mulcl 9877 ax-i2m1 9883 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-neg 10148 df-sgns 29057 |
This theorem is referenced by: (None) |
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