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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 32812 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) |
6 | c0ex 11207 | . . . . 5 ⊢ 0 ∈ V | |
7 | 6 | tpid2 4767 | . . . 4 ⊢ 0 ∈ {-1, 0, 1} |
8 | 1ex 11209 | . . . . . 6 ⊢ 1 ∈ V | |
9 | 8 | tpid3 4770 | . . . . 5 ⊢ 1 ∈ {-1, 0, 1} |
10 | negex 11457 | . . . . . 6 ⊢ -1 ∈ V | |
11 | 10 | tpid1 4765 | . . . . 5 ⊢ -1 ∈ {-1, 0, 1} |
12 | 9, 11 | ifcli 4568 | . . . 4 ⊢ if( 0 < 𝑥, 1, -1) ∈ {-1, 0, 1} |
13 | 7, 12 | ifcli 4568 | . . 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 7108 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑆:𝐵⟶{-1, 0, 1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ifcif 4521 {ctp 4625 class class class wbr 5139 ⟶wf 6530 ‘cfv 6534 0cc0 11107 1c1 11108 -cneg 11444 Basecbs 17149 0gc0g 17390 ltcplt 18269 sgnscsgns 32810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-mulcl 11169 ax-i2m1 11175 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-neg 11446 df-sgns 32811 |
This theorem is referenced by: (None) |
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