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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 32876 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) |
6 | c0ex 11233 | . . . . 5 ⊢ 0 ∈ V | |
7 | 6 | tpid2 4771 | . . . 4 ⊢ 0 ∈ {-1, 0, 1} |
8 | 1ex 11235 | . . . . . 6 ⊢ 1 ∈ V | |
9 | 8 | tpid3 4774 | . . . . 5 ⊢ 1 ∈ {-1, 0, 1} |
10 | negex 11483 | . . . . . 6 ⊢ -1 ∈ V | |
11 | 10 | tpid1 4769 | . . . . 5 ⊢ -1 ∈ {-1, 0, 1} |
12 | 9, 11 | ifcli 4572 | . . . 4 ⊢ if( 0 < 𝑥, 1, -1) ∈ {-1, 0, 1} |
13 | 7, 12 | ifcli 4572 | . . 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 7121 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑆:𝐵⟶{-1, 0, 1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ifcif 4525 {ctp 4629 class class class wbr 5143 ⟶wf 6539 ‘cfv 6543 0cc0 11133 1c1 11134 -cneg 11470 Basecbs 17174 0gc0g 17415 ltcplt 18294 sgnscsgns 32874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5424 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-mulcl 11195 ax-i2m1 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-neg 11472 df-sgns 32875 |
This theorem is referenced by: (None) |
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