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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sgnsval | Structured version Visualization version GIF version | ||
| Description: The sign value. (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 | 
|---|---|
| sgnsval | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑆‘𝑋) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sgnsval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
| 2 | sgnsval.0 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
| 3 | sgnsval.l | . . . 4 ⊢ < = (lt‘𝑅) | |
| 4 | sgnsval.s | . . . 4 ⊢ 𝑆 = (sgns‘𝑅) | |
| 5 | 1, 2, 3, 4 | sgnsv 33181 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) | 
| 6 | 5 | adantr 480 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) | 
| 7 | eqeq1 2740 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0 )) | |
| 8 | breq2 5146 | . . . . 5 ⊢ (𝑥 = 𝑋 → ( 0 < 𝑥 ↔ 0 < 𝑋)) | |
| 9 | 8 | ifbid 4548 | . . . 4 ⊢ (𝑥 = 𝑋 → if( 0 < 𝑥, 1, -1) = if( 0 < 𝑋, 1, -1)) | 
| 10 | 7, 9 | ifbieq2d 4551 | . . 3 ⊢ (𝑥 = 𝑋 → if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1))) | 
| 11 | 10 | adantl 481 | . 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 = 𝑋) → if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1))) | 
| 12 | simpr 484 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
| 13 | c0ex 11256 | . . . 4 ⊢ 0 ∈ V | |
| 14 | 13 | a1i 11 | . . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 = 0 ) → 0 ∈ V) | 
| 15 | 1ex 11258 | . . . . 5 ⊢ 1 ∈ V | |
| 16 | 15 | a1i 11 | . . . 4 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑋 = 0 ) ∧ 0 < 𝑋) → 1 ∈ V) | 
| 17 | negex 11507 | . . . . 5 ⊢ -1 ∈ V | |
| 18 | 17 | a1i 11 | . . . 4 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑋 = 0 ) ∧ ¬ 0 < 𝑋) → -1 ∈ V) | 
| 19 | 16, 18 | ifclda 4560 | . . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑋 = 0 ) → if( 0 < 𝑋, 1, -1) ∈ V) | 
| 20 | 14, 19 | ifclda 4560 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)) ∈ V) | 
| 21 | 6, 11, 12, 20 | fvmptd 7022 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑆‘𝑋) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ifcif 4524 class class class wbr 5142 ↦ cmpt 5224 ‘cfv 6560 0cc0 11156 1c1 11157 -cneg 11494 Basecbs 17248 0gc0g 17485 ltcplt 18355 sgnscsgns 33179 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-mulcl 11218 ax-i2m1 11224 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-neg 11496 df-sgns 33180 | 
| This theorem is referenced by: (None) | 
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