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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sgnsv | Structured version Visualization version GIF version | ||
| Description: The sign mapping. (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 | 
|---|---|
| sgnsv | ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sgnsval.s | . 2 ⊢ 𝑆 = (sgns‘𝑅) | |
| 2 | elex 3500 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
| 3 | fveq2 6905 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
| 4 | sgnsval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
| 5 | 3, 4 | eqtr4di 2794 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) | 
| 6 | fveq2 6905 | . . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) | |
| 7 | sgnsval.0 | . . . . . . . . 9 ⊢ 0 = (0g‘𝑅) | |
| 8 | 6, 7 | eqtr4di 2794 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) | 
| 9 | 8 | adantr 480 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (0g‘𝑟) = 0 ) | 
| 10 | 9 | eqeq2d 2747 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (𝑥 = (0g‘𝑟) ↔ 𝑥 = 0 )) | 
| 11 | fveq2 6905 | . . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (lt‘𝑟) = (lt‘𝑅)) | |
| 12 | sgnsval.l | . . . . . . . . . 10 ⊢ < = (lt‘𝑅) | |
| 13 | 11, 12 | eqtr4di 2794 | . . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (lt‘𝑟) = < ) | 
| 14 | 13 | adantr 480 | . . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (lt‘𝑟) = < ) | 
| 15 | eqidd 2737 | . . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → 𝑥 = 𝑥) | |
| 16 | 9, 14, 15 | breq123d 5156 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → ((0g‘𝑟)(lt‘𝑟)𝑥 ↔ 0 < 𝑥)) | 
| 17 | 16 | ifbid 4548 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1) = if( 0 < 𝑥, 1, -1)) | 
| 18 | 10, 17 | ifbieq2d 4551 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1)) = if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))) | 
| 19 | 5, 18 | mpteq12dva 5230 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1))) = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) | 
| 20 | df-sgns 33180 | . . . 4 ⊢ sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1)))) | |
| 21 | 19, 20, 4 | mptfvmpt 7249 | . . 3 ⊢ (𝑅 ∈ V → (sgns‘𝑅) = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) | 
| 22 | 2, 21 | syl 17 | . 2 ⊢ (𝑅 ∈ 𝑉 → (sgns‘𝑅) = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) | 
| 23 | 1, 22 | eqtrid 2788 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → 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 | 
| 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-sgns 33180 | 
| This theorem is referenced by: sgnsval 33182 sgnsf 33183 | 
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