Mathbox for Thierry Arnoux |
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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 3416 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
3 | fveq2 6695 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
4 | sgnsval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 3, 4 | eqtr4di 2789 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
6 | fveq2 6695 | . . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) | |
7 | sgnsval.0 | . . . . . . . . 9 ⊢ 0 = (0g‘𝑅) | |
8 | 6, 7 | eqtr4di 2789 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) |
9 | 8 | adantr 484 | . . . . . . 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 6695 | . . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (lt‘𝑟) = (lt‘𝑅)) | |
12 | sgnsval.l | . . . . . . . . . 10 ⊢ < = (lt‘𝑅) | |
13 | 11, 12 | eqtr4di 2789 | . . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (lt‘𝑟) = < ) |
14 | 13 | adantr 484 | . . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (lt‘𝑟) = < ) |
15 | eqidd 2737 | . . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → 𝑥 = 𝑥) | |
16 | 9, 14, 15 | breq123d 5053 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → ((0g‘𝑟)(lt‘𝑟)𝑥 ↔ 0 < 𝑥)) |
17 | 16 | ifbid 4448 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1) = if( 0 < 𝑥, 1, -1)) |
18 | 10, 17 | ifbieq2d 4451 | . . . . 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 5124 | . . . 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 31099 | . . . 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 7022 | . . 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 | syl5eq 2783 | 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 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ifcif 4425 class class class wbr 5039 ↦ cmpt 5120 ‘cfv 6358 0cc0 10694 1c1 10695 -cneg 11028 Basecbs 16666 0gc0g 16898 ltcplt 17769 sgnscsgns 31098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-sgns 31099 |
This theorem is referenced by: sgnsval 31101 sgnsf 31102 |
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