Mathbox for Thierry Arnoux |
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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 30729 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) |
6 | 5 | adantr 481 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) |
7 | eqeq1 2822 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 = 0 ↔ 𝑋 = 0 )) | |
8 | breq2 5061 | . . . . 5 ⊢ (𝑥 = 𝑋 → ( 0 < 𝑥 ↔ 0 < 𝑋)) | |
9 | 8 | ifbid 4485 | . . . 4 ⊢ (𝑥 = 𝑋 → if( 0 < 𝑥, 1, -1) = if( 0 < 𝑋, 1, -1)) |
10 | 7, 9 | ifbieq2d 4488 | . . 3 ⊢ (𝑥 = 𝑋 → if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1))) |
11 | 10 | adantl 482 | . 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 485 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
13 | c0ex 10623 | . . . 4 ⊢ 0 ∈ V | |
14 | 13 | a1i 11 | . . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 = 0 ) → 0 ∈ V) |
15 | 1ex 10625 | . . . . 5 ⊢ 1 ∈ V | |
16 | 15 | a1i 11 | . . . 4 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑋 = 0 ) ∧ 0 < 𝑋) → 1 ∈ V) |
17 | negex 10872 | . . . . 5 ⊢ -1 ∈ V | |
18 | 17 | a1i 11 | . . . 4 ⊢ ((((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑋 = 0 ) ∧ ¬ 0 < 𝑋) → -1 ∈ V) |
19 | 16, 18 | ifclda 4497 | . . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑋 = 0 ) → if( 0 < 𝑋, 1, -1) ∈ V) |
20 | 14, 19 | ifclda 4497 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)) ∈ V) |
21 | 6, 11, 12, 20 | fvmptd 6767 | 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 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ifcif 4463 class class class wbr 5057 ↦ cmpt 5137 ‘cfv 6348 0cc0 10525 1c1 10526 -cneg 10859 Basecbs 16471 0gc0g 16701 ltcplt 17539 sgnscsgns 30727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-mulcl 10587 ax-i2m1 10593 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-neg 10861 df-sgns 30728 |
This theorem is referenced by: (None) |
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