Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 3512 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
3 | fveq2 6670 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
4 | sgnsval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
5 | 3, 4 | syl6eqr 2874 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
6 | fveq2 6670 | . . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) | |
7 | sgnsval.0 | . . . . . . . . 9 ⊢ 0 = (0g‘𝑅) | |
8 | 6, 7 | syl6eqr 2874 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) |
9 | 8 | adantr 483 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (0g‘𝑟) = 0 ) |
10 | 9 | eqeq2d 2832 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (𝑥 = (0g‘𝑟) ↔ 𝑥 = 0 )) |
11 | fveq2 6670 | . . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (lt‘𝑟) = (lt‘𝑅)) | |
12 | sgnsval.l | . . . . . . . . . 10 ⊢ < = (lt‘𝑅) | |
13 | 11, 12 | syl6eqr 2874 | . . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (lt‘𝑟) = < ) |
14 | 13 | adantr 483 | . . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (lt‘𝑟) = < ) |
15 | eqidd 2822 | . . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → 𝑥 = 𝑥) | |
16 | 9, 14, 15 | breq123d 5080 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → ((0g‘𝑟)(lt‘𝑟)𝑥 ↔ 0 < 𝑥)) |
17 | 16 | ifbid 4489 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1) = if( 0 < 𝑥, 1, -1)) |
18 | 10, 17 | ifbieq2d 4492 | . . . . 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 5150 | . . . 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 30801 | . . . 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 6990 | . . 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 2868 | 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 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ifcif 4467 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 0cc0 10537 1c1 10538 -cneg 10871 Basecbs 16483 0gc0g 16713 ltcplt 17551 sgnscsgns 30800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-sgns 30801 |
This theorem is referenced by: sgnsval 30803 sgnsf 30804 |
Copyright terms: Public domain | W3C validator |