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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > signswrid | Structured version Visualization version GIF version |
Description: The zero-skipping operation propagages nonzeros. (Contributed by Thierry Arnoux, 11-Oct-2018.) |
Ref | Expression |
---|---|
signsw.p | ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) |
signsw.w | ⊢ 𝑊 = {〈(Base‘ndx), {-1, 0, 1}〉, 〈(+g‘ndx), ⨣ 〉} |
Ref | Expression |
---|---|
signswrid | ⊢ (𝑋 ∈ {-1, 0, 1} → (𝑋 ⨣ 0) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 10624 | . . . 4 ⊢ 0 ∈ V | |
2 | 1 | tpid2 4666 | . . 3 ⊢ 0 ∈ {-1, 0, 1} |
3 | signsw.p | . . . 4 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) | |
4 | 3 | signspval 31932 | . . 3 ⊢ ((𝑋 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑋 ⨣ 0) = if(0 = 0, 𝑋, 0)) |
5 | 2, 4 | mpan2 690 | . 2 ⊢ (𝑋 ∈ {-1, 0, 1} → (𝑋 ⨣ 0) = if(0 = 0, 𝑋, 0)) |
6 | eqid 2798 | . . 3 ⊢ 0 = 0 | |
7 | iftrue 4431 | . . 3 ⊢ (0 = 0 → if(0 = 0, 𝑋, 0) = 𝑋) | |
8 | 6, 7 | mp1i 13 | . 2 ⊢ (𝑋 ∈ {-1, 0, 1} → if(0 = 0, 𝑋, 0) = 𝑋) |
9 | 5, 8 | eqtrd 2833 | 1 ⊢ (𝑋 ∈ {-1, 0, 1} → (𝑋 ⨣ 0) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ifcif 4425 {cpr 4527 {ctp 4529 〈cop 4531 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 0cc0 10526 1c1 10527 -cneg 10860 ndxcnx 16472 Basecbs 16475 +gcplusg 16557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-mulcl 10588 ax-i2m1 10594 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 |
This theorem is referenced by: signstfveq0 31957 |
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