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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 11280 | . . . 4 ⊢ 0 ∈ V | |
2 | 1 | tpid2 4795 | . . 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 34521 | . . 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 2734 | . . 3 ⊢ 0 = 0 | |
7 | iftrue 4554 | . . 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 2774 | 1 ⊢ (𝑋 ∈ {-1, 0, 1} → (𝑋 ⨣ 0) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 ifcif 4548 {cpr 4650 {ctp 4652 〈cop 4654 ‘cfv 6572 (class class class)co 7445 ∈ cmpo 7447 0cc0 11180 1c1 11181 -cneg 11517 ndxcnx 17235 Basecbs 17253 +gcplusg 17306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-mulcl 11242 ax-i2m1 11248 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-iota 6524 df-fun 6574 df-fv 6580 df-ov 7448 df-oprab 7449 df-mpo 7450 |
This theorem is referenced by: signstfveq0 34546 |
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