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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 11149 | . . . 4 ⊢ 0 ∈ V | |
2 | 1 | tpid2 4731 | . . 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 33164 | . . 3 ⊢ ((𝑋 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑋 ⨣ 0) = if(0 = 0, 𝑋, 0)) |
5 | 2, 4 | mpan2 689 | . 2 ⊢ (𝑋 ∈ {-1, 0, 1} → (𝑋 ⨣ 0) = if(0 = 0, 𝑋, 0)) |
6 | eqid 2736 | . . 3 ⊢ 0 = 0 | |
7 | iftrue 4492 | . . 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 2776 | 1 ⊢ (𝑋 ∈ {-1, 0, 1} → (𝑋 ⨣ 0) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ifcif 4486 {cpr 4588 {ctp 4590 〈cop 4592 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 0cc0 11051 1c1 11052 -cneg 11386 ndxcnx 17065 Basecbs 17083 +gcplusg 17133 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-mulcl 11113 ax-i2m1 11119 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-iota 6448 df-fun 6498 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 |
This theorem is referenced by: signstfveq0 33189 |
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