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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 10638 | . . . 4 ⊢ 0 ∈ V | |
2 | 1 | tpid2 4709 | . . 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 31826 | . . 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 2824 | . . 3 ⊢ 0 = 0 | |
7 | iftrue 4476 | . . 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 2859 | 1 ⊢ (𝑋 ∈ {-1, 0, 1} → (𝑋 ⨣ 0) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ifcif 4470 {cpr 4572 {ctp 4574 〈cop 4576 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 0cc0 10540 1c1 10541 -cneg 10874 ndxcnx 16483 Basecbs 16486 +gcplusg 16568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-mulcl 10602 ax-i2m1 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 |
This theorem is referenced by: signstfveq0 31851 |
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