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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > signswn0 | 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 |
---|---|
signswn0 | ⊢ (((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) ∧ 𝑋 ≠ 0) → (𝑋 ⨣ 𝑌) ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | signsw.p | . . . 4 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) | |
2 | 1 | signspval 31500 | . . 3 ⊢ ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → (𝑋 ⨣ 𝑌) = if(𝑌 = 0, 𝑋, 𝑌)) |
3 | 2 | adantr 473 | . 2 ⊢ (((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) ∧ 𝑋 ≠ 0) → (𝑋 ⨣ 𝑌) = if(𝑌 = 0, 𝑋, 𝑌)) |
4 | neeq1 3022 | . . 3 ⊢ (𝑋 = if(𝑌 = 0, 𝑋, 𝑌) → (𝑋 ≠ 0 ↔ if(𝑌 = 0, 𝑋, 𝑌) ≠ 0)) | |
5 | neeq1 3022 | . . 3 ⊢ (𝑌 = if(𝑌 = 0, 𝑋, 𝑌) → (𝑌 ≠ 0 ↔ if(𝑌 = 0, 𝑋, 𝑌) ≠ 0)) | |
6 | simplr 757 | . . 3 ⊢ ((((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) ∧ 𝑋 ≠ 0) ∧ 𝑌 = 0) → 𝑋 ≠ 0) | |
7 | simpr 477 | . . . 4 ⊢ ((((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) ∧ 𝑋 ≠ 0) ∧ ¬ 𝑌 = 0) → ¬ 𝑌 = 0) | |
8 | 7 | neqned 2967 | . . 3 ⊢ ((((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) ∧ 𝑋 ≠ 0) ∧ ¬ 𝑌 = 0) → 𝑌 ≠ 0) |
9 | 4, 5, 6, 8 | ifbothda 4381 | . 2 ⊢ (((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) ∧ 𝑋 ≠ 0) → if(𝑌 = 0, 𝑋, 𝑌) ≠ 0) |
10 | 3, 9 | eqnetrd 3027 | 1 ⊢ (((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) ∧ 𝑋 ≠ 0) → (𝑋 ⨣ 𝑌) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 ifcif 4344 {cpr 4437 {ctp 4439 〈cop 4441 ‘cfv 6185 (class class class)co 6974 ∈ cmpo 6976 0cc0 10333 1c1 10334 -cneg 10669 ndxcnx 16334 Basecbs 16337 +gcplusg 16419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-iota 6149 df-fun 6187 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 |
This theorem is referenced by: signstfvneq0 31521 |
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