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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > signspval | Structured version Visualization version GIF version |
Description: The value of the skipping 0 sign operation. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
Ref | Expression |
---|---|
signsw.p | ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) |
Ref | Expression |
---|---|
signspval | ⊢ ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → (𝑋 ⨣ 𝑌) = if(𝑌 = 0, 𝑋, 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifcl 4350 | . 2 ⊢ ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → if(𝑌 = 0, 𝑋, 𝑌) ∈ {-1, 0, 1}) | |
2 | ifeq1 4310 | . . 3 ⊢ (𝑎 = 𝑋 → if(𝑏 = 0, 𝑎, 𝑏) = if(𝑏 = 0, 𝑋, 𝑏)) | |
3 | eqeq1 2829 | . . . 4 ⊢ (𝑏 = 𝑌 → (𝑏 = 0 ↔ 𝑌 = 0)) | |
4 | id 22 | . . . 4 ⊢ (𝑏 = 𝑌 → 𝑏 = 𝑌) | |
5 | 3, 4 | ifbieq2d 4331 | . . 3 ⊢ (𝑏 = 𝑌 → if(𝑏 = 0, 𝑋, 𝑏) = if(𝑌 = 0, 𝑋, 𝑌)) |
6 | signsw.p | . . 3 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) | |
7 | 2, 5, 6 | ovmpt2g 7055 | . 2 ⊢ ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1} ∧ if(𝑌 = 0, 𝑋, 𝑌) ∈ {-1, 0, 1}) → (𝑋 ⨣ 𝑌) = if(𝑌 = 0, 𝑋, 𝑌)) |
8 | 1, 7 | mpd3an3 1592 | 1 ⊢ ((𝑋 ∈ {-1, 0, 1} ∧ 𝑌 ∈ {-1, 0, 1}) → (𝑋 ⨣ 𝑌) = if(𝑌 = 0, 𝑋, 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ifcif 4306 {ctp 4401 (class class class)co 6905 ↦ cmpt2 6907 0cc0 10252 1c1 10253 -cneg 10586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 |
This theorem is referenced by: signsw0glem 31177 signswmnd 31181 signswrid 31182 signswlid 31183 signswn0 31184 signswch 31185 |
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