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Mirrors > Home > MPE Home > Th. List > Mathboxes > signsw0glem | Structured version Visualization version GIF version |
Description: Neutral element property of ⨣. (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 |
---|---|
signsw0glem | ⊢ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 10629 | . . . . . 6 ⊢ 0 ∈ V | |
2 | 1 | tpid2 4700 | . . . . 5 ⊢ 0 ∈ {-1, 0, 1} |
3 | signsw.p | . . . . . 6 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) | |
4 | 3 | signspval 31817 | . . . . 5 ⊢ ((0 ∈ {-1, 0, 1} ∧ 𝑢 ∈ {-1, 0, 1}) → (0 ⨣ 𝑢) = if(𝑢 = 0, 0, 𝑢)) |
5 | 2, 4 | mpan 688 | . . . 4 ⊢ (𝑢 ∈ {-1, 0, 1} → (0 ⨣ 𝑢) = if(𝑢 = 0, 0, 𝑢)) |
6 | iftrue 4473 | . . . . . 6 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 0) | |
7 | id 22 | . . . . . 6 ⊢ (𝑢 = 0 → 𝑢 = 0) | |
8 | 6, 7 | eqtr4d 2859 | . . . . 5 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢) |
9 | iffalse 4476 | . . . . 5 ⊢ (¬ 𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢) | |
10 | 8, 9 | pm2.61i 184 | . . . 4 ⊢ if(𝑢 = 0, 0, 𝑢) = 𝑢 |
11 | 5, 10 | syl6eq 2872 | . . 3 ⊢ (𝑢 ∈ {-1, 0, 1} → (0 ⨣ 𝑢) = 𝑢) |
12 | 3 | signspval 31817 | . . . . 5 ⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑢 ⨣ 0) = if(0 = 0, 𝑢, 0)) |
13 | 2, 12 | mpan2 689 | . . . 4 ⊢ (𝑢 ∈ {-1, 0, 1} → (𝑢 ⨣ 0) = if(0 = 0, 𝑢, 0)) |
14 | eqid 2821 | . . . . 5 ⊢ 0 = 0 | |
15 | 14 | iftruei 4474 | . . . 4 ⊢ if(0 = 0, 𝑢, 0) = 𝑢 |
16 | 13, 15 | syl6eq 2872 | . . 3 ⊢ (𝑢 ∈ {-1, 0, 1} → (𝑢 ⨣ 0) = 𝑢) |
17 | 11, 16 | jca 514 | . 2 ⊢ (𝑢 ∈ {-1, 0, 1} → ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)) |
18 | 17 | rgen 3148 | 1 ⊢ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ifcif 4467 {ctp 4565 (class class class)co 7150 ∈ cmpo 7152 0cc0 10531 1c1 10532 -cneg 10865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-mulcl 10593 ax-i2m1 10599 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 |
This theorem is referenced by: signsw0g 31821 signswmnd 31822 |
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