![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 11210 | . . . . . 6 ⊢ 0 ∈ V | |
2 | 1 | tpid2 4774 | . . . . 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 33632 | . . . . 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 4534 | . . . . . 6 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 0) | |
7 | id 22 | . . . . . 6 ⊢ (𝑢 = 0 → 𝑢 = 0) | |
8 | 6, 7 | eqtr4d 2775 | . . . . 5 ⊢ (𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢) |
9 | iffalse 4537 | . . . . 5 ⊢ (¬ 𝑢 = 0 → if(𝑢 = 0, 0, 𝑢) = 𝑢) | |
10 | 8, 9 | pm2.61i 182 | . . . 4 ⊢ if(𝑢 = 0, 0, 𝑢) = 𝑢 |
11 | 5, 10 | eqtrdi 2788 | . . 3 ⊢ (𝑢 ∈ {-1, 0, 1} → (0 ⨣ 𝑢) = 𝑢) |
12 | 3 | signspval 33632 | . . . . 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 2732 | . . . . 5 ⊢ 0 = 0 | |
15 | 14 | iftruei 4535 | . . . 4 ⊢ if(0 = 0, 𝑢, 0) = 𝑢 |
16 | 13, 15 | eqtrdi 2788 | . . 3 ⊢ (𝑢 ∈ {-1, 0, 1} → (𝑢 ⨣ 0) = 𝑢) |
17 | 11, 16 | jca 512 | . 2 ⊢ (𝑢 ∈ {-1, 0, 1} → ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)) |
18 | 17 | rgen 3063 | 1 ⊢ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ifcif 4528 {ctp 4632 (class class class)co 7411 ∈ cmpo 7413 0cc0 11112 1c1 11113 -cneg 11447 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-mulcl 11174 ax-i2m1 11180 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 |
This theorem is referenced by: signsw0g 33636 signswmnd 33637 |
Copyright terms: Public domain | W3C validator |