Step | Hyp | Ref
| Expression |
1 | | signsw.p |
. . . . . 6
⊢ ⨣ =
(𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦
if(𝑏 = 0, 𝑎, 𝑏)) |
2 | 1 | signspval 30629 |
. . . . 5
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) →
(𝑢 ⨣ 𝑣) = if(𝑣 = 0, 𝑢, 𝑣)) |
3 | | ifcl 4130 |
. . . . 5
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) →
if(𝑣 = 0, 𝑢, 𝑣) ∈ {-1, 0, 1}) |
4 | 2, 3 | eqeltrd 2701 |
. . . 4
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) →
(𝑢 ⨣ 𝑣) ∈ {-1, 0, 1}) |
5 | 1 | signspval 30629 |
. . . . . . . . . . . . 13
⊢ (((𝑢 ⨣ 𝑣) ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤)) |
6 | 4, 5 | stoic3 1701 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) →
((𝑢 ⨣ 𝑣) ⨣ 𝑤) = if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤)) |
7 | | iftrue 4092 |
. . . . . . . . . . . 12
⊢ (𝑤 = 0 → if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤) = (𝑢 ⨣ 𝑣)) |
8 | 6, 7 | sylan9eq 2676 |
. . . . . . . . . . 11
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ 𝑣)) |
9 | 8 | adantr 481 |
. . . . . . . . . 10
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ 𝑣)) |
10 | 2 | 3adant3 1081 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) →
(𝑢 ⨣ 𝑣) = if(𝑣 = 0, 𝑢, 𝑣)) |
11 | 10 | ad2antrr 762 |
. . . . . . . . . 10
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑢 ⨣ 𝑣) = if(𝑣 = 0, 𝑢, 𝑣)) |
12 | | iftrue 4092 |
. . . . . . . . . . 11
⊢ (𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑢) |
13 | 12 | adantl 482 |
. . . . . . . . . 10
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → if(𝑣 = 0, 𝑢, 𝑣) = 𝑢) |
14 | 9, 11, 13 | 3eqtrd 2660 |
. . . . . . . . 9
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = 𝑢) |
15 | | simp1 1061 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → 𝑢 ∈ {-1, 0,
1}) |
16 | 1 | signspval 30629 |
. . . . . . . . . . . . . 14
⊢ ((𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) →
(𝑣 ⨣ 𝑤) = if(𝑤 = 0, 𝑣, 𝑤)) |
17 | 16 | 3adant1 1079 |
. . . . . . . . . . . . 13
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) →
(𝑣 ⨣ 𝑤) = if(𝑤 = 0, 𝑣, 𝑤)) |
18 | | simpl2 1065 |
. . . . . . . . . . . . . 14
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → 𝑣 ∈ {-1, 0, 1}) |
19 | | simpl3 1066 |
. . . . . . . . . . . . . 14
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → 𝑤 ∈ {-1, 0,
1}) |
20 | 18, 19 | ifclda 4120 |
. . . . . . . . . . . . 13
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) →
if(𝑤 = 0, 𝑣, 𝑤) ∈ {-1, 0, 1}) |
21 | 17, 20 | eqeltrd 2701 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) →
(𝑣 ⨣ 𝑤) ∈ {-1, 0, 1}) |
22 | 1 | signspval 30629 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ (𝑣 ⨣ 𝑤) ∈ {-1, 0, 1}) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤))) |
23 | 15, 21, 22 | syl2anc 693 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) →
(𝑢 ⨣ (𝑣 ⨣ 𝑤)) = if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤))) |
24 | 23 | ad2antrr 762 |
. . . . . . . . . 10
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤))) |
25 | | iftrue 4092 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑣) |
26 | 17, 25 | sylan9eq 2676 |
. . . . . . . . . . . 12
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → (𝑣 ⨣ 𝑤) = 𝑣) |
27 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑣 = 0 → 𝑣 = 0) |
28 | 26, 27 | sylan9eq 2676 |
. . . . . . . . . . 11
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑣 ⨣ 𝑤) = 0) |
29 | 28 | iftrued 4094 |
. . . . . . . . . 10
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤)) = 𝑢) |
30 | 24, 29 | eqtrd 2656 |
. . . . . . . . 9
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = 𝑢) |
31 | 14, 30 | eqtr4d 2659 |
. . . . . . . 8
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))) |
32 | 6 | ad2antrr 762 |
. . . . . . . . . 10
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤)) |
33 | 7 | ad2antlr 763 |
. . . . . . . . . 10
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤) = (𝑢 ⨣ 𝑣)) |
34 | 10 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 ⨣ 𝑣) = if(𝑣 = 0, 𝑢, 𝑣)) |
35 | | iffalse 4095 |
. . . . . . . . . . . 12
⊢ (¬
𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑣) |
36 | 35 | adantl 482 |
. . . . . . . . . . 11
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → if(𝑣 = 0, 𝑢, 𝑣) = 𝑣) |
37 | 34, 36 | eqtrd 2656 |
. . . . . . . . . 10
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 ⨣ 𝑣) = 𝑣) |
38 | 32, 33, 37 | 3eqtrd 2660 |
. . . . . . . . 9
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = 𝑣) |
39 | 23 | ad2antrr 762 |
. . . . . . . . . 10
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤))) |
40 | | simpr 477 |
. . . . . . . . . . . 12
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ¬ 𝑣 = 0) |
41 | 17 | ad2antrr 762 |
. . . . . . . . . . . . . 14
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑣 ⨣ 𝑤) = if(𝑤 = 0, 𝑣, 𝑤)) |
42 | 25 | ad2antlr 763 |
. . . . . . . . . . . . . 14
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → if(𝑤 = 0, 𝑣, 𝑤) = 𝑣) |
43 | 41, 42 | eqtrd 2656 |
. . . . . . . . . . . . 13
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑣 ⨣ 𝑤) = 𝑣) |
44 | 43 | eqeq1d 2624 |
. . . . . . . . . . . 12
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑣 ⨣ 𝑤) = 0 ↔ 𝑣 = 0)) |
45 | 40, 44 | mtbird 315 |
. . . . . . . . . . 11
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ¬ (𝑣 ⨣ 𝑤) = 0) |
46 | 45 | iffalsed 4097 |
. . . . . . . . . 10
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤)) = (𝑣 ⨣ 𝑤)) |
47 | 39, 46, 43 | 3eqtrd 2660 |
. . . . . . . . 9
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = 𝑣) |
48 | 38, 47 | eqtr4d 2659 |
. . . . . . . 8
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))) |
49 | 31, 48 | pm2.61dan 832 |
. . . . . . 7
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))) |
50 | | iffalse 4095 |
. . . . . . . . 9
⊢ (¬
𝑤 = 0 → if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤) = 𝑤) |
51 | 6, 50 | sylan9eq 2676 |
. . . . . . . 8
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = 𝑤) |
52 | 23 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤))) |
53 | | simpr 477 |
. . . . . . . . . . 11
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → ¬ 𝑤 = 0) |
54 | | iffalse 4095 |
. . . . . . . . . . . . 13
⊢ (¬
𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑤) |
55 | 17, 54 | sylan9eq 2676 |
. . . . . . . . . . . 12
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → (𝑣 ⨣ 𝑤) = 𝑤) |
56 | 55 | eqeq1d 2624 |
. . . . . . . . . . 11
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → ((𝑣 ⨣ 𝑤) = 0 ↔ 𝑤 = 0)) |
57 | 53, 56 | mtbird 315 |
. . . . . . . . . 10
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → ¬ (𝑣 ⨣ 𝑤) = 0) |
58 | 57 | iffalsed 4097 |
. . . . . . . . 9
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → if((𝑣 ⨣ 𝑤) = 0, 𝑢, (𝑣 ⨣ 𝑤)) = (𝑣 ⨣ 𝑤)) |
59 | 52, 58, 55 | 3eqtrd 2660 |
. . . . . . . 8
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = 𝑤) |
60 | 51, 59 | eqtr4d 2659 |
. . . . . . 7
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))) |
61 | 49, 60 | pm2.61dan 832 |
. . . . . 6
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) →
((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))) |
62 | 61 | 3expa 1265 |
. . . . 5
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) ∧ 𝑤 ∈ {-1, 0, 1}) →
((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))) |
63 | 62 | ralrimiva 2966 |
. . . 4
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) →
∀𝑤 ∈ {-1, 0, 1}
((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))) |
64 | 4, 63 | jca 554 |
. . 3
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) →
((𝑢 ⨣ 𝑣) ∈ {-1, 0, 1} ∧ ∀𝑤 ∈ {-1, 0, 1} ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤)))) |
65 | 64 | rgen2a 2977 |
. 2
⊢
∀𝑢 ∈
{-1, 0, 1}∀𝑣 ∈
{-1, 0, 1} ((𝑢 ⨣ 𝑣) ∈ {-1, 0, 1} ∧
∀𝑤 ∈ {-1, 0, 1}
((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))) |
66 | | c0ex 10034 |
. . . 4
⊢ 0 ∈
V |
67 | 66 | tpid2 4304 |
. . 3
⊢ 0 ∈
{-1, 0, 1} |
68 | 1 | signsw0glem 30630 |
. . 3
⊢
∀𝑢 ∈
{-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢) |
69 | | oveq1 6657 |
. . . . . . 7
⊢ (𝑒 = 0 → (𝑒 ⨣ 𝑢) = (0 ⨣ 𝑢)) |
70 | 69 | eqeq1d 2624 |
. . . . . 6
⊢ (𝑒 = 0 → ((𝑒 ⨣ 𝑢) = 𝑢 ↔ (0 ⨣ 𝑢) = 𝑢)) |
71 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑒 = 0 → (𝑢 ⨣ 𝑒) = (𝑢 ⨣
0)) |
72 | 71 | eqeq1d 2624 |
. . . . . 6
⊢ (𝑒 = 0 → ((𝑢 ⨣ 𝑒) = 𝑢 ↔ (𝑢 ⨣ 0) = 𝑢)) |
73 | 70, 72 | anbi12d 747 |
. . . . 5
⊢ (𝑒 = 0 → (((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢) ↔ ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢))) |
74 | 73 | ralbidv 2986 |
. . . 4
⊢ (𝑒 = 0 → (∀𝑢 ∈ {-1, 0, 1} ((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢) ↔ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢))) |
75 | 74 | rspcev 3309 |
. . 3
⊢ ((0
∈ {-1, 0, 1} ∧ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)) → ∃𝑒 ∈ {-1, 0, 1}∀𝑢 ∈ {-1, 0, 1} ((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢)) |
76 | 67, 68, 75 | mp2an 708 |
. 2
⊢
∃𝑒 ∈ {-1,
0, 1}∀𝑢 ∈ {-1,
0, 1} ((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢) |
77 | | signsw.w |
. . . 4
⊢ 𝑊 = {〈(Base‘ndx), {-1,
0, 1}〉, 〈(+g‘ndx), ⨣
〉} |
78 | 1, 77 | signswbase 30631 |
. . 3
⊢ {-1, 0,
1} = (Base‘𝑊) |
79 | 1, 77 | signswplusg 30632 |
. . 3
⊢ ⨣ =
(+g‘𝑊) |
80 | 78, 79 | ismnd 17297 |
. 2
⊢ (𝑊 ∈ Mnd ↔
(∀𝑢 ∈ {-1, 0,
1}∀𝑣 ∈ {-1, 0,
1} ((𝑢 ⨣ 𝑣) ∈ {-1, 0, 1} ∧ ∀𝑤 ∈ {-1, 0, 1} ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))) ∧ ∃𝑒 ∈ {-1, 0, 1}∀𝑢 ∈ {-1, 0, 1} ((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢))) |
81 | 65, 76, 80 | mpbir2an 955 |
1
⊢ 𝑊 ∈ Mnd |