| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | signsw.p | . . . . . 6
⊢  ⨣ =
(𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦
if(𝑏 = 0, 𝑎, 𝑏)) | 
| 2 | 1 | signspval 34567 | . . . . 5
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) →
(𝑢 ⨣ 𝑣) = if(𝑣 = 0, 𝑢, 𝑣)) | 
| 3 |  | ifcl 4571 | . . . . 5
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) →
if(𝑣 = 0, 𝑢, 𝑣) ∈ {-1, 0, 1}) | 
| 4 | 2, 3 | eqeltrd 2841 | . . . 4
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) →
(𝑢 ⨣ 𝑣) ∈ {-1, 0, 1}) | 
| 5 | 1 | signspval 34567 | . . . . . . . . . . . . 13
⊢ (((𝑢 ⨣ 𝑣) ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤)) | 
| 6 | 4, 5 | stoic3 1776 | . . . . . . . . . . . 12
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) →
((𝑢 ⨣ 𝑣) ⨣ 𝑤) = if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤)) | 
| 7 |  | iftrue 4531 | . . . . . . . . . . . 12
⊢ (𝑤 = 0 → if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤) = (𝑢 ⨣ 𝑣)) | 
| 8 | 6, 7 | sylan9eq 2797 | . . . . . . . . . . 11
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ 𝑣)) | 
| 9 | 8 | adantr 480 | . . . . . . . . . 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 1133 | . . . . . . . . . . 11
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) →
(𝑢 ⨣ 𝑣) = if(𝑣 = 0, 𝑢, 𝑣)) | 
| 11 | 10 | ad2antrr 726 | . . . . . . . . . 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 4531 | . . . . . . . . . . 11
⊢ (𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑢) | 
| 13 | 12 | adantl 481 | . . . . . . . . . 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 2781 | . . . . . . . . 9
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = 𝑢) | 
| 15 |  | simp1 1137 | . . . . . . . . . . . 12
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) → 𝑢 ∈ {-1, 0,
1}) | 
| 16 | 1 | signspval 34567 | . . . . . . . . . . . . . 14
⊢ ((𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) →
(𝑣 ⨣ 𝑤) = if(𝑤 = 0, 𝑣, 𝑤)) | 
| 17 | 16 | 3adant1 1131 | . . . . . . . . . . . . 13
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) →
(𝑣 ⨣ 𝑤) = if(𝑤 = 0, 𝑣, 𝑤)) | 
| 18 |  | simpl2 1193 | . . . . . . . . . . . . . 14
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → 𝑣 ∈ {-1, 0, 1}) | 
| 19 |  | simpl3 1194 | . . . . . . . . . . . . . 14
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → 𝑤 ∈ {-1, 0,
1}) | 
| 20 | 18, 19 | ifclda 4561 | . . . . . . . . . . . . 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 2841 | . . . . . . . . . . . 12
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) →
(𝑣 ⨣ 𝑤) ∈ {-1, 0, 1}) | 
| 22 | 1 | signspval 34567 | . . . . . . . . . . . 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 584 | . . . . . . . . . . 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 726 | . . . . . . . . . 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 4531 | . . . . . . . . . . . . 13
⊢ (𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑣) | 
| 26 | 17, 25 | sylan9eq 2797 | . . . . . . . . . . . 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 2797 | . . . . . . . . . . 11
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ 𝑣 = 0) → (𝑣 ⨣ 𝑤) = 0) | 
| 29 | 28 | iftrued 4533 | . . . . . . . . . 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 2777 | . . . . . . . . 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 2780 | . . . . . . . 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 726 | . . . . . . . . . 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 727 | . . . . . . . . . 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 726 | . . . . . . . . . . 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 4534 | . . . . . . . . . . . 12
⊢ (¬
𝑣 = 0 → if(𝑣 = 0, 𝑢, 𝑣) = 𝑣) | 
| 36 | 35 | adantl 481 | . . . . . . . . . . 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 2777 | . . . . . . . . . 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 2781 | . . . . . . . . 9
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = 𝑣) | 
| 39 | 23 | ad2antrr 726 | . . . . . . . . . 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 484 | . . . . . . . . . . . 12
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ¬ 𝑣 = 0) | 
| 41 | 17 | ad2antrr 726 | . . . . . . . . . . . . . 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 727 | . . . . . . . . . . . . . 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 2777 | . . . . . . . . . . . . 13
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → (𝑣 ⨣ 𝑤) = 𝑣) | 
| 44 | 43 | eqeq1d 2739 | . . . . . . . . . . . 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 325 | . . . . . . . . . . 11
⊢ ((((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) ∧ ¬ 𝑣 = 0) → ¬ (𝑣 ⨣ 𝑤) = 0) | 
| 46 | 45 | iffalsed 4536 | . . . . . . . . . 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 2781 | . . . . . . . . 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 2780 | . . . . . . . 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 813 | . . . . . . 7
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ 𝑤 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))) | 
| 50 |  | iffalse 4534 | . . . . . . . . 9
⊢ (¬
𝑤 = 0 → if(𝑤 = 0, (𝑢 ⨣ 𝑣), 𝑤) = 𝑤) | 
| 51 | 6, 50 | sylan9eq 2797 | . . . . . . . 8
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → ((𝑢 ⨣ 𝑣) ⨣ 𝑤) = 𝑤) | 
| 52 | 23 | adantr 480 | . . . . . . . . 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 484 | . . . . . . . . . . 11
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → ¬ 𝑤 = 0) | 
| 54 |  | iffalse 4534 | . . . . . . . . . . . . 13
⊢ (¬
𝑤 = 0 → if(𝑤 = 0, 𝑣, 𝑤) = 𝑤) | 
| 55 | 17, 54 | sylan9eq 2797 | . . . . . . . . . . . 12
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → (𝑣 ⨣ 𝑤) = 𝑤) | 
| 56 | 55 | eqeq1d 2739 | . . . . . . . . . . 11
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → ((𝑣 ⨣ 𝑤) = 0 ↔ 𝑤 = 0)) | 
| 57 | 53, 56 | mtbird 325 | . . . . . . . . . 10
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → ¬ (𝑣 ⨣ 𝑤) = 0) | 
| 58 | 57 | iffalsed 4536 | . . . . . . . . 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 2781 | . . . . . . . 8
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) ∧ ¬
𝑤 = 0) → (𝑢 ⨣ (𝑣 ⨣ 𝑤)) = 𝑤) | 
| 60 | 51, 59 | eqtr4d 2780 | . . . . . . 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 813 | . . . . . 6
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1} ∧ 𝑤 ∈ {-1, 0, 1}) →
((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))) | 
| 62 | 61 | 3expa 1119 | . . . . 5
⊢ (((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) ∧ 𝑤 ∈ {-1, 0, 1}) →
((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))) | 
| 63 | 62 | ralrimiva 3146 | . . . 4
⊢ ((𝑢 ∈ {-1, 0, 1} ∧ 𝑣 ∈ {-1, 0, 1}) →
∀𝑤 ∈ {-1, 0, 1}
((𝑢 ⨣ 𝑣) ⨣ 𝑤) = (𝑢 ⨣ (𝑣 ⨣ 𝑤))) | 
| 64 | 4, 63 | jca 511 | . . 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 | rgen2 3199 | . 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 11255 | . . . 4
⊢ 0 ∈
V | 
| 67 | 66 | tpid2 4770 | . . 3
⊢ 0 ∈
{-1, 0, 1} | 
| 68 | 1 | signsw0glem 34568 | . . 3
⊢
∀𝑢 ∈
{-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢) | 
| 69 |  | oveq1 7438 | . . . . . 6
⊢ (𝑒 = 0 → (𝑒 ⨣ 𝑢) = (0 ⨣ 𝑢)) | 
| 70 | 69 | eqeq1d 2739 | . . . . 5
⊢ (𝑒 = 0 → ((𝑒 ⨣ 𝑢) = 𝑢 ↔ (0 ⨣ 𝑢) = 𝑢)) | 
| 71 | 70 | ovanraleqv 7455 | . . . 4
⊢ (𝑒 = 0 → (∀𝑢 ∈ {-1, 0, 1} ((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢) ↔ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢))) | 
| 72 | 71 | rspcev 3622 | . . 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)) | 
| 73 | 67, 68, 72 | mp2an 692 | . 2
⊢
∃𝑒 ∈ {-1,
0, 1}∀𝑢 ∈ {-1,
0, 1} ((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢) | 
| 74 |  | signsw.w | . . . 4
⊢ 𝑊 = {〈(Base‘ndx), {-1,
0, 1}〉, 〈(+g‘ndx), ⨣
〉} | 
| 75 | 1, 74 | signswbase 34569 | . . 3
⊢ {-1, 0,
1} = (Base‘𝑊) | 
| 76 | 1, 74 | signswplusg 34570 | . . 3
⊢  ⨣ =
(+g‘𝑊) | 
| 77 | 75, 76 | ismnd 18750 | . 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))) | 
| 78 | 65, 73, 77 | mpbir2an 711 | 1
⊢ 𝑊 ∈ Mnd |