Proof of Theorem sgrp2rid2
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | prid1g 4759 | . . . 4
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) | 
| 2 |  | mgm2nsgrp.s | . . . 4
⊢ 𝑆 = {𝐴, 𝐵} | 
| 3 | 1, 2 | eleqtrrdi 2851 | . . 3
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑆) | 
| 4 |  | prid2g 4760 | . . . 4
⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵}) | 
| 5 | 4, 2 | eleqtrrdi 2851 | . . 3
⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑆) | 
| 6 |  | simpl 482 | . . . . 5
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆) | 
| 7 |  | mgm2nsgrp.b | . . . . . 6
⊢
(Base‘𝑀) =
𝑆 | 
| 8 |  | sgrp2nmnd.o | . . . . . 6
⊢
(+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)) | 
| 9 |  | sgrp2nmnd.p | . . . . . 6
⊢  ⚬ =
(+g‘𝑀) | 
| 10 | 2, 7, 8, 9 | sgrp2nmndlem2 18938 | . . . . 5
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐴 ⚬ 𝐴) = 𝐴) | 
| 11 | 6, 10 | syldan 591 | . . . 4
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ 𝐴) = 𝐴) | 
| 12 |  | oveq1 7439 | . . . . . . 7
⊢ (𝐴 = 𝐵 → (𝐴 ⚬ 𝐴) = (𝐵 ⚬ 𝐴)) | 
| 13 |  | id 22 | . . . . . . 7
⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | 
| 14 | 12, 13 | eqeq12d 2752 | . . . . . 6
⊢ (𝐴 = 𝐵 → ((𝐴 ⚬ 𝐴) = 𝐴 ↔ (𝐵 ⚬ 𝐴) = 𝐵)) | 
| 15 | 11, 14 | imbitrid 244 | . . . . 5
⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵 ⚬ 𝐴) = 𝐵)) | 
| 16 |  | simprl 770 | . . . . . . 7
⊢ ((¬
𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐴 ∈ 𝑆) | 
| 17 |  | simprr 772 | . . . . . . 7
⊢ ((¬
𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐵 ∈ 𝑆) | 
| 18 |  | neqne 2947 | . . . . . . . 8
⊢ (¬
𝐴 = 𝐵 → 𝐴 ≠ 𝐵) | 
| 19 | 18 | adantr 480 | . . . . . . 7
⊢ ((¬
𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → 𝐴 ≠ 𝐵) | 
| 20 | 2, 7, 8, 9 | sgrp2nmndlem3 18939 | . . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐴) = 𝐵) | 
| 21 | 16, 17, 19, 20 | syl3anc 1372 | . . . . . 6
⊢ ((¬
𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐵 ⚬ 𝐴) = 𝐵) | 
| 22 | 21 | ex 412 | . . . . 5
⊢ (¬
𝐴 = 𝐵 → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵 ⚬ 𝐴) = 𝐵)) | 
| 23 | 15, 22 | pm2.61i 182 | . . . 4
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵 ⚬ 𝐴) = 𝐵) | 
| 24 | 2, 7, 8, 9 | sgrp2nmndlem2 18938 | . . . . 5
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ⚬ 𝐵) = 𝐴) | 
| 25 | 13, 13 | oveq12d 7450 | . . . . . . . 8
⊢ (𝐴 = 𝐵 → (𝐴 ⚬ 𝐴) = (𝐵 ⚬ 𝐵)) | 
| 26 | 25, 13 | eqeq12d 2752 | . . . . . . 7
⊢ (𝐴 = 𝐵 → ((𝐴 ⚬ 𝐴) = 𝐴 ↔ (𝐵 ⚬ 𝐵) = 𝐵)) | 
| 27 | 11, 26 | imbitrid 244 | . . . . . 6
⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵 ⚬ 𝐵) = 𝐵)) | 
| 28 | 2, 7, 8, 9 | sgrp2nmndlem3 18939 | . . . . . . . 8
⊢ ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐵) = 𝐵) | 
| 29 | 17, 17, 19, 28 | syl3anc 1372 | . . . . . . 7
⊢ ((¬
𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐵 ⚬ 𝐵) = 𝐵) | 
| 30 | 29 | ex 412 | . . . . . 6
⊢ (¬
𝐴 = 𝐵 → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵 ⚬ 𝐵) = 𝐵)) | 
| 31 | 27, 30 | pm2.61i 182 | . . . . 5
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵 ⚬ 𝐵) = 𝐵) | 
| 32 | 24, 31 | jca 511 | . . . 4
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴 ⚬ 𝐵) = 𝐴 ∧ (𝐵 ⚬ 𝐵) = 𝐵)) | 
| 33 | 11, 23, 32 | jca31 514 | . . 3
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (((𝐴 ⚬ 𝐴) = 𝐴 ∧ (𝐵 ⚬ 𝐴) = 𝐵) ∧ ((𝐴 ⚬ 𝐵) = 𝐴 ∧ (𝐵 ⚬ 𝐵) = 𝐵))) | 
| 34 | 3, 5, 33 | syl2an 596 | . 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (((𝐴 ⚬ 𝐴) = 𝐴 ∧ (𝐵 ⚬ 𝐴) = 𝐵) ∧ ((𝐴 ⚬ 𝐵) = 𝐴 ∧ (𝐵 ⚬ 𝐵) = 𝐵))) | 
| 35 | 2 | raleqi 3323 | . . . . 5
⊢
(∀𝑦 ∈
𝑆 (𝑦 ⚬ 𝑥) = 𝑦 ↔ ∀𝑦 ∈ {𝐴, 𝐵} (𝑦 ⚬ 𝑥) = 𝑦) | 
| 36 |  | oveq1 7439 | . . . . . . 7
⊢ (𝑦 = 𝐴 → (𝑦 ⚬ 𝑥) = (𝐴 ⚬ 𝑥)) | 
| 37 |  | id 22 | . . . . . . 7
⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴) | 
| 38 | 36, 37 | eqeq12d 2752 | . . . . . 6
⊢ (𝑦 = 𝐴 → ((𝑦 ⚬ 𝑥) = 𝑦 ↔ (𝐴 ⚬ 𝑥) = 𝐴)) | 
| 39 |  | oveq1 7439 | . . . . . . 7
⊢ (𝑦 = 𝐵 → (𝑦 ⚬ 𝑥) = (𝐵 ⚬ 𝑥)) | 
| 40 |  | id 22 | . . . . . . 7
⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵) | 
| 41 | 39, 40 | eqeq12d 2752 | . . . . . 6
⊢ (𝑦 = 𝐵 → ((𝑦 ⚬ 𝑥) = 𝑦 ↔ (𝐵 ⚬ 𝑥) = 𝐵)) | 
| 42 | 38, 41 | ralprg 4695 | . . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑦 ∈ {𝐴, 𝐵} (𝑦 ⚬ 𝑥) = 𝑦 ↔ ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵))) | 
| 43 | 35, 42 | bitrid 283 | . . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑦 ∈ 𝑆 (𝑦 ⚬ 𝑥) = 𝑦 ↔ ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵))) | 
| 44 | 43 | ralbidv 3177 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑦 ⚬ 𝑥) = 𝑦 ↔ ∀𝑥 ∈ 𝑆 ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵))) | 
| 45 | 2 | raleqi 3323 | . . . 4
⊢
(∀𝑥 ∈
𝑆 ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵)) | 
| 46 |  | oveq2 7440 | . . . . . . 7
⊢ (𝑥 = 𝐴 → (𝐴 ⚬ 𝑥) = (𝐴 ⚬ 𝐴)) | 
| 47 | 46 | eqeq1d 2738 | . . . . . 6
⊢ (𝑥 = 𝐴 → ((𝐴 ⚬ 𝑥) = 𝐴 ↔ (𝐴 ⚬ 𝐴) = 𝐴)) | 
| 48 |  | oveq2 7440 | . . . . . . 7
⊢ (𝑥 = 𝐴 → (𝐵 ⚬ 𝑥) = (𝐵 ⚬ 𝐴)) | 
| 49 | 48 | eqeq1d 2738 | . . . . . 6
⊢ (𝑥 = 𝐴 → ((𝐵 ⚬ 𝑥) = 𝐵 ↔ (𝐵 ⚬ 𝐴) = 𝐵)) | 
| 50 | 47, 49 | anbi12d 632 | . . . . 5
⊢ (𝑥 = 𝐴 → (((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵) ↔ ((𝐴 ⚬ 𝐴) = 𝐴 ∧ (𝐵 ⚬ 𝐴) = 𝐵))) | 
| 51 |  | oveq2 7440 | . . . . . . 7
⊢ (𝑥 = 𝐵 → (𝐴 ⚬ 𝑥) = (𝐴 ⚬ 𝐵)) | 
| 52 | 51 | eqeq1d 2738 | . . . . . 6
⊢ (𝑥 = 𝐵 → ((𝐴 ⚬ 𝑥) = 𝐴 ↔ (𝐴 ⚬ 𝐵) = 𝐴)) | 
| 53 |  | oveq2 7440 | . . . . . . 7
⊢ (𝑥 = 𝐵 → (𝐵 ⚬ 𝑥) = (𝐵 ⚬ 𝐵)) | 
| 54 | 53 | eqeq1d 2738 | . . . . . 6
⊢ (𝑥 = 𝐵 → ((𝐵 ⚬ 𝑥) = 𝐵 ↔ (𝐵 ⚬ 𝐵) = 𝐵)) | 
| 55 | 52, 54 | anbi12d 632 | . . . . 5
⊢ (𝑥 = 𝐵 → (((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵) ↔ ((𝐴 ⚬ 𝐵) = 𝐴 ∧ (𝐵 ⚬ 𝐵) = 𝐵))) | 
| 56 | 50, 55 | ralprg 4695 | . . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵} ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵) ↔ (((𝐴 ⚬ 𝐴) = 𝐴 ∧ (𝐵 ⚬ 𝐴) = 𝐵) ∧ ((𝐴 ⚬ 𝐵) = 𝐴 ∧ (𝐵 ⚬ 𝐵) = 𝐵)))) | 
| 57 | 45, 56 | bitrid 283 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑆 ((𝐴 ⚬ 𝑥) = 𝐴 ∧ (𝐵 ⚬ 𝑥) = 𝐵) ↔ (((𝐴 ⚬ 𝐴) = 𝐴 ∧ (𝐵 ⚬ 𝐴) = 𝐵) ∧ ((𝐴 ⚬ 𝐵) = 𝐴 ∧ (𝐵 ⚬ 𝐵) = 𝐵)))) | 
| 58 | 44, 57 | bitrd 279 | . 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑦 ⚬ 𝑥) = 𝑦 ↔ (((𝐴 ⚬ 𝐴) = 𝐴 ∧ (𝐵 ⚬ 𝐴) = 𝐵) ∧ ((𝐴 ⚬ 𝐵) = 𝐴 ∧ (𝐵 ⚬ 𝐵) = 𝐵)))) | 
| 59 | 34, 58 | mpbird 257 | 1
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑦 ⚬ 𝑥) = 𝑦) |