| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ismndd.c | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) | 
| 2 | 1 | 3expb 1121 | . . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵) | 
| 3 |  | simpll 767 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝜑) | 
| 4 |  | simplrl 777 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑥 ∈ 𝐵) | 
| 5 |  | simplrr 778 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑦 ∈ 𝐵) | 
| 6 |  | simpr 484 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵) | 
| 7 |  | ismndd.a | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | 
| 8 | 3, 4, 5, 6, 7 | syl13anc 1374 | . . . . . 6
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | 
| 9 | 8 | ralrimiva 3146 | . . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | 
| 10 | 2, 9 | jca 511 | . . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))) | 
| 11 | 10 | ralrimivva 3202 | . . 3
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))) | 
| 12 |  | ismndd.b | . . . 4
⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | 
| 13 |  | ismndd.p | . . . . . . . 8
⊢ (𝜑 → + =
(+g‘𝐺)) | 
| 14 | 13 | oveqd 7448 | . . . . . . 7
⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦)) | 
| 15 | 14, 12 | eleq12d 2835 | . . . . . 6
⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))) | 
| 16 |  | eqidd 2738 | . . . . . . . . 9
⊢ (𝜑 → 𝑧 = 𝑧) | 
| 17 | 13, 14, 16 | oveq123d 7452 | . . . . . . . 8
⊢ (𝜑 → ((𝑥 + 𝑦) + 𝑧) = ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧)) | 
| 18 |  | eqidd 2738 | . . . . . . . . 9
⊢ (𝜑 → 𝑥 = 𝑥) | 
| 19 | 13 | oveqd 7448 | . . . . . . . . 9
⊢ (𝜑 → (𝑦 + 𝑧) = (𝑦(+g‘𝐺)𝑧)) | 
| 20 | 13, 18, 19 | oveq123d 7452 | . . . . . . . 8
⊢ (𝜑 → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))) | 
| 21 | 17, 20 | eqeq12d 2753 | . . . . . . 7
⊢ (𝜑 → (((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ↔ ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))) | 
| 22 | 12, 21 | raleqbidv 3346 | . . . . . 6
⊢ (𝜑 → (∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))) | 
| 23 | 15, 22 | anbi12d 632 | . . . . 5
⊢ (𝜑 → (((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ↔ ((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))))) | 
| 24 | 12, 23 | raleqbidv 3346 | . . . 4
⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ↔ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))))) | 
| 25 | 12, 24 | raleqbidv 3346 | . . 3
⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))))) | 
| 26 | 11, 25 | mpbid 232 | . 2
⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))) | 
| 27 |  | ismndd.z | . . . 4
⊢ (𝜑 → 0 ∈ 𝐵) | 
| 28 | 27, 12 | eleqtrd 2843 | . . 3
⊢ (𝜑 → 0 ∈ (Base‘𝐺)) | 
| 29 | 12 | eleq2d 2827 | . . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐺))) | 
| 30 | 29 | biimpar 477 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ 𝐵) | 
| 31 | 13 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → + =
(+g‘𝐺)) | 
| 32 | 31 | oveqd 7448 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = ( 0 (+g‘𝐺)𝑥)) | 
| 33 |  | ismndd.i | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) | 
| 34 | 32, 33 | eqtr3d 2779 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑥) = 𝑥) | 
| 35 | 31 | oveqd 7448 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = (𝑥(+g‘𝐺) 0 )) | 
| 36 |  | ismndd.j | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥) | 
| 37 | 35, 36 | eqtr3d 2779 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝐺) 0 ) = 𝑥) | 
| 38 | 34, 37 | jca 511 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥)) | 
| 39 | 30, 38 | syldan 591 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥)) | 
| 40 | 39 | ralrimiva 3146 | . . 3
⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)(( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥)) | 
| 41 |  | oveq1 7438 | . . . . . 6
⊢ (𝑢 = 0 → (𝑢(+g‘𝐺)𝑥) = ( 0 (+g‘𝐺)𝑥)) | 
| 42 | 41 | eqeq1d 2739 | . . . . 5
⊢ (𝑢 = 0 → ((𝑢(+g‘𝐺)𝑥) = 𝑥 ↔ ( 0 (+g‘𝐺)𝑥) = 𝑥)) | 
| 43 | 42 | ovanraleqv 7455 | . . . 4
⊢ (𝑢 = 0 → (∀𝑥 ∈ (Base‘𝐺)((𝑢(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑢) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐺)(( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥))) | 
| 44 | 43 | rspcev 3622 | . . 3
⊢ (( 0 ∈
(Base‘𝐺) ∧
∀𝑥 ∈
(Base‘𝐺)(( 0
(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥)) → ∃𝑢 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑢(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑢) = 𝑥)) | 
| 45 | 28, 40, 44 | syl2anc 584 | . 2
⊢ (𝜑 → ∃𝑢 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑢(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑢) = 𝑥)) | 
| 46 |  | eqid 2737 | . . 3
⊢
(Base‘𝐺) =
(Base‘𝐺) | 
| 47 |  | eqid 2737 | . . 3
⊢
(+g‘𝐺) = (+g‘𝐺) | 
| 48 | 46, 47 | ismnd 18750 | . 2
⊢ (𝐺 ∈ Mnd ↔
(∀𝑥 ∈
(Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))) ∧ ∃𝑢 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑢(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑢) = 𝑥))) | 
| 49 | 26, 45, 48 | sylanbrc 583 | 1
⊢ (𝜑 → 𝐺 ∈ Mnd) |