Proof of Theorem mgpress
| Step | Hyp | Ref
 | Expression | 
| 1 |   | mgpress.2 | 
. . . . 5
⊢ 𝑀 = (mulGrp‘𝑅) | 
| 2 |   | eqid 2196 | 
. . . . 5
⊢
(.r‘𝑅) = (.r‘𝑅) | 
| 3 | 1, 2 | mgpvalg 13479 | 
. . . 4
⊢ (𝑅 ∈ 𝑉 → 𝑀 = (𝑅 sSet 〈(+g‘ndx),
(.r‘𝑅)〉)) | 
| 4 | 3 | adantr 276 | 
. . 3
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝑀 = (𝑅 sSet 〈(+g‘ndx),
(.r‘𝑅)〉)) | 
| 5 | 4 | oveq1d 5937 | 
. 2
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑅))〉) = ((𝑅 sSet 〈(+g‘ndx),
(.r‘𝑅)〉) sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑅))〉)) | 
| 6 | 1 | mgpex 13481 | 
. . . 4
⊢ (𝑅 ∈ 𝑉 → 𝑀 ∈ V) | 
| 7 |   | ressvalsets 12742 | 
. . . 4
⊢ ((𝑀 ∈ V ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (𝑀 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑀))〉)) | 
| 8 | 6, 7 | sylan 283 | 
. . 3
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (𝑀 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑀))〉)) | 
| 9 |   | eqid 2196 | 
. . . . . . . 8
⊢
(Base‘𝑅) =
(Base‘𝑅) | 
| 10 | 1, 9 | mgpbasg 13482 | 
. . . . . . 7
⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑀)) | 
| 11 | 10 | adantr 276 | 
. . . . . 6
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (Base‘𝑅) = (Base‘𝑀)) | 
| 12 | 11 | ineq2d 3364 | 
. . . . 5
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐴 ∩ (Base‘𝑅)) = (𝐴 ∩ (Base‘𝑀))) | 
| 13 | 12 | opeq2d 3815 | 
. . . 4
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑅))〉 =
〈(Base‘ndx), (𝐴
∩ (Base‘𝑀))〉) | 
| 14 | 13 | oveq2d 5938 | 
. . 3
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑅))〉) = (𝑀 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑀))〉)) | 
| 15 | 8, 14 | eqtr4d 2232 | 
. 2
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (𝑀 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑅))〉)) | 
| 16 |   | mgpress.1 | 
. . . . 5
⊢ 𝑆 = (𝑅 ↾s 𝐴) | 
| 17 |   | ressvalsets 12742 | 
. . . . 5
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑅 ↾s 𝐴) = (𝑅 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑅))〉)) | 
| 18 | 16, 17 | eqtrid 2241 | 
. . . 4
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝑆 = (𝑅 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑅))〉)) | 
| 19 | 16, 2 | ressmulrg 12822 | 
. . . . . . 7
⊢ ((𝐴 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (.r‘𝑅) = (.r‘𝑆)) | 
| 20 | 19 | ancoms 268 | 
. . . . . 6
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (.r‘𝑅) = (.r‘𝑆)) | 
| 21 | 20 | eqcomd 2202 | 
. . . . 5
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (.r‘𝑆) = (.r‘𝑅)) | 
| 22 | 21 | opeq2d 3815 | 
. . . 4
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 〈(+g‘ndx),
(.r‘𝑆)〉 = 〈(+g‘ndx),
(.r‘𝑅)〉) | 
| 23 | 18, 22 | oveq12d 5940 | 
. . 3
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑆 sSet 〈(+g‘ndx),
(.r‘𝑆)〉) = ((𝑅 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑅))〉) sSet
〈(+g‘ndx), (.r‘𝑅)〉)) | 
| 24 |   | ressex 12743 | 
. . . . 5
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑅 ↾s 𝐴) ∈ V) | 
| 25 | 16, 24 | eqeltrid 2283 | 
. . . 4
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝑆 ∈ V) | 
| 26 |   | eqid 2196 | 
. . . . 5
⊢
(mulGrp‘𝑆) =
(mulGrp‘𝑆) | 
| 27 |   | eqid 2196 | 
. . . . 5
⊢
(.r‘𝑆) = (.r‘𝑆) | 
| 28 | 26, 27 | mgpvalg 13479 | 
. . . 4
⊢ (𝑆 ∈ V →
(mulGrp‘𝑆) = (𝑆 sSet
〈(+g‘ndx), (.r‘𝑆)〉)) | 
| 29 | 25, 28 | syl 14 | 
. . 3
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (mulGrp‘𝑆) = (𝑆 sSet 〈(+g‘ndx),
(.r‘𝑆)〉)) | 
| 30 |   | plusgslid 12790 | 
. . . . . 6
⊢
(+g = Slot (+g‘ndx) ∧
(+g‘ndx) ∈ ℕ) | 
| 31 | 30 | simpri 113 | 
. . . . 5
⊢
(+g‘ndx) ∈ ℕ | 
| 32 | 31 | a1i 9 | 
. . . 4
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (+g‘ndx) ∈
ℕ) | 
| 33 |   | basendxnn 12734 | 
. . . . 5
⊢
(Base‘ndx) ∈ ℕ | 
| 34 | 33 | a1i 9 | 
. . . 4
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (Base‘ndx) ∈
ℕ) | 
| 35 |   | simpl 109 | 
. . . 4
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝑅 ∈ 𝑉) | 
| 36 |   | basendxnplusgndx 12802 | 
. . . . . 6
⊢
(Base‘ndx) ≠ (+g‘ndx) | 
| 37 | 36 | necomi 2452 | 
. . . . 5
⊢
(+g‘ndx) ≠ (Base‘ndx) | 
| 38 | 37 | a1i 9 | 
. . . 4
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (+g‘ndx) ≠
(Base‘ndx)) | 
| 39 |   | mulrslid 12809 | 
. . . . . 6
⊢
(.r = Slot (.r‘ndx) ∧
(.r‘ndx) ∈ ℕ) | 
| 40 | 39 | slotex 12705 | 
. . . . 5
⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) ∈ V) | 
| 41 | 40 | adantr 276 | 
. . . 4
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (.r‘𝑅) ∈ V) | 
| 42 |   | inex1g 4169 | 
. . . . 5
⊢ (𝐴 ∈ 𝑊 → (𝐴 ∩ (Base‘𝑅)) ∈ V) | 
| 43 | 42 | adantl 277 | 
. . . 4
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐴 ∩ (Base‘𝑅)) ∈ V) | 
| 44 | 32, 34, 35, 38, 41, 43 | setscomd 12719 | 
. . 3
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝑅 sSet 〈(+g‘ndx),
(.r‘𝑅)〉) sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑅))〉) = ((𝑅 sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑅))〉) sSet
〈(+g‘ndx), (.r‘𝑅)〉)) | 
| 45 | 23, 29, 44 | 3eqtr4d 2239 | 
. 2
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (mulGrp‘𝑆) = ((𝑅 sSet 〈(+g‘ndx),
(.r‘𝑅)〉) sSet 〈(Base‘ndx), (𝐴 ∩ (Base‘𝑅))〉)) | 
| 46 | 5, 15, 45 | 3eqtr4d 2239 | 
1
⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (mulGrp‘𝑆)) |