| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | marepvfval.q | . 2
⊢ 𝑄 = (𝑁 matRepV 𝑅) | 
| 2 |  | marepvfval.b | . . . . . 6
⊢ 𝐵 = (Base‘𝐴) | 
| 3 | 2 | fvexi 6919 | . . . . 5
⊢ 𝐵 ∈ V | 
| 4 |  | marepvfval.v | . . . . . . 7
⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) | 
| 5 | 4 | ovexi 7466 | . . . . . 6
⊢ 𝑉 ∈ V | 
| 6 | 5 | a1i 11 | . . . . 5
⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝑉 ∈ V) | 
| 7 |  | mpoexga 8103 | . . . . 5
⊢ ((𝐵 ∈ V ∧ 𝑉 ∈ V) → (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) ∈ V) | 
| 8 | 3, 6, 7 | sylancr 587 | . . . 4
⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) ∈ V) | 
| 9 |  | oveq12 7441 | . . . . . . . . 9
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅)) | 
| 10 |  | marepvfval.a | . . . . . . . . 9
⊢ 𝐴 = (𝑁 Mat 𝑅) | 
| 11 | 9, 10 | eqtr4di 2794 | . . . . . . . 8
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴) | 
| 12 | 11 | fveq2d 6909 | . . . . . . 7
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴)) | 
| 13 | 12, 2 | eqtr4di 2794 | . . . . . 6
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵) | 
| 14 |  | fveq2 6905 | . . . . . . . . 9
⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | 
| 15 | 14 | adantl 481 | . . . . . . . 8
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = (Base‘𝑅)) | 
| 16 |  | simpl 482 | . . . . . . . 8
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁) | 
| 17 | 15, 16 | oveq12d 7450 | . . . . . . 7
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((Base‘𝑟) ↑m 𝑛) = ((Base‘𝑅) ↑m 𝑁)) | 
| 18 | 17, 4 | eqtr4di 2794 | . . . . . 6
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((Base‘𝑟) ↑m 𝑛) = 𝑉) | 
| 19 |  | eqidd 2737 | . . . . . . . 8
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)) = if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))) | 
| 20 | 16, 16, 19 | mpoeq123dv 7509 | . . . . . . 7
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))) | 
| 21 | 16, 20 | mpteq12dv 5232 | . . . . . 6
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))) = (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) | 
| 22 | 13, 18, 21 | mpoeq123dv 7509 | . . . . 5
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))))) | 
| 23 |  | df-marepv 22566 | . . . . 5
⊢  matRepV
= (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))))) | 
| 24 | 22, 23 | ovmpoga 7588 | . . . 4
⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V ∧ (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) ∈ V) → (𝑁 matRepV 𝑅) = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))))) | 
| 25 | 8, 24 | mpd3an3 1463 | . . 3
⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRepV 𝑅) = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))))) | 
| 26 | 23 | mpondm0 7674 | . . . 4
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRepV 𝑅) = ∅) | 
| 27 | 10 | fveq2i 6908 | . . . . . . . 8
⊢
(Base‘𝐴) =
(Base‘(𝑁 Mat 𝑅)) | 
| 28 | 2, 27 | eqtri 2764 | . . . . . . 7
⊢ 𝐵 = (Base‘(𝑁 Mat 𝑅)) | 
| 29 |  | matbas0pc 22414 | . . . . . . 7
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) →
(Base‘(𝑁 Mat 𝑅)) = ∅) | 
| 30 | 28, 29 | eqtrid 2788 | . . . . . 6
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅) | 
| 31 | 30 | orcd 873 | . . . . 5
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝐵 = ∅ ∨ 𝑉 = ∅)) | 
| 32 |  | 0mpo0 7517 | . . . . 5
⊢ ((𝐵 = ∅ ∨ 𝑉 = ∅) → (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) = ∅) | 
| 33 | 31, 32 | syl 17 | . . . 4
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) = ∅) | 
| 34 | 26, 33 | eqtr4d 2779 | . . 3
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 matRepV 𝑅) = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))))) | 
| 35 | 25, 34 | pm2.61i 182 | . 2
⊢ (𝑁 matRepV 𝑅) = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) | 
| 36 | 1, 35 | eqtri 2764 | 1
⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) |