Step | Hyp | Ref
| Expression |
1 | | submafval.q |
. 2
⊢ 𝑄 = (𝑁 subMat 𝑅) |
2 | | oveq12 7222 |
. . . . . . . 8
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅)) |
3 | | submafval.a |
. . . . . . . 8
⊢ 𝐴 = (𝑁 Mat 𝑅) |
4 | 2, 3 | eqtr4di 2796 |
. . . . . . 7
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴) |
5 | 4 | fveq2d 6721 |
. . . . . 6
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴)) |
6 | | submafval.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝐴) |
7 | 5, 6 | eqtr4di 2796 |
. . . . 5
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵) |
8 | | simpl 486 |
. . . . . 6
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁) |
9 | | difeq1 4030 |
. . . . . . . 8
⊢ (𝑛 = 𝑁 → (𝑛 ∖ {𝑘}) = (𝑁 ∖ {𝑘})) |
10 | 9 | adantr 484 |
. . . . . . 7
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 ∖ {𝑘}) = (𝑁 ∖ {𝑘})) |
11 | | difeq1 4030 |
. . . . . . . 8
⊢ (𝑛 = 𝑁 → (𝑛 ∖ {𝑙}) = (𝑁 ∖ {𝑙})) |
12 | 11 | adantr 484 |
. . . . . . 7
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 ∖ {𝑙}) = (𝑁 ∖ {𝑙})) |
13 | | eqidd 2738 |
. . . . . . 7
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑖𝑚𝑗) = (𝑖𝑚𝑗)) |
14 | 10, 12, 13 | mpoeq123dv 7286 |
. . . . . 6
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)) = (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))) |
15 | 8, 8, 14 | mpoeq123dv 7286 |
. . . . 5
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))) |
16 | 7, 15 | mpteq12dv 5140 |
. . . 4
⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))) = (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))) |
17 | | df-subma 21474 |
. . . 4
⊢ subMat =
(𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))) |
18 | 6 | fvexi 6731 |
. . . . 5
⊢ 𝐵 ∈ V |
19 | 18 | mptex 7039 |
. . . 4
⊢ (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))) ∈ V |
20 | 16, 17, 19 | ovmpoa 7364 |
. . 3
⊢ ((𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 subMat 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))) |
21 | 17 | mpondm0 7446 |
. . . . 5
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 subMat 𝑅) = ∅) |
22 | | mpt0 6520 |
. . . . 5
⊢ (𝑚 ∈ ∅ ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))) = ∅ |
23 | 21, 22 | eqtr4di 2796 |
. . . 4
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 subMat 𝑅) = (𝑚 ∈ ∅ ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))) |
24 | 3 | fveq2i 6720 |
. . . . . . 7
⊢
(Base‘𝐴) =
(Base‘(𝑁 Mat 𝑅)) |
25 | 6, 24 | eqtri 2765 |
. . . . . 6
⊢ 𝐵 = (Base‘(𝑁 Mat 𝑅)) |
26 | | matbas0pc 21306 |
. . . . . 6
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) →
(Base‘(𝑁 Mat 𝑅)) = ∅) |
27 | 25, 26 | syl5eq 2790 |
. . . . 5
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅) |
28 | 27 | mpteq1d 5144 |
. . . 4
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))) = (𝑚 ∈ ∅ ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))) |
29 | 23, 28 | eqtr4d 2780 |
. . 3
⊢ (¬
(𝑁 ∈ V ∧ 𝑅 ∈ V) → (𝑁 subMat 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))) |
30 | 20, 29 | pm2.61i 185 |
. 2
⊢ (𝑁 subMat 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))) |
31 | 1, 30 | eqtri 2765 |
1
⊢ 𝑄 = (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))) |