Step |
Hyp |
Ref |
Expression |
1 |
|
marepvfval.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
marepvfval.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
3 |
|
marepvfval.q |
⊢ 𝑄 = ( 𝑁 matRepV 𝑅 ) |
4 |
|
marepvfval.v |
⊢ 𝑉 = ( ( Base ‘ 𝑅 ) ↑m 𝑁 ) |
5 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
6 |
4
|
ovexi |
⊢ 𝑉 ∈ V |
7 |
6
|
a1i |
⊢ ( ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → 𝑉 ∈ V ) |
8 |
|
mpoexga |
⊢ ( ( 𝐵 ∈ V ∧ 𝑉 ∈ V ) → ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ∈ V ) |
9 |
5 7 8
|
sylancr |
⊢ ( ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ∈ V ) |
10 |
|
oveq12 |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑛 Mat 𝑟 ) = ( 𝑁 Mat 𝑅 ) ) |
11 |
10 1
|
eqtr4di |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑛 Mat 𝑟 ) = 𝐴 ) |
12 |
11
|
fveq2d |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( Base ‘ ( 𝑛 Mat 𝑟 ) ) = ( Base ‘ 𝐴 ) ) |
13 |
12 2
|
eqtr4di |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( Base ‘ ( 𝑛 Mat 𝑟 ) ) = 𝐵 ) |
14 |
|
fveq2 |
⊢ ( 𝑟 = 𝑅 → ( Base ‘ 𝑟 ) = ( Base ‘ 𝑅 ) ) |
15 |
14
|
adantl |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( Base ‘ 𝑟 ) = ( Base ‘ 𝑅 ) ) |
16 |
|
simpl |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → 𝑛 = 𝑁 ) |
17 |
15 16
|
oveq12d |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) = ( ( Base ‘ 𝑅 ) ↑m 𝑁 ) ) |
18 |
17 4
|
eqtr4di |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( ( Base ‘ 𝑟 ) ↑m 𝑛 ) = 𝑉 ) |
19 |
|
eqidd |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) = if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) |
20 |
16 16 19
|
mpoeq123dv |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑖 ∈ 𝑛 , 𝑗 ∈ 𝑛 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) = ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) |
21 |
16 20
|
mpteq12dv |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑘 ∈ 𝑛 ↦ ( 𝑖 ∈ 𝑛 , 𝑗 ∈ 𝑛 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) = ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) |
22 |
13 18 21
|
mpoeq123dv |
⊢ ( ( 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 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ) |
23 |
|
df-marepv |
⊢ 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 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ) |
24 |
22 23
|
ovmpoga |
⊢ ( ( 𝑁 ∈ V ∧ 𝑅 ∈ V ∧ ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ∈ V ) → ( 𝑁 matRepV 𝑅 ) = ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ) |
25 |
9 24
|
mpd3an3 |
⊢ ( ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 matRepV 𝑅 ) = ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ) |
26 |
23
|
mpondm0 |
⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 matRepV 𝑅 ) = ∅ ) |
27 |
1
|
fveq2i |
⊢ ( Base ‘ 𝐴 ) = ( Base ‘ ( 𝑁 Mat 𝑅 ) ) |
28 |
2 27
|
eqtri |
⊢ 𝐵 = ( Base ‘ ( 𝑁 Mat 𝑅 ) ) |
29 |
|
matbas0pc |
⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( Base ‘ ( 𝑁 Mat 𝑅 ) ) = ∅ ) |
30 |
28 29
|
syl5eq |
⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → 𝐵 = ∅ ) |
31 |
30
|
orcd |
⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐵 = ∅ ∨ 𝑉 = ∅ ) ) |
32 |
|
0mpo0 |
⊢ ( ( 𝐵 = ∅ ∨ 𝑉 = ∅ ) → ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ∅ ) |
33 |
31 32
|
syl |
⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) = ∅ ) |
34 |
26 33
|
eqtr4d |
⊢ ( ¬ ( 𝑁 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑁 matRepV 𝑅 ) = ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) ) |
35 |
25 34
|
pm2.61i |
⊢ ( 𝑁 matRepV 𝑅 ) = ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) |
36 |
3 35
|
eqtri |
⊢ 𝑄 = ( 𝑚 ∈ 𝐵 , 𝑣 ∈ 𝑉 ↦ ( 𝑘 ∈ 𝑁 ↦ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑘 , ( 𝑣 ‘ 𝑖 ) , ( 𝑖 𝑚 𝑗 ) ) ) ) ) |