Step |
Hyp |
Ref |
Expression |
1 |
|
cpmat.s |
⊢ 𝑆 = ( 𝑁 ConstPolyMat 𝑅 ) |
2 |
|
cpmat.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
3 |
|
cpmat.c |
⊢ 𝐶 = ( 𝑁 Mat 𝑃 ) |
4 |
|
cpmat.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
5 |
|
df-cpmat |
⊢ ConstPolyMat = ( 𝑛 ∈ Fin , 𝑟 ∈ V ↦ { 𝑚 ∈ ( Base ‘ ( 𝑛 Mat ( Poly1 ‘ 𝑟 ) ) ) ∣ ∀ 𝑖 ∈ 𝑛 ∀ 𝑗 ∈ 𝑛 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑟 ) } ) |
6 |
5
|
a1i |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ConstPolyMat = ( 𝑛 ∈ Fin , 𝑟 ∈ V ↦ { 𝑚 ∈ ( Base ‘ ( 𝑛 Mat ( Poly1 ‘ 𝑟 ) ) ) ∣ ∀ 𝑖 ∈ 𝑛 ∀ 𝑗 ∈ 𝑛 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑟 ) } ) ) |
7 |
|
simpl |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → 𝑛 = 𝑁 ) |
8 |
|
fveq2 |
⊢ ( 𝑟 = 𝑅 → ( Poly1 ‘ 𝑟 ) = ( Poly1 ‘ 𝑅 ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( Poly1 ‘ 𝑟 ) = ( Poly1 ‘ 𝑅 ) ) |
10 |
7 9
|
oveq12d |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 𝑛 Mat ( Poly1 ‘ 𝑟 ) ) = ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) ) |
11 |
10
|
fveq2d |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( Base ‘ ( 𝑛 Mat ( Poly1 ‘ 𝑟 ) ) ) = ( Base ‘ ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) ) ) |
12 |
2
|
oveq2i |
⊢ ( 𝑁 Mat 𝑃 ) = ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) |
13 |
3 12
|
eqtri |
⊢ 𝐶 = ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) |
14 |
13
|
fveq2i |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) ) |
15 |
4 14
|
eqtri |
⊢ 𝐵 = ( Base ‘ ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) ) |
16 |
11 15
|
eqtr4di |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( Base ‘ ( 𝑛 Mat ( Poly1 ‘ 𝑟 ) ) ) = 𝐵 ) |
17 |
|
fveq2 |
⊢ ( 𝑟 = 𝑅 → ( 0g ‘ 𝑟 ) = ( 0g ‘ 𝑅 ) ) |
18 |
17
|
adantl |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( 0g ‘ 𝑟 ) = ( 0g ‘ 𝑅 ) ) |
19 |
18
|
eqeq2d |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑟 ) ↔ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) ) ) |
20 |
19
|
ralbidv |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑟 ) ↔ ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) ) ) |
21 |
7 20
|
raleqbidv |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( ∀ 𝑗 ∈ 𝑛 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑟 ) ↔ ∀ 𝑗 ∈ 𝑁 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) ) ) |
22 |
7 21
|
raleqbidv |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → ( ∀ 𝑖 ∈ 𝑛 ∀ 𝑗 ∈ 𝑛 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑟 ) ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) ) ) |
23 |
16 22
|
rabeqbidv |
⊢ ( ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) → { 𝑚 ∈ ( Base ‘ ( 𝑛 Mat ( Poly1 ‘ 𝑟 ) ) ) ∣ ∀ 𝑖 ∈ 𝑛 ∀ 𝑗 ∈ 𝑛 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑟 ) } = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) } ) |
24 |
23
|
adantl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) ∧ ( 𝑛 = 𝑁 ∧ 𝑟 = 𝑅 ) ) → { 𝑚 ∈ ( Base ‘ ( 𝑛 Mat ( Poly1 ‘ 𝑟 ) ) ) ∣ ∀ 𝑖 ∈ 𝑛 ∀ 𝑗 ∈ 𝑛 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑟 ) } = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) } ) |
25 |
|
simpl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝑁 ∈ Fin ) |
26 |
|
elex |
⊢ ( 𝑅 ∈ 𝑉 → 𝑅 ∈ V ) |
27 |
26
|
adantl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝑅 ∈ V ) |
28 |
4
|
fvexi |
⊢ 𝐵 ∈ V |
29 |
|
rabexg |
⊢ ( 𝐵 ∈ V → { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) } ∈ V ) |
30 |
28 29
|
mp1i |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) } ∈ V ) |
31 |
6 24 25 27 30
|
ovmpod |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑁 ConstPolyMat 𝑅 ) = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) } ) |
32 |
1 31
|
syl5eq |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝑆 = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑘 ∈ ℕ ( ( coe1 ‘ ( 𝑖 𝑚 𝑗 ) ) ‘ 𝑘 ) = ( 0g ‘ 𝑅 ) } ) |