Step |
Hyp |
Ref |
Expression |
1 |
|
m2cpm.s |
⊢ 𝑆 = ( 𝑁 ConstPolyMat 𝑅 ) |
2 |
|
m2cpm.t |
⊢ 𝑇 = ( 𝑁 matToPolyMat 𝑅 ) |
3 |
|
m2cpm.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
4 |
|
m2cpm.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
5 |
|
eqid |
⊢ ( Poly1 ‘ 𝑅 ) = ( Poly1 ‘ 𝑅 ) |
6 |
|
eqid |
⊢ ( algSc ‘ ( Poly1 ‘ 𝑅 ) ) = ( algSc ‘ ( Poly1 ‘ 𝑅 ) ) |
7 |
2 3 4 5 6
|
mat2pmatvalel |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑖 ( 𝑇 ‘ 𝑀 ) 𝑗 ) = ( ( algSc ‘ ( Poly1 ‘ 𝑅 ) ) ‘ ( 𝑖 𝑀 𝑗 ) ) ) |
8 |
7
|
adantr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝑖 ( 𝑇 ‘ 𝑀 ) 𝑗 ) = ( ( algSc ‘ ( Poly1 ‘ 𝑅 ) ) ‘ ( 𝑖 𝑀 𝑗 ) ) ) |
9 |
8
|
fveq2d |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑛 ∈ ℕ ) → ( coe1 ‘ ( 𝑖 ( 𝑇 ‘ 𝑀 ) 𝑗 ) ) = ( coe1 ‘ ( ( algSc ‘ ( Poly1 ‘ 𝑅 ) ) ‘ ( 𝑖 𝑀 𝑗 ) ) ) ) |
10 |
9
|
fveq1d |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑛 ∈ ℕ ) → ( ( coe1 ‘ ( 𝑖 ( 𝑇 ‘ 𝑀 ) 𝑗 ) ) ‘ 𝑛 ) = ( ( coe1 ‘ ( ( algSc ‘ ( Poly1 ‘ 𝑅 ) ) ‘ ( 𝑖 𝑀 𝑗 ) ) ) ‘ 𝑛 ) ) |
11 |
|
simpl2 |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑅 ∈ Ring ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
13 |
|
simprl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑖 ∈ 𝑁 ) |
14 |
|
simprr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑗 ∈ 𝑁 ) |
15 |
|
simpl3 |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑀 ∈ 𝐵 ) |
16 |
3 12 4 13 14 15
|
matecld |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑖 𝑀 𝑗 ) ∈ ( Base ‘ 𝑅 ) ) |
17 |
11 16
|
jca |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑅 ∈ Ring ∧ ( 𝑖 𝑀 𝑗 ) ∈ ( Base ‘ 𝑅 ) ) ) |
18 |
17
|
adantr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝑅 ∈ Ring ∧ ( 𝑖 𝑀 𝑗 ) ∈ ( Base ‘ 𝑅 ) ) ) |
19 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
20 |
5 6 12 19
|
coe1scl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑖 𝑀 𝑗 ) ∈ ( Base ‘ 𝑅 ) ) → ( coe1 ‘ ( ( algSc ‘ ( Poly1 ‘ 𝑅 ) ) ‘ ( 𝑖 𝑀 𝑗 ) ) ) = ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 = 0 , ( 𝑖 𝑀 𝑗 ) , ( 0g ‘ 𝑅 ) ) ) ) |
21 |
18 20
|
syl |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑛 ∈ ℕ ) → ( coe1 ‘ ( ( algSc ‘ ( Poly1 ‘ 𝑅 ) ) ‘ ( 𝑖 𝑀 𝑗 ) ) ) = ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 = 0 , ( 𝑖 𝑀 𝑗 ) , ( 0g ‘ 𝑅 ) ) ) ) |
22 |
|
eqeq1 |
⊢ ( 𝑘 = 𝑛 → ( 𝑘 = 0 ↔ 𝑛 = 0 ) ) |
23 |
22
|
ifbid |
⊢ ( 𝑘 = 𝑛 → if ( 𝑘 = 0 , ( 𝑖 𝑀 𝑗 ) , ( 0g ‘ 𝑅 ) ) = if ( 𝑛 = 0 , ( 𝑖 𝑀 𝑗 ) , ( 0g ‘ 𝑅 ) ) ) |
24 |
23
|
adantl |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 = 𝑛 ) → if ( 𝑘 = 0 , ( 𝑖 𝑀 𝑗 ) , ( 0g ‘ 𝑅 ) ) = if ( 𝑛 = 0 , ( 𝑖 𝑀 𝑗 ) , ( 0g ‘ 𝑅 ) ) ) |
25 |
|
nnnn0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 ) |
26 |
25
|
adantl |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℕ0 ) |
27 |
|
ovex |
⊢ ( 𝑖 𝑀 𝑗 ) ∈ V |
28 |
|
fvex |
⊢ ( 0g ‘ 𝑅 ) ∈ V |
29 |
27 28
|
ifex |
⊢ if ( 𝑛 = 0 , ( 𝑖 𝑀 𝑗 ) , ( 0g ‘ 𝑅 ) ) ∈ V |
30 |
29
|
a1i |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑛 ∈ ℕ ) → if ( 𝑛 = 0 , ( 𝑖 𝑀 𝑗 ) , ( 0g ‘ 𝑅 ) ) ∈ V ) |
31 |
21 24 26 30
|
fvmptd |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑛 ∈ ℕ ) → ( ( coe1 ‘ ( ( algSc ‘ ( Poly1 ‘ 𝑅 ) ) ‘ ( 𝑖 𝑀 𝑗 ) ) ) ‘ 𝑛 ) = if ( 𝑛 = 0 , ( 𝑖 𝑀 𝑗 ) , ( 0g ‘ 𝑅 ) ) ) |
32 |
|
nnne0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ≠ 0 ) |
33 |
32
|
neneqd |
⊢ ( 𝑛 ∈ ℕ → ¬ 𝑛 = 0 ) |
34 |
33
|
adantl |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑛 ∈ ℕ ) → ¬ 𝑛 = 0 ) |
35 |
34
|
iffalsed |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑛 ∈ ℕ ) → if ( 𝑛 = 0 , ( 𝑖 𝑀 𝑗 ) , ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
36 |
10 31 35
|
3eqtrd |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑛 ∈ ℕ ) → ( ( coe1 ‘ ( 𝑖 ( 𝑇 ‘ 𝑀 ) 𝑗 ) ) ‘ 𝑛 ) = ( 0g ‘ 𝑅 ) ) |
37 |
36
|
ralrimiva |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ∀ 𝑛 ∈ ℕ ( ( coe1 ‘ ( 𝑖 ( 𝑇 ‘ 𝑀 ) 𝑗 ) ) ‘ 𝑛 ) = ( 0g ‘ 𝑅 ) ) |
38 |
37
|
ralrimivva |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑛 ∈ ℕ ( ( coe1 ‘ ( 𝑖 ( 𝑇 ‘ 𝑀 ) 𝑗 ) ) ‘ 𝑛 ) = ( 0g ‘ 𝑅 ) ) |
39 |
|
eqid |
⊢ ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) = ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) |
40 |
2 3 4 5 39
|
mat2pmatbas |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( 𝑇 ‘ 𝑀 ) ∈ ( Base ‘ ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) ) ) |
41 |
|
eqid |
⊢ ( Base ‘ ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) ) = ( Base ‘ ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) ) |
42 |
1 5 39 41
|
cpmatel |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ ( 𝑇 ‘ 𝑀 ) ∈ ( Base ‘ ( 𝑁 Mat ( Poly1 ‘ 𝑅 ) ) ) ) → ( ( 𝑇 ‘ 𝑀 ) ∈ 𝑆 ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑛 ∈ ℕ ( ( coe1 ‘ ( 𝑖 ( 𝑇 ‘ 𝑀 ) 𝑗 ) ) ‘ 𝑛 ) = ( 0g ‘ 𝑅 ) ) ) |
43 |
40 42
|
syld3an3 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( ( 𝑇 ‘ 𝑀 ) ∈ 𝑆 ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∀ 𝑛 ∈ ℕ ( ( coe1 ‘ ( 𝑖 ( 𝑇 ‘ 𝑀 ) 𝑗 ) ) ‘ 𝑛 ) = ( 0g ‘ 𝑅 ) ) ) |
44 |
38 43
|
mpbird |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( 𝑇 ‘ 𝑀 ) ∈ 𝑆 ) |