Step |
Hyp |
Ref |
Expression |
1 |
|
mat2pmatbas.t |
⊢ 𝑇 = ( 𝑁 matToPolyMat 𝑅 ) |
2 |
|
mat2pmatbas.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
3 |
|
mat2pmatbas.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
4 |
|
mat2pmatbas.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
5 |
|
mat2pmatbas.c |
⊢ 𝐶 = ( 𝑁 Mat 𝑃 ) |
6 |
|
mat2pmatbas0.h |
⊢ 𝐻 = ( Base ‘ 𝐶 ) |
7 |
|
simpl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑁 ∈ Fin ) |
8 |
|
simpr |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑅 ∈ Ring ) |
9 |
2
|
matring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐴 ∈ Ring ) |
10 |
|
eqid |
⊢ ( 1r ‘ 𝐴 ) = ( 1r ‘ 𝐴 ) |
11 |
3 10
|
ringidcl |
⊢ ( 𝐴 ∈ Ring → ( 1r ‘ 𝐴 ) ∈ 𝐵 ) |
12 |
9 11
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 1r ‘ 𝐴 ) ∈ 𝐵 ) |
13 |
7 8 12
|
3jca |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ ( 1r ‘ 𝐴 ) ∈ 𝐵 ) ) |
14 |
|
eqid |
⊢ ( algSc ‘ 𝑃 ) = ( algSc ‘ 𝑃 ) |
15 |
1 2 3 4 14
|
mat2pmatvalel |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ ( 1r ‘ 𝐴 ) ∈ 𝐵 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑖 ( 𝑇 ‘ ( 1r ‘ 𝐴 ) ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ ( 𝑖 ( 1r ‘ 𝐴 ) 𝑗 ) ) ) |
16 |
13 15
|
sylan |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑖 ( 𝑇 ‘ ( 1r ‘ 𝐴 ) ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ ( 𝑖 ( 1r ‘ 𝐴 ) 𝑗 ) ) ) |
17 |
|
fvif |
⊢ ( ( algSc ‘ 𝑃 ) ‘ if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) = if ( 𝑖 = 𝑗 , ( ( algSc ‘ 𝑃 ) ‘ ( 1r ‘ 𝑅 ) ) , ( ( algSc ‘ 𝑃 ) ‘ ( 0g ‘ 𝑅 ) ) ) |
18 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
19 |
|
eqid |
⊢ ( 1r ‘ 𝑃 ) = ( 1r ‘ 𝑃 ) |
20 |
4 14 18 19
|
ply1scl1 |
⊢ ( 𝑅 ∈ Ring → ( ( algSc ‘ 𝑃 ) ‘ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑃 ) ) |
21 |
20
|
ad2antlr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( ( algSc ‘ 𝑃 ) ‘ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑃 ) ) |
22 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
23 |
|
eqid |
⊢ ( 0g ‘ 𝑃 ) = ( 0g ‘ 𝑃 ) |
24 |
4 14 22 23
|
ply1scl0 |
⊢ ( 𝑅 ∈ Ring → ( ( algSc ‘ 𝑃 ) ‘ ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑃 ) ) |
25 |
24
|
ad2antlr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( ( algSc ‘ 𝑃 ) ‘ ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑃 ) ) |
26 |
21 25
|
ifeq12d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → if ( 𝑖 = 𝑗 , ( ( algSc ‘ 𝑃 ) ‘ ( 1r ‘ 𝑅 ) ) , ( ( algSc ‘ 𝑃 ) ‘ ( 0g ‘ 𝑅 ) ) ) = if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑃 ) , ( 0g ‘ 𝑃 ) ) ) |
27 |
17 26
|
eqtrid |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( ( algSc ‘ 𝑃 ) ‘ if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) = if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑃 ) , ( 0g ‘ 𝑃 ) ) ) |
28 |
7
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑁 ∈ Fin ) |
29 |
8
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑅 ∈ Ring ) |
30 |
|
simpl |
⊢ ( ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) → 𝑖 ∈ 𝑁 ) |
31 |
30
|
adantl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑖 ∈ 𝑁 ) |
32 |
|
simpr |
⊢ ( ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) → 𝑗 ∈ 𝑁 ) |
33 |
32
|
adantl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑗 ∈ 𝑁 ) |
34 |
2 18 22 28 29 31 33 10
|
mat1ov |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑖 ( 1r ‘ 𝐴 ) 𝑗 ) = if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) |
35 |
34
|
fveq2d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( ( algSc ‘ 𝑃 ) ‘ ( 𝑖 ( 1r ‘ 𝐴 ) 𝑗 ) ) = ( ( algSc ‘ 𝑃 ) ‘ if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) |
36 |
4
|
ply1ring |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring ) |
37 |
36
|
ad2antlr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑃 ∈ Ring ) |
38 |
|
eqid |
⊢ ( 1r ‘ 𝐶 ) = ( 1r ‘ 𝐶 ) |
39 |
5 19 23 28 37 31 33 38
|
mat1ov |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑖 ( 1r ‘ 𝐶 ) 𝑗 ) = if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑃 ) , ( 0g ‘ 𝑃 ) ) ) |
40 |
27 35 39
|
3eqtr4d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( ( algSc ‘ 𝑃 ) ‘ ( 𝑖 ( 1r ‘ 𝐴 ) 𝑗 ) ) = ( 𝑖 ( 1r ‘ 𝐶 ) 𝑗 ) ) |
41 |
16 40
|
eqtrd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑖 ( 𝑇 ‘ ( 1r ‘ 𝐴 ) ) 𝑗 ) = ( 𝑖 ( 1r ‘ 𝐶 ) 𝑗 ) ) |
42 |
41
|
ralrimivva |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ( 𝑇 ‘ ( 1r ‘ 𝐴 ) ) 𝑗 ) = ( 𝑖 ( 1r ‘ 𝐶 ) 𝑗 ) ) |
43 |
1 2 3 4 5 6
|
mat2pmatbas0 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ ( 1r ‘ 𝐴 ) ∈ 𝐵 ) → ( 𝑇 ‘ ( 1r ‘ 𝐴 ) ) ∈ 𝐻 ) |
44 |
13 43
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑇 ‘ ( 1r ‘ 𝐴 ) ) ∈ 𝐻 ) |
45 |
4 5
|
pmatring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐶 ∈ Ring ) |
46 |
6 38
|
ringidcl |
⊢ ( 𝐶 ∈ Ring → ( 1r ‘ 𝐶 ) ∈ 𝐻 ) |
47 |
45 46
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 1r ‘ 𝐶 ) ∈ 𝐻 ) |
48 |
5 6
|
eqmat |
⊢ ( ( ( 𝑇 ‘ ( 1r ‘ 𝐴 ) ) ∈ 𝐻 ∧ ( 1r ‘ 𝐶 ) ∈ 𝐻 ) → ( ( 𝑇 ‘ ( 1r ‘ 𝐴 ) ) = ( 1r ‘ 𝐶 ) ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ( 𝑇 ‘ ( 1r ‘ 𝐴 ) ) 𝑗 ) = ( 𝑖 ( 1r ‘ 𝐶 ) 𝑗 ) ) ) |
49 |
44 47 48
|
syl2anc |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( ( 𝑇 ‘ ( 1r ‘ 𝐴 ) ) = ( 1r ‘ 𝐶 ) ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ( 𝑇 ‘ ( 1r ‘ 𝐴 ) ) 𝑗 ) = ( 𝑖 ( 1r ‘ 𝐶 ) 𝑗 ) ) ) |
50 |
42 49
|
mpbird |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑇 ‘ ( 1r ‘ 𝐴 ) ) = ( 1r ‘ 𝐶 ) ) |