Step |
Hyp |
Ref |
Expression |
1 |
|
madurid.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
madurid.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
3 |
|
madurid.j |
⊢ 𝐽 = ( 𝑁 maAdju 𝑅 ) |
4 |
|
madurid.d |
⊢ 𝐷 = ( 𝑁 maDet 𝑅 ) |
5 |
|
madurid.i |
⊢ 1 = ( 1r ‘ 𝐴 ) |
6 |
|
madurid.t |
⊢ · = ( .r ‘ 𝐴 ) |
7 |
|
madurid.s |
⊢ ∙ = ( ·𝑠 ‘ 𝐴 ) |
8 |
|
eqid |
⊢ ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) = ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
10 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
11 |
|
simpr |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → 𝑅 ∈ CRing ) |
12 |
1 2
|
matrcl |
⊢ ( 𝑀 ∈ 𝐵 → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) ) |
13 |
12
|
simpld |
⊢ ( 𝑀 ∈ 𝐵 → 𝑁 ∈ Fin ) |
14 |
13
|
adantr |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → 𝑁 ∈ Fin ) |
15 |
1 9 2
|
matbas2i |
⊢ ( 𝑀 ∈ 𝐵 → 𝑀 ∈ ( ( Base ‘ 𝑅 ) ↑m ( 𝑁 × 𝑁 ) ) ) |
16 |
15
|
adantr |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → 𝑀 ∈ ( ( Base ‘ 𝑅 ) ↑m ( 𝑁 × 𝑁 ) ) ) |
17 |
1 3 2
|
maduf |
⊢ ( 𝑅 ∈ CRing → 𝐽 : 𝐵 ⟶ 𝐵 ) |
18 |
17
|
adantl |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → 𝐽 : 𝐵 ⟶ 𝐵 ) |
19 |
|
simpl |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → 𝑀 ∈ 𝐵 ) |
20 |
18 19
|
ffvelrnd |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → ( 𝐽 ‘ 𝑀 ) ∈ 𝐵 ) |
21 |
1 9 2
|
matbas2i |
⊢ ( ( 𝐽 ‘ 𝑀 ) ∈ 𝐵 → ( 𝐽 ‘ 𝑀 ) ∈ ( ( Base ‘ 𝑅 ) ↑m ( 𝑁 × 𝑁 ) ) ) |
22 |
20 21
|
syl |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → ( 𝐽 ‘ 𝑀 ) ∈ ( ( Base ‘ 𝑅 ) ↑m ( 𝑁 × 𝑁 ) ) ) |
23 |
8 9 10 11 14 14 14 16 22
|
mamuval |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → ( 𝑀 ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) ( 𝐽 ‘ 𝑀 ) ) = ( 𝑎 ∈ 𝑁 , 𝑏 ∈ 𝑁 ↦ ( 𝑅 Σg ( 𝑐 ∈ 𝑁 ↦ ( ( 𝑎 𝑀 𝑐 ) ( .r ‘ 𝑅 ) ( 𝑐 ( 𝐽 ‘ 𝑀 ) 𝑏 ) ) ) ) ) ) |
24 |
1 8
|
matmulr |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) = ( .r ‘ 𝐴 ) ) |
25 |
13 24
|
sylan |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) = ( .r ‘ 𝐴 ) ) |
26 |
25 6
|
eqtr4di |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) = · ) |
27 |
26
|
oveqd |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → ( 𝑀 ( 𝑅 maMul 〈 𝑁 , 𝑁 , 𝑁 〉 ) ( 𝐽 ‘ 𝑀 ) ) = ( 𝑀 · ( 𝐽 ‘ 𝑀 ) ) ) |
28 |
|
simp1l |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) → 𝑀 ∈ 𝐵 ) |
29 |
|
simp1r |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) → 𝑅 ∈ CRing ) |
30 |
|
elmapi |
⊢ ( 𝑀 ∈ ( ( Base ‘ 𝑅 ) ↑m ( 𝑁 × 𝑁 ) ) → 𝑀 : ( 𝑁 × 𝑁 ) ⟶ ( Base ‘ 𝑅 ) ) |
31 |
16 30
|
syl |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → 𝑀 : ( 𝑁 × 𝑁 ) ⟶ ( Base ‘ 𝑅 ) ) |
32 |
31
|
3ad2ant1 |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) → 𝑀 : ( 𝑁 × 𝑁 ) ⟶ ( Base ‘ 𝑅 ) ) |
33 |
32
|
adantr |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ 𝑐 ∈ 𝑁 ) → 𝑀 : ( 𝑁 × 𝑁 ) ⟶ ( Base ‘ 𝑅 ) ) |
34 |
|
simpl2 |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ 𝑐 ∈ 𝑁 ) → 𝑎 ∈ 𝑁 ) |
35 |
|
simpr |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ 𝑐 ∈ 𝑁 ) → 𝑐 ∈ 𝑁 ) |
36 |
33 34 35
|
fovrnd |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ 𝑐 ∈ 𝑁 ) → ( 𝑎 𝑀 𝑐 ) ∈ ( Base ‘ 𝑅 ) ) |
37 |
|
simp3 |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) → 𝑏 ∈ 𝑁 ) |
38 |
1 3 2 4 10 9 28 29 36 37
|
madugsum |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) → ( 𝑅 Σg ( 𝑐 ∈ 𝑁 ↦ ( ( 𝑎 𝑀 𝑐 ) ( .r ‘ 𝑅 ) ( 𝑐 ( 𝐽 ‘ 𝑀 ) 𝑏 ) ) ) ) = ( 𝐷 ‘ ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) ) ) |
39 |
|
iftrue |
⊢ ( 𝑎 = 𝑏 → if ( 𝑎 = 𝑏 , ( 𝐷 ‘ 𝑀 ) , ( 0g ‘ 𝑅 ) ) = ( 𝐷 ‘ 𝑀 ) ) |
40 |
39
|
adantl |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 = 𝑏 ) → if ( 𝑎 = 𝑏 , ( 𝐷 ‘ 𝑀 ) , ( 0g ‘ 𝑅 ) ) = ( 𝐷 ‘ 𝑀 ) ) |
41 |
31
|
ffnd |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → 𝑀 Fn ( 𝑁 × 𝑁 ) ) |
42 |
|
fnov |
⊢ ( 𝑀 Fn ( 𝑁 × 𝑁 ) ↔ 𝑀 = ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ ( 𝑑 𝑀 𝑐 ) ) ) |
43 |
41 42
|
sylib |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → 𝑀 = ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ ( 𝑑 𝑀 𝑐 ) ) ) |
44 |
43
|
adantr |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 = 𝑏 ) → 𝑀 = ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ ( 𝑑 𝑀 𝑐 ) ) ) |
45 |
|
equtr2 |
⊢ ( ( 𝑎 = 𝑏 ∧ 𝑑 = 𝑏 ) → 𝑎 = 𝑑 ) |
46 |
45
|
oveq1d |
⊢ ( ( 𝑎 = 𝑏 ∧ 𝑑 = 𝑏 ) → ( 𝑎 𝑀 𝑐 ) = ( 𝑑 𝑀 𝑐 ) ) |
47 |
46
|
ifeq1da |
⊢ ( 𝑎 = 𝑏 → if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) = if ( 𝑑 = 𝑏 , ( 𝑑 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) |
48 |
|
ifid |
⊢ if ( 𝑑 = 𝑏 , ( 𝑑 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) = ( 𝑑 𝑀 𝑐 ) |
49 |
47 48
|
eqtrdi |
⊢ ( 𝑎 = 𝑏 → if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) = ( 𝑑 𝑀 𝑐 ) ) |
50 |
49
|
adantl |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 = 𝑏 ) → if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) = ( 𝑑 𝑀 𝑐 ) ) |
51 |
50
|
mpoeq3dv |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 = 𝑏 ) → ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) = ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ ( 𝑑 𝑀 𝑐 ) ) ) |
52 |
44 51
|
eqtr4d |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 = 𝑏 ) → 𝑀 = ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) ) |
53 |
52
|
fveq2d |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 = 𝑏 ) → ( 𝐷 ‘ 𝑀 ) = ( 𝐷 ‘ ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) ) ) |
54 |
40 53
|
eqtr2d |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 = 𝑏 ) → ( 𝐷 ‘ ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) ) = if ( 𝑎 = 𝑏 , ( 𝐷 ‘ 𝑀 ) , ( 0g ‘ 𝑅 ) ) ) |
55 |
54
|
3ad2antl1 |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ 𝑎 = 𝑏 ) → ( 𝐷 ‘ ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) ) = if ( 𝑎 = 𝑏 , ( 𝐷 ‘ 𝑀 ) , ( 0g ‘ 𝑅 ) ) ) |
56 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
57 |
|
simpl1r |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) → 𝑅 ∈ CRing ) |
58 |
14
|
3ad2ant1 |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) → 𝑁 ∈ Fin ) |
59 |
58
|
adantr |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) → 𝑁 ∈ Fin ) |
60 |
32
|
ad2antrr |
⊢ ( ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) ∧ 𝑐 ∈ 𝑁 ) → 𝑀 : ( 𝑁 × 𝑁 ) ⟶ ( Base ‘ 𝑅 ) ) |
61 |
|
simpll2 |
⊢ ( ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) ∧ 𝑐 ∈ 𝑁 ) → 𝑎 ∈ 𝑁 ) |
62 |
|
simpr |
⊢ ( ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) ∧ 𝑐 ∈ 𝑁 ) → 𝑐 ∈ 𝑁 ) |
63 |
60 61 62
|
fovrnd |
⊢ ( ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) ∧ 𝑐 ∈ 𝑁 ) → ( 𝑎 𝑀 𝑐 ) ∈ ( Base ‘ 𝑅 ) ) |
64 |
32
|
adantr |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) → 𝑀 : ( 𝑁 × 𝑁 ) ⟶ ( Base ‘ 𝑅 ) ) |
65 |
64
|
fovrnda |
⊢ ( ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) ∧ ( 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁 ) ) → ( 𝑑 𝑀 𝑐 ) ∈ ( Base ‘ 𝑅 ) ) |
66 |
65
|
3impb |
⊢ ( ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ∈ 𝑁 ) → ( 𝑑 𝑀 𝑐 ) ∈ ( Base ‘ 𝑅 ) ) |
67 |
|
simpl3 |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) → 𝑏 ∈ 𝑁 ) |
68 |
|
simpl2 |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) → 𝑎 ∈ 𝑁 ) |
69 |
|
neqne |
⊢ ( ¬ 𝑎 = 𝑏 → 𝑎 ≠ 𝑏 ) |
70 |
69
|
necomd |
⊢ ( ¬ 𝑎 = 𝑏 → 𝑏 ≠ 𝑎 ) |
71 |
70
|
adantl |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) → 𝑏 ≠ 𝑎 ) |
72 |
4 9 56 57 59 63 66 67 68 71
|
mdetralt2 |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) → ( 𝐷 ‘ ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , if ( 𝑑 = 𝑎 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) ) ) = ( 0g ‘ 𝑅 ) ) |
73 |
|
ifid |
⊢ if ( 𝑑 = 𝑎 , ( 𝑑 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) = ( 𝑑 𝑀 𝑐 ) |
74 |
|
oveq1 |
⊢ ( 𝑑 = 𝑎 → ( 𝑑 𝑀 𝑐 ) = ( 𝑎 𝑀 𝑐 ) ) |
75 |
74
|
adantl |
⊢ ( ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) ∧ 𝑑 = 𝑎 ) → ( 𝑑 𝑀 𝑐 ) = ( 𝑎 𝑀 𝑐 ) ) |
76 |
75
|
ifeq1da |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) → if ( 𝑑 = 𝑎 , ( 𝑑 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) = if ( 𝑑 = 𝑎 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) |
77 |
73 76
|
syl5eqr |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) → ( 𝑑 𝑀 𝑐 ) = if ( 𝑑 = 𝑎 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) |
78 |
77
|
ifeq2d |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) → if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) = if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , if ( 𝑑 = 𝑎 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) ) |
79 |
78
|
mpoeq3dv |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) → ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) = ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , if ( 𝑑 = 𝑎 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) ) ) |
80 |
79
|
fveq2d |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) → ( 𝐷 ‘ ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) ) = ( 𝐷 ‘ ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , if ( 𝑑 = 𝑎 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) ) ) ) |
81 |
|
iffalse |
⊢ ( ¬ 𝑎 = 𝑏 → if ( 𝑎 = 𝑏 , ( 𝐷 ‘ 𝑀 ) , ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
82 |
81
|
adantl |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) → if ( 𝑎 = 𝑏 , ( 𝐷 ‘ 𝑀 ) , ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
83 |
72 80 82
|
3eqtr4d |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) ∧ ¬ 𝑎 = 𝑏 ) → ( 𝐷 ‘ ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) ) = if ( 𝑎 = 𝑏 , ( 𝐷 ‘ 𝑀 ) , ( 0g ‘ 𝑅 ) ) ) |
84 |
55 83
|
pm2.61dan |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) → ( 𝐷 ‘ ( 𝑑 ∈ 𝑁 , 𝑐 ∈ 𝑁 ↦ if ( 𝑑 = 𝑏 , ( 𝑎 𝑀 𝑐 ) , ( 𝑑 𝑀 𝑐 ) ) ) ) = if ( 𝑎 = 𝑏 , ( 𝐷 ‘ 𝑀 ) , ( 0g ‘ 𝑅 ) ) ) |
85 |
38 84
|
eqtrd |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁 ) → ( 𝑅 Σg ( 𝑐 ∈ 𝑁 ↦ ( ( 𝑎 𝑀 𝑐 ) ( .r ‘ 𝑅 ) ( 𝑐 ( 𝐽 ‘ 𝑀 ) 𝑏 ) ) ) ) = if ( 𝑎 = 𝑏 , ( 𝐷 ‘ 𝑀 ) , ( 0g ‘ 𝑅 ) ) ) |
86 |
85
|
mpoeq3dva |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → ( 𝑎 ∈ 𝑁 , 𝑏 ∈ 𝑁 ↦ ( 𝑅 Σg ( 𝑐 ∈ 𝑁 ↦ ( ( 𝑎 𝑀 𝑐 ) ( .r ‘ 𝑅 ) ( 𝑐 ( 𝐽 ‘ 𝑀 ) 𝑏 ) ) ) ) ) = ( 𝑎 ∈ 𝑁 , 𝑏 ∈ 𝑁 ↦ if ( 𝑎 = 𝑏 , ( 𝐷 ‘ 𝑀 ) , ( 0g ‘ 𝑅 ) ) ) ) |
87 |
5
|
oveq2i |
⊢ ( ( 𝐷 ‘ 𝑀 ) ∙ 1 ) = ( ( 𝐷 ‘ 𝑀 ) ∙ ( 1r ‘ 𝐴 ) ) |
88 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
89 |
88
|
adantl |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → 𝑅 ∈ Ring ) |
90 |
4 1 2 9
|
mdetf |
⊢ ( 𝑅 ∈ CRing → 𝐷 : 𝐵 ⟶ ( Base ‘ 𝑅 ) ) |
91 |
90
|
adantl |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → 𝐷 : 𝐵 ⟶ ( Base ‘ 𝑅 ) ) |
92 |
91 19
|
ffvelrnd |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → ( 𝐷 ‘ 𝑀 ) ∈ ( Base ‘ 𝑅 ) ) |
93 |
1 9 7 56
|
matsc |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ ( 𝐷 ‘ 𝑀 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝐷 ‘ 𝑀 ) ∙ ( 1r ‘ 𝐴 ) ) = ( 𝑎 ∈ 𝑁 , 𝑏 ∈ 𝑁 ↦ if ( 𝑎 = 𝑏 , ( 𝐷 ‘ 𝑀 ) , ( 0g ‘ 𝑅 ) ) ) ) |
94 |
14 89 92 93
|
syl3anc |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → ( ( 𝐷 ‘ 𝑀 ) ∙ ( 1r ‘ 𝐴 ) ) = ( 𝑎 ∈ 𝑁 , 𝑏 ∈ 𝑁 ↦ if ( 𝑎 = 𝑏 , ( 𝐷 ‘ 𝑀 ) , ( 0g ‘ 𝑅 ) ) ) ) |
95 |
87 94
|
syl5eq |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → ( ( 𝐷 ‘ 𝑀 ) ∙ 1 ) = ( 𝑎 ∈ 𝑁 , 𝑏 ∈ 𝑁 ↦ if ( 𝑎 = 𝑏 , ( 𝐷 ‘ 𝑀 ) , ( 0g ‘ 𝑅 ) ) ) ) |
96 |
86 95
|
eqtr4d |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → ( 𝑎 ∈ 𝑁 , 𝑏 ∈ 𝑁 ↦ ( 𝑅 Σg ( 𝑐 ∈ 𝑁 ↦ ( ( 𝑎 𝑀 𝑐 ) ( .r ‘ 𝑅 ) ( 𝑐 ( 𝐽 ‘ 𝑀 ) 𝑏 ) ) ) ) ) = ( ( 𝐷 ‘ 𝑀 ) ∙ 1 ) ) |
97 |
23 27 96
|
3eqtr3d |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing ) → ( 𝑀 · ( 𝐽 ‘ 𝑀 ) ) = ( ( 𝐷 ‘ 𝑀 ) ∙ 1 ) ) |