Step |
Hyp |
Ref |
Expression |
1 |
|
mamumat1cl.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
mamumat1cl.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
3 |
|
mamumat1cl.o |
⊢ 1 = ( 1r ‘ 𝑅 ) |
4 |
|
mamumat1cl.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
5 |
|
mamumat1cl.i |
⊢ 𝐼 = ( 𝑖 ∈ 𝑀 , 𝑗 ∈ 𝑀 ↦ if ( 𝑖 = 𝑗 , 1 , 0 ) ) |
6 |
|
mamumat1cl.m |
⊢ ( 𝜑 → 𝑀 ∈ Fin ) |
7 |
|
mamulid.n |
⊢ ( 𝜑 → 𝑁 ∈ Fin ) |
8 |
|
mamurid.f |
⊢ 𝐹 = ( 𝑅 maMul 〈 𝑁 , 𝑀 , 𝑀 〉 ) |
9 |
|
mamurid.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝐵 ↑m ( 𝑁 × 𝑀 ) ) ) |
10 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
11 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → 𝑅 ∈ Ring ) |
12 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → 𝑁 ∈ Fin ) |
13 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → 𝑀 ∈ Fin ) |
14 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → 𝑋 ∈ ( 𝐵 ↑m ( 𝑁 × 𝑀 ) ) ) |
15 |
1 2 3 4 5 6
|
mamumat1cl |
⊢ ( 𝜑 → 𝐼 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑀 ) ) ) |
16 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → 𝐼 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑀 ) ) ) |
17 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → 𝑙 ∈ 𝑁 ) |
18 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → 𝑚 ∈ 𝑀 ) |
19 |
8 1 10 11 12 13 13 14 16 17 18
|
mamufv |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → ( 𝑙 ( 𝑋 𝐹 𝐼 ) 𝑚 ) = ( 𝑅 Σg ( 𝑘 ∈ 𝑀 ↦ ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) ) ) ) |
20 |
|
ringmnd |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd ) |
21 |
11 20
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → 𝑅 ∈ Mnd ) |
22 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ) → 𝑅 ∈ Ring ) |
23 |
|
elmapi |
⊢ ( 𝑋 ∈ ( 𝐵 ↑m ( 𝑁 × 𝑀 ) ) → 𝑋 : ( 𝑁 × 𝑀 ) ⟶ 𝐵 ) |
24 |
9 23
|
syl |
⊢ ( 𝜑 → 𝑋 : ( 𝑁 × 𝑀 ) ⟶ 𝐵 ) |
25 |
24
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ) → 𝑋 : ( 𝑁 × 𝑀 ) ⟶ 𝐵 ) |
26 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ) → 𝑙 ∈ 𝑁 ) |
27 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ) → 𝑘 ∈ 𝑀 ) |
28 |
25 26 27
|
fovrnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ) → ( 𝑙 𝑋 𝑘 ) ∈ 𝐵 ) |
29 |
|
elmapi |
⊢ ( 𝐼 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑀 ) ) → 𝐼 : ( 𝑀 × 𝑀 ) ⟶ 𝐵 ) |
30 |
15 29
|
syl |
⊢ ( 𝜑 → 𝐼 : ( 𝑀 × 𝑀 ) ⟶ 𝐵 ) |
31 |
30
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ) → 𝐼 : ( 𝑀 × 𝑀 ) ⟶ 𝐵 ) |
32 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ) → 𝑚 ∈ 𝑀 ) |
33 |
31 27 32
|
fovrnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ) → ( 𝑘 𝐼 𝑚 ) ∈ 𝐵 ) |
34 |
1 10
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑙 𝑋 𝑘 ) ∈ 𝐵 ∧ ( 𝑘 𝐼 𝑚 ) ∈ 𝐵 ) → ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) ∈ 𝐵 ) |
35 |
22 28 33 34
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ) → ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) ∈ 𝐵 ) |
36 |
35
|
fmpttd |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → ( 𝑘 ∈ 𝑀 ↦ ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) ) : 𝑀 ⟶ 𝐵 ) |
37 |
|
simp2 |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ∧ 𝑘 ≠ 𝑚 ) → 𝑘 ∈ 𝑀 ) |
38 |
32
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ∧ 𝑘 ≠ 𝑚 ) → 𝑚 ∈ 𝑀 ) |
39 |
1 2 3 4 5 6
|
mat1comp |
⊢ ( ( 𝑘 ∈ 𝑀 ∧ 𝑚 ∈ 𝑀 ) → ( 𝑘 𝐼 𝑚 ) = if ( 𝑘 = 𝑚 , 1 , 0 ) ) |
40 |
37 38 39
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ∧ 𝑘 ≠ 𝑚 ) → ( 𝑘 𝐼 𝑚 ) = if ( 𝑘 = 𝑚 , 1 , 0 ) ) |
41 |
|
ifnefalse |
⊢ ( 𝑘 ≠ 𝑚 → if ( 𝑘 = 𝑚 , 1 , 0 ) = 0 ) |
42 |
41
|
3ad2ant3 |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ∧ 𝑘 ≠ 𝑚 ) → if ( 𝑘 = 𝑚 , 1 , 0 ) = 0 ) |
43 |
40 42
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ∧ 𝑘 ≠ 𝑚 ) → ( 𝑘 𝐼 𝑚 ) = 0 ) |
44 |
43
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ∧ 𝑘 ≠ 𝑚 ) → ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) = ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) 0 ) ) |
45 |
1 10 4
|
ringrz |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑙 𝑋 𝑘 ) ∈ 𝐵 ) → ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) 0 ) = 0 ) |
46 |
22 28 45
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ) → ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) 0 ) = 0 ) |
47 |
46
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ∧ 𝑘 ≠ 𝑚 ) → ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) 0 ) = 0 ) |
48 |
44 47
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) ∧ 𝑘 ∈ 𝑀 ∧ 𝑘 ≠ 𝑚 ) → ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) = 0 ) |
49 |
48 13
|
suppsssn |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → ( ( 𝑘 ∈ 𝑀 ↦ ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) ) supp 0 ) ⊆ { 𝑚 } ) |
50 |
1 4 21 13 18 36 49
|
gsumpt |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → ( 𝑅 Σg ( 𝑘 ∈ 𝑀 ↦ ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) ) ) = ( ( 𝑘 ∈ 𝑀 ↦ ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) ) ‘ 𝑚 ) ) |
51 |
|
oveq2 |
⊢ ( 𝑘 = 𝑚 → ( 𝑙 𝑋 𝑘 ) = ( 𝑙 𝑋 𝑚 ) ) |
52 |
|
oveq1 |
⊢ ( 𝑘 = 𝑚 → ( 𝑘 𝐼 𝑚 ) = ( 𝑚 𝐼 𝑚 ) ) |
53 |
51 52
|
oveq12d |
⊢ ( 𝑘 = 𝑚 → ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) = ( ( 𝑙 𝑋 𝑚 ) ( .r ‘ 𝑅 ) ( 𝑚 𝐼 𝑚 ) ) ) |
54 |
|
eqid |
⊢ ( 𝑘 ∈ 𝑀 ↦ ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) ) = ( 𝑘 ∈ 𝑀 ↦ ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) ) |
55 |
|
ovex |
⊢ ( ( 𝑙 𝑋 𝑚 ) ( .r ‘ 𝑅 ) ( 𝑚 𝐼 𝑚 ) ) ∈ V |
56 |
53 54 55
|
fvmpt |
⊢ ( 𝑚 ∈ 𝑀 → ( ( 𝑘 ∈ 𝑀 ↦ ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) ) ‘ 𝑚 ) = ( ( 𝑙 𝑋 𝑚 ) ( .r ‘ 𝑅 ) ( 𝑚 𝐼 𝑚 ) ) ) |
57 |
56
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → ( ( 𝑘 ∈ 𝑀 ↦ ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) ) ‘ 𝑚 ) = ( ( 𝑙 𝑋 𝑚 ) ( .r ‘ 𝑅 ) ( 𝑚 𝐼 𝑚 ) ) ) |
58 |
|
equequ1 |
⊢ ( 𝑖 = 𝑚 → ( 𝑖 = 𝑗 ↔ 𝑚 = 𝑗 ) ) |
59 |
58
|
ifbid |
⊢ ( 𝑖 = 𝑚 → if ( 𝑖 = 𝑗 , 1 , 0 ) = if ( 𝑚 = 𝑗 , 1 , 0 ) ) |
60 |
|
equequ2 |
⊢ ( 𝑗 = 𝑚 → ( 𝑚 = 𝑗 ↔ 𝑚 = 𝑚 ) ) |
61 |
60
|
ifbid |
⊢ ( 𝑗 = 𝑚 → if ( 𝑚 = 𝑗 , 1 , 0 ) = if ( 𝑚 = 𝑚 , 1 , 0 ) ) |
62 |
|
eqid |
⊢ 𝑚 = 𝑚 |
63 |
62
|
iftruei |
⊢ if ( 𝑚 = 𝑚 , 1 , 0 ) = 1 |
64 |
61 63
|
eqtrdi |
⊢ ( 𝑗 = 𝑚 → if ( 𝑚 = 𝑗 , 1 , 0 ) = 1 ) |
65 |
3
|
fvexi |
⊢ 1 ∈ V |
66 |
59 64 5 65
|
ovmpo |
⊢ ( ( 𝑚 ∈ 𝑀 ∧ 𝑚 ∈ 𝑀 ) → ( 𝑚 𝐼 𝑚 ) = 1 ) |
67 |
66
|
anidms |
⊢ ( 𝑚 ∈ 𝑀 → ( 𝑚 𝐼 𝑚 ) = 1 ) |
68 |
67
|
oveq2d |
⊢ ( 𝑚 ∈ 𝑀 → ( ( 𝑙 𝑋 𝑚 ) ( .r ‘ 𝑅 ) ( 𝑚 𝐼 𝑚 ) ) = ( ( 𝑙 𝑋 𝑚 ) ( .r ‘ 𝑅 ) 1 ) ) |
69 |
68
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → ( ( 𝑙 𝑋 𝑚 ) ( .r ‘ 𝑅 ) ( 𝑚 𝐼 𝑚 ) ) = ( ( 𝑙 𝑋 𝑚 ) ( .r ‘ 𝑅 ) 1 ) ) |
70 |
24
|
fovrnda |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → ( 𝑙 𝑋 𝑚 ) ∈ 𝐵 ) |
71 |
1 10 3
|
ringridm |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑙 𝑋 𝑚 ) ∈ 𝐵 ) → ( ( 𝑙 𝑋 𝑚 ) ( .r ‘ 𝑅 ) 1 ) = ( 𝑙 𝑋 𝑚 ) ) |
72 |
11 70 71
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → ( ( 𝑙 𝑋 𝑚 ) ( .r ‘ 𝑅 ) 1 ) = ( 𝑙 𝑋 𝑚 ) ) |
73 |
57 69 72
|
3eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → ( ( 𝑘 ∈ 𝑀 ↦ ( ( 𝑙 𝑋 𝑘 ) ( .r ‘ 𝑅 ) ( 𝑘 𝐼 𝑚 ) ) ) ‘ 𝑚 ) = ( 𝑙 𝑋 𝑚 ) ) |
74 |
19 50 73
|
3eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑙 ∈ 𝑁 ∧ 𝑚 ∈ 𝑀 ) ) → ( 𝑙 ( 𝑋 𝐹 𝐼 ) 𝑚 ) = ( 𝑙 𝑋 𝑚 ) ) |
75 |
74
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑙 ∈ 𝑁 ∀ 𝑚 ∈ 𝑀 ( 𝑙 ( 𝑋 𝐹 𝐼 ) 𝑚 ) = ( 𝑙 𝑋 𝑚 ) ) |
76 |
1 2 8 7 6 6 9 15
|
mamucl |
⊢ ( 𝜑 → ( 𝑋 𝐹 𝐼 ) ∈ ( 𝐵 ↑m ( 𝑁 × 𝑀 ) ) ) |
77 |
|
elmapi |
⊢ ( ( 𝑋 𝐹 𝐼 ) ∈ ( 𝐵 ↑m ( 𝑁 × 𝑀 ) ) → ( 𝑋 𝐹 𝐼 ) : ( 𝑁 × 𝑀 ) ⟶ 𝐵 ) |
78 |
76 77
|
syl |
⊢ ( 𝜑 → ( 𝑋 𝐹 𝐼 ) : ( 𝑁 × 𝑀 ) ⟶ 𝐵 ) |
79 |
78
|
ffnd |
⊢ ( 𝜑 → ( 𝑋 𝐹 𝐼 ) Fn ( 𝑁 × 𝑀 ) ) |
80 |
24
|
ffnd |
⊢ ( 𝜑 → 𝑋 Fn ( 𝑁 × 𝑀 ) ) |
81 |
|
eqfnov2 |
⊢ ( ( ( 𝑋 𝐹 𝐼 ) Fn ( 𝑁 × 𝑀 ) ∧ 𝑋 Fn ( 𝑁 × 𝑀 ) ) → ( ( 𝑋 𝐹 𝐼 ) = 𝑋 ↔ ∀ 𝑙 ∈ 𝑁 ∀ 𝑚 ∈ 𝑀 ( 𝑙 ( 𝑋 𝐹 𝐼 ) 𝑚 ) = ( 𝑙 𝑋 𝑚 ) ) ) |
82 |
79 80 81
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑋 𝐹 𝐼 ) = 𝑋 ↔ ∀ 𝑙 ∈ 𝑁 ∀ 𝑚 ∈ 𝑀 ( 𝑙 ( 𝑋 𝐹 𝐼 ) 𝑚 ) = ( 𝑙 𝑋 𝑚 ) ) ) |
83 |
75 82
|
mpbird |
⊢ ( 𝜑 → ( 𝑋 𝐹 𝐼 ) = 𝑋 ) |