Step |
Hyp |
Ref |
Expression |
1 |
|
1mavmul.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
1mavmul.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
3 |
|
1mavmul.t |
⊢ · = ( 𝑅 maVecMul 〈 𝑁 , 𝑁 〉 ) |
4 |
|
1mavmul.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
5 |
|
1mavmul.n |
⊢ ( 𝜑 → 𝑁 ∈ Fin ) |
6 |
|
1mavmul.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝐵 ↑m 𝑁 ) ) |
7 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝐴 ) = ( Base ‘ 𝐴 ) |
9 |
1
|
fveq2i |
⊢ ( 1r ‘ 𝐴 ) = ( 1r ‘ ( 𝑁 Mat 𝑅 ) ) |
10 |
1 8 9
|
mat1bas |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ) → ( 1r ‘ 𝐴 ) ∈ ( Base ‘ 𝐴 ) ) |
11 |
4 5 10
|
syl2anc |
⊢ ( 𝜑 → ( 1r ‘ 𝐴 ) ∈ ( Base ‘ 𝐴 ) ) |
12 |
1 3 2 7 4 5 11 6
|
mavmulval |
⊢ ( 𝜑 → ( ( 1r ‘ 𝐴 ) · 𝑌 ) = ( 𝑖 ∈ 𝑁 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 ( 1r ‘ 𝐴 ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) ) ) ) ) |
13 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
14 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
15 |
1 13 14
|
mat1 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 1r ‘ 𝐴 ) = ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) |
16 |
5 4 15
|
syl2anc |
⊢ ( 𝜑 → ( 1r ‘ 𝐴 ) = ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) |
17 |
16
|
oveqdr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) → ( 𝑖 ( 1r ‘ 𝐴 ) 𝑗 ) = ( 𝑖 ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) 𝑗 ) ) |
18 |
17
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) → ( ( 𝑖 ( 1r ‘ 𝐴 ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = ( ( 𝑖 ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) ) |
19 |
18
|
mpteq2dv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) → ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 ( 1r ‘ 𝐴 ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) ) ) |
20 |
19
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) → ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 ( 1r ‘ 𝐴 ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) ) ) ) |
21 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) = ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) |
22 |
|
eqeq12 |
⊢ ( ( 𝑥 = 𝑖 ∧ 𝑦 = 𝑗 ) → ( 𝑥 = 𝑦 ↔ 𝑖 = 𝑗 ) ) |
23 |
22
|
ifbid |
⊢ ( ( 𝑥 = 𝑖 ∧ 𝑦 = 𝑗 ) → if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) = if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) |
24 |
23
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) ∧ ( 𝑥 = 𝑖 ∧ 𝑦 = 𝑗 ) ) → if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) = if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) |
25 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) → 𝑖 ∈ 𝑁 ) |
26 |
25
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → 𝑖 ∈ 𝑁 ) |
27 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → 𝑗 ∈ 𝑁 ) |
28 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → ( 1r ‘ 𝑅 ) ∈ V ) |
29 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → ( 0g ‘ 𝑅 ) ∈ V ) |
30 |
28 29
|
ifcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ∈ V ) |
31 |
21 24 26 27 30
|
ovmpod |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → ( 𝑖 ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) 𝑗 ) = if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) |
32 |
31
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → ( ( 𝑖 ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = ( if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) ) |
33 |
|
iftrue |
⊢ ( 𝑖 = 𝑗 → if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
34 |
33
|
adantr |
⊢ ( ( 𝑖 = 𝑗 ∧ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) ) → if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
35 |
34
|
oveq1d |
⊢ ( ( 𝑖 = 𝑗 ∧ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) ) → ( if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) ) |
36 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) → 𝑅 ∈ Ring ) |
37 |
36
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → 𝑅 ∈ Ring ) |
38 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
39 |
38
|
a1i |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
40 |
39 5
|
elmapd |
⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝐵 ↑m 𝑁 ) ↔ 𝑌 : 𝑁 ⟶ 𝐵 ) ) |
41 |
|
ffvelrn |
⊢ ( ( 𝑌 : 𝑁 ⟶ 𝐵 ∧ 𝑗 ∈ 𝑁 ) → ( 𝑌 ‘ 𝑗 ) ∈ 𝐵 ) |
42 |
41
|
ex |
⊢ ( 𝑌 : 𝑁 ⟶ 𝐵 → ( 𝑗 ∈ 𝑁 → ( 𝑌 ‘ 𝑗 ) ∈ 𝐵 ) ) |
43 |
40 42
|
syl6bi |
⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝐵 ↑m 𝑁 ) → ( 𝑗 ∈ 𝑁 → ( 𝑌 ‘ 𝑗 ) ∈ 𝐵 ) ) ) |
44 |
6 43
|
mpd |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑁 → ( 𝑌 ‘ 𝑗 ) ∈ 𝐵 ) ) |
45 |
44
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) → ( 𝑗 ∈ 𝑁 → ( 𝑌 ‘ 𝑗 ) ∈ 𝐵 ) ) |
46 |
45
|
imp |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → ( 𝑌 ‘ 𝑗 ) ∈ 𝐵 ) |
47 |
2 7 13
|
ringlidm |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑌 ‘ 𝑗 ) ∈ 𝐵 ) → ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = ( 𝑌 ‘ 𝑗 ) ) |
48 |
37 46 47
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = ( 𝑌 ‘ 𝑗 ) ) |
49 |
48
|
adantl |
⊢ ( ( 𝑖 = 𝑗 ∧ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) ) → ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = ( 𝑌 ‘ 𝑗 ) ) |
50 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( 𝑌 ‘ 𝑗 ) = ( 𝑌 ‘ 𝑖 ) ) |
51 |
50
|
equcoms |
⊢ ( 𝑖 = 𝑗 → ( 𝑌 ‘ 𝑗 ) = ( 𝑌 ‘ 𝑖 ) ) |
52 |
51
|
adantr |
⊢ ( ( 𝑖 = 𝑗 ∧ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑌 ‘ 𝑗 ) = ( 𝑌 ‘ 𝑖 ) ) |
53 |
35 49 52
|
3eqtrd |
⊢ ( ( 𝑖 = 𝑗 ∧ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) ) → ( if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = ( 𝑌 ‘ 𝑖 ) ) |
54 |
|
iftrue |
⊢ ( 𝑗 = 𝑖 → if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) = ( 𝑌 ‘ 𝑖 ) ) |
55 |
54
|
equcoms |
⊢ ( 𝑖 = 𝑗 → if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) = ( 𝑌 ‘ 𝑖 ) ) |
56 |
55
|
adantr |
⊢ ( ( 𝑖 = 𝑗 ∧ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) ) → if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) = ( 𝑌 ‘ 𝑖 ) ) |
57 |
53 56
|
eqtr4d |
⊢ ( ( 𝑖 = 𝑗 ∧ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) ) → ( if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) ) |
58 |
|
iffalse |
⊢ ( ¬ 𝑖 = 𝑗 → if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
59 |
58
|
oveq1d |
⊢ ( ¬ 𝑖 = 𝑗 → ( if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = ( ( 0g ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) ) |
60 |
59
|
adantr |
⊢ ( ( ¬ 𝑖 = 𝑗 ∧ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) ) → ( if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = ( ( 0g ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) ) |
61 |
2 7 14
|
ringlz |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑌 ‘ 𝑗 ) ∈ 𝐵 ) → ( ( 0g ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = ( 0g ‘ 𝑅 ) ) |
62 |
37 46 61
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → ( ( 0g ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = ( 0g ‘ 𝑅 ) ) |
63 |
62
|
adantl |
⊢ ( ( ¬ 𝑖 = 𝑗 ∧ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) ) → ( ( 0g ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = ( 0g ‘ 𝑅 ) ) |
64 |
|
eqcom |
⊢ ( 𝑖 = 𝑗 ↔ 𝑗 = 𝑖 ) |
65 |
|
iffalse |
⊢ ( ¬ 𝑗 = 𝑖 → if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
66 |
64 65
|
sylnbi |
⊢ ( ¬ 𝑖 = 𝑗 → if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
67 |
66
|
eqcomd |
⊢ ( ¬ 𝑖 = 𝑗 → ( 0g ‘ 𝑅 ) = if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) ) |
68 |
67
|
adantr |
⊢ ( ( ¬ 𝑖 = 𝑗 ∧ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) ) → ( 0g ‘ 𝑅 ) = if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) ) |
69 |
60 63 68
|
3eqtrd |
⊢ ( ( ¬ 𝑖 = 𝑗 ∧ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) ) → ( if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) ) |
70 |
57 69
|
pm2.61ian |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → ( if ( 𝑖 = 𝑗 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) ) |
71 |
32 70
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → ( ( 𝑖 ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) = if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) ) |
72 |
71
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) → ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) ) ) |
73 |
72
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) → ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 ( 𝑥 ∈ 𝑁 , 𝑦 ∈ 𝑁 ↦ if ( 𝑥 = 𝑦 , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) ) ) ) |
74 |
|
ringmnd |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd ) |
75 |
4 74
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Mnd ) |
76 |
75
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) → 𝑅 ∈ Mnd ) |
77 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) → 𝑁 ∈ Fin ) |
78 |
|
eqid |
⊢ ( 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) ) = ( 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) ) |
79 |
|
ffvelrn |
⊢ ( ( 𝑌 : 𝑁 ⟶ 𝐵 ∧ 𝑖 ∈ 𝑁 ) → ( 𝑌 ‘ 𝑖 ) ∈ 𝐵 ) |
80 |
79 2
|
eleqtrdi |
⊢ ( ( 𝑌 : 𝑁 ⟶ 𝐵 ∧ 𝑖 ∈ 𝑁 ) → ( 𝑌 ‘ 𝑖 ) ∈ ( Base ‘ 𝑅 ) ) |
81 |
80
|
ex |
⊢ ( 𝑌 : 𝑁 ⟶ 𝐵 → ( 𝑖 ∈ 𝑁 → ( 𝑌 ‘ 𝑖 ) ∈ ( Base ‘ 𝑅 ) ) ) |
82 |
40 81
|
syl6bi |
⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝐵 ↑m 𝑁 ) → ( 𝑖 ∈ 𝑁 → ( 𝑌 ‘ 𝑖 ) ∈ ( Base ‘ 𝑅 ) ) ) ) |
83 |
6 82
|
mpd |
⊢ ( 𝜑 → ( 𝑖 ∈ 𝑁 → ( 𝑌 ‘ 𝑖 ) ∈ ( Base ‘ 𝑅 ) ) ) |
84 |
83
|
imp |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) → ( 𝑌 ‘ 𝑖 ) ∈ ( Base ‘ 𝑅 ) ) |
85 |
14 76 77 25 78 84
|
gsummptif1n0 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) → ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ if ( 𝑗 = 𝑖 , ( 𝑌 ‘ 𝑖 ) , ( 0g ‘ 𝑅 ) ) ) ) = ( 𝑌 ‘ 𝑖 ) ) |
86 |
20 73 85
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑁 ) → ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 ( 1r ‘ 𝐴 ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) ) ) = ( 𝑌 ‘ 𝑖 ) ) |
87 |
86
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑖 ∈ 𝑁 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 ( 1r ‘ 𝐴 ) 𝑗 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ 𝑗 ) ) ) ) ) = ( 𝑖 ∈ 𝑁 ↦ ( 𝑌 ‘ 𝑖 ) ) ) |
88 |
|
ffn |
⊢ ( 𝑌 : 𝑁 ⟶ 𝐵 → 𝑌 Fn 𝑁 ) |
89 |
40 88
|
syl6bi |
⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝐵 ↑m 𝑁 ) → 𝑌 Fn 𝑁 ) ) |
90 |
6 89
|
mpd |
⊢ ( 𝜑 → 𝑌 Fn 𝑁 ) |
91 |
|
eqcom |
⊢ ( ( 𝑖 ∈ 𝑁 ↦ ( 𝑌 ‘ 𝑖 ) ) = 𝑌 ↔ 𝑌 = ( 𝑖 ∈ 𝑁 ↦ ( 𝑌 ‘ 𝑖 ) ) ) |
92 |
|
dffn5 |
⊢ ( 𝑌 Fn 𝑁 ↔ 𝑌 = ( 𝑖 ∈ 𝑁 ↦ ( 𝑌 ‘ 𝑖 ) ) ) |
93 |
91 92
|
bitr4i |
⊢ ( ( 𝑖 ∈ 𝑁 ↦ ( 𝑌 ‘ 𝑖 ) ) = 𝑌 ↔ 𝑌 Fn 𝑁 ) |
94 |
90 93
|
sylibr |
⊢ ( 𝜑 → ( 𝑖 ∈ 𝑁 ↦ ( 𝑌 ‘ 𝑖 ) ) = 𝑌 ) |
95 |
12 87 94
|
3eqtrd |
⊢ ( 𝜑 → ( ( 1r ‘ 𝐴 ) · 𝑌 ) = 𝑌 ) |