Step |
Hyp |
Ref |
Expression |
1 |
|
fedgmul.a |
⊢ 𝐴 = ( ( subringAlg ‘ 𝐸 ) ‘ 𝑉 ) |
2 |
|
fedgmul.b |
⊢ 𝐵 = ( ( subringAlg ‘ 𝐸 ) ‘ 𝑈 ) |
3 |
|
fedgmul.c |
⊢ 𝐶 = ( ( subringAlg ‘ 𝐹 ) ‘ 𝑉 ) |
4 |
|
fedgmul.f |
⊢ 𝐹 = ( 𝐸 ↾s 𝑈 ) |
5 |
|
fedgmul.k |
⊢ 𝐾 = ( 𝐸 ↾s 𝑉 ) |
6 |
|
fedgmul.1 |
⊢ ( 𝜑 → 𝐸 ∈ DivRing ) |
7 |
|
fedgmul.2 |
⊢ ( 𝜑 → 𝐹 ∈ DivRing ) |
8 |
|
fedgmul.3 |
⊢ ( 𝜑 → 𝐾 ∈ DivRing ) |
9 |
|
fedgmul.4 |
⊢ ( 𝜑 → 𝑈 ∈ ( SubRing ‘ 𝐸 ) ) |
10 |
|
fedgmul.5 |
⊢ ( 𝜑 → 𝑉 ∈ ( SubRing ‘ 𝐹 ) ) |
11 |
|
fedgmullem.d |
⊢ 𝐷 = ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( 𝑖 ( .r ‘ 𝐸 ) 𝑗 ) ) |
12 |
|
fedgmullem.h |
⊢ 𝐻 = ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( ( 𝐺 ‘ 𝑗 ) ‘ 𝑖 ) ) |
13 |
|
fedgmullem.x |
⊢ ( 𝜑 → 𝑋 ∈ ( LBasis ‘ 𝐶 ) ) |
14 |
|
fedgmullem.y |
⊢ ( 𝜑 → 𝑌 ∈ ( LBasis ‘ 𝐵 ) ) |
15 |
|
fedgmullem2.1 |
⊢ ( 𝜑 → 𝑊 ∈ ( Base ‘ ( ( Scalar ‘ 𝐴 ) freeLMod ( 𝑌 × 𝑋 ) ) ) ) |
16 |
|
fedgmullem2.2 |
⊢ ( 𝜑 → ( 𝐴 Σg ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) ) = ( 0g ‘ 𝐴 ) ) |
17 |
4
|
subsubrg |
⊢ ( 𝑈 ∈ ( SubRing ‘ 𝐸 ) → ( 𝑉 ∈ ( SubRing ‘ 𝐹 ) ↔ ( 𝑉 ∈ ( SubRing ‘ 𝐸 ) ∧ 𝑉 ⊆ 𝑈 ) ) ) |
18 |
17
|
biimpa |
⊢ ( ( 𝑈 ∈ ( SubRing ‘ 𝐸 ) ∧ 𝑉 ∈ ( SubRing ‘ 𝐹 ) ) → ( 𝑉 ∈ ( SubRing ‘ 𝐸 ) ∧ 𝑉 ⊆ 𝑈 ) ) |
19 |
9 10 18
|
syl2anc |
⊢ ( 𝜑 → ( 𝑉 ∈ ( SubRing ‘ 𝐸 ) ∧ 𝑉 ⊆ 𝑈 ) ) |
20 |
19
|
simpld |
⊢ ( 𝜑 → 𝑉 ∈ ( SubRing ‘ 𝐸 ) ) |
21 |
1 5
|
sralvec |
⊢ ( ( 𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ ( SubRing ‘ 𝐸 ) ) → 𝐴 ∈ LVec ) |
22 |
6 8 20 21
|
syl3anc |
⊢ ( 𝜑 → 𝐴 ∈ LVec ) |
23 |
|
lveclmod |
⊢ ( 𝐴 ∈ LVec → 𝐴 ∈ LMod ) |
24 |
22 23
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ LMod ) |
25 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
26 |
|
eqid |
⊢ ( LBasis ‘ 𝐶 ) = ( LBasis ‘ 𝐶 ) |
27 |
25 26
|
lbsss |
⊢ ( 𝑋 ∈ ( LBasis ‘ 𝐶 ) → 𝑋 ⊆ ( Base ‘ 𝐶 ) ) |
28 |
13 27
|
syl |
⊢ ( 𝜑 → 𝑋 ⊆ ( Base ‘ 𝐶 ) ) |
29 |
|
eqid |
⊢ ( Base ‘ 𝐸 ) = ( Base ‘ 𝐸 ) |
30 |
29
|
subrgss |
⊢ ( 𝑈 ∈ ( SubRing ‘ 𝐸 ) → 𝑈 ⊆ ( Base ‘ 𝐸 ) ) |
31 |
9 30
|
syl |
⊢ ( 𝜑 → 𝑈 ⊆ ( Base ‘ 𝐸 ) ) |
32 |
4 29
|
ressbas2 |
⊢ ( 𝑈 ⊆ ( Base ‘ 𝐸 ) → 𝑈 = ( Base ‘ 𝐹 ) ) |
33 |
31 32
|
syl |
⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝐹 ) ) |
34 |
3
|
a1i |
⊢ ( 𝜑 → 𝐶 = ( ( subringAlg ‘ 𝐹 ) ‘ 𝑉 ) ) |
35 |
|
eqid |
⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 ) |
36 |
35
|
subrgss |
⊢ ( 𝑉 ∈ ( SubRing ‘ 𝐹 ) → 𝑉 ⊆ ( Base ‘ 𝐹 ) ) |
37 |
10 36
|
syl |
⊢ ( 𝜑 → 𝑉 ⊆ ( Base ‘ 𝐹 ) ) |
38 |
34 37
|
srabase |
⊢ ( 𝜑 → ( Base ‘ 𝐹 ) = ( Base ‘ 𝐶 ) ) |
39 |
33 38
|
eqtrd |
⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝐶 ) ) |
40 |
39 31
|
eqsstrrd |
⊢ ( 𝜑 → ( Base ‘ 𝐶 ) ⊆ ( Base ‘ 𝐸 ) ) |
41 |
1
|
a1i |
⊢ ( 𝜑 → 𝐴 = ( ( subringAlg ‘ 𝐸 ) ‘ 𝑉 ) ) |
42 |
29
|
subrgss |
⊢ ( 𝑉 ∈ ( SubRing ‘ 𝐸 ) → 𝑉 ⊆ ( Base ‘ 𝐸 ) ) |
43 |
20 42
|
syl |
⊢ ( 𝜑 → 𝑉 ⊆ ( Base ‘ 𝐸 ) ) |
44 |
41 43
|
srabase |
⊢ ( 𝜑 → ( Base ‘ 𝐸 ) = ( Base ‘ 𝐴 ) ) |
45 |
40 44
|
sseqtrd |
⊢ ( 𝜑 → ( Base ‘ 𝐶 ) ⊆ ( Base ‘ 𝐴 ) ) |
46 |
28 45
|
sstrd |
⊢ ( 𝜑 → 𝑋 ⊆ ( Base ‘ 𝐴 ) ) |
47 |
41 9 43
|
srasubrg |
⊢ ( 𝜑 → 𝑈 ∈ ( SubRing ‘ 𝐴 ) ) |
48 |
|
subrgsubg |
⊢ ( 𝑈 ∈ ( SubRing ‘ 𝐴 ) → 𝑈 ∈ ( SubGrp ‘ 𝐴 ) ) |
49 |
47 48
|
syl |
⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝐴 ) ) |
50 |
1 6 20
|
drgextvsca |
⊢ ( 𝜑 → ( .r ‘ 𝐸 ) = ( ·𝑠 ‘ 𝐴 ) ) |
51 |
50
|
oveqdr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑈 ) ) → ( 𝑥 ( .r ‘ 𝐸 ) 𝑦 ) = ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑦 ) ) |
52 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑈 ) ) → 𝑈 ∈ ( SubRing ‘ 𝐸 ) ) |
53 |
19
|
simprd |
⊢ ( 𝜑 → 𝑉 ⊆ 𝑈 ) |
54 |
53
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑈 ) ) → 𝑉 ⊆ 𝑈 ) |
55 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑈 ) ) → 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) |
56 |
|
ressabs |
⊢ ( ( 𝑈 ∈ ( SubRing ‘ 𝐸 ) ∧ 𝑉 ⊆ 𝑈 ) → ( ( 𝐸 ↾s 𝑈 ) ↾s 𝑉 ) = ( 𝐸 ↾s 𝑉 ) ) |
57 |
9 53 56
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐸 ↾s 𝑈 ) ↾s 𝑉 ) = ( 𝐸 ↾s 𝑉 ) ) |
58 |
4
|
oveq1i |
⊢ ( 𝐹 ↾s 𝑉 ) = ( ( 𝐸 ↾s 𝑈 ) ↾s 𝑉 ) |
59 |
57 58 5
|
3eqtr4g |
⊢ ( 𝜑 → ( 𝐹 ↾s 𝑉 ) = 𝐾 ) |
60 |
34 37
|
srasca |
⊢ ( 𝜑 → ( 𝐹 ↾s 𝑉 ) = ( Scalar ‘ 𝐶 ) ) |
61 |
59 60
|
eqtr3d |
⊢ ( 𝜑 → 𝐾 = ( Scalar ‘ 𝐶 ) ) |
62 |
61
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ ( Scalar ‘ 𝐶 ) ) ) |
63 |
5 29
|
ressbas2 |
⊢ ( 𝑉 ⊆ ( Base ‘ 𝐸 ) → 𝑉 = ( Base ‘ 𝐾 ) ) |
64 |
43 63
|
syl |
⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝐾 ) ) |
65 |
41 43
|
srasca |
⊢ ( 𝜑 → ( 𝐸 ↾s 𝑉 ) = ( Scalar ‘ 𝐴 ) ) |
66 |
5 65
|
syl5eq |
⊢ ( 𝜑 → 𝐾 = ( Scalar ‘ 𝐴 ) ) |
67 |
59 60 66
|
3eqtr3rd |
⊢ ( 𝜑 → ( Scalar ‘ 𝐴 ) = ( Scalar ‘ 𝐶 ) ) |
68 |
67
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐴 ) ) = ( Base ‘ ( Scalar ‘ 𝐶 ) ) ) |
69 |
62 64 68
|
3eqtr4d |
⊢ ( 𝜑 → 𝑉 = ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) |
70 |
69
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑈 ) ) → 𝑉 = ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) |
71 |
55 70
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑈 ) ) → 𝑥 ∈ 𝑉 ) |
72 |
54 71
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑈 ) ) → 𝑥 ∈ 𝑈 ) |
73 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑈 ) ) → 𝑦 ∈ 𝑈 ) |
74 |
|
eqid |
⊢ ( .r ‘ 𝐸 ) = ( .r ‘ 𝐸 ) |
75 |
74
|
subrgmcl |
⊢ ( ( 𝑈 ∈ ( SubRing ‘ 𝐸 ) ∧ 𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈 ) → ( 𝑥 ( .r ‘ 𝐸 ) 𝑦 ) ∈ 𝑈 ) |
76 |
52 72 73 75
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑈 ) ) → ( 𝑥 ( .r ‘ 𝐸 ) 𝑦 ) ∈ 𝑈 ) |
77 |
51 76
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑈 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑦 ) ∈ 𝑈 ) |
78 |
77
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∀ 𝑦 ∈ 𝑈 ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑦 ) ∈ 𝑈 ) |
79 |
|
eqid |
⊢ ( Scalar ‘ 𝐴 ) = ( Scalar ‘ 𝐴 ) |
80 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝐴 ) ) = ( Base ‘ ( Scalar ‘ 𝐴 ) ) |
81 |
|
eqid |
⊢ ( Base ‘ 𝐴 ) = ( Base ‘ 𝐴 ) |
82 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝐴 ) = ( ·𝑠 ‘ 𝐴 ) |
83 |
|
eqid |
⊢ ( LSubSp ‘ 𝐴 ) = ( LSubSp ‘ 𝐴 ) |
84 |
79 80 81 82 83
|
islss4 |
⊢ ( 𝐴 ∈ LMod → ( 𝑈 ∈ ( LSubSp ‘ 𝐴 ) ↔ ( 𝑈 ∈ ( SubGrp ‘ 𝐴 ) ∧ ∀ 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∀ 𝑦 ∈ 𝑈 ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑦 ) ∈ 𝑈 ) ) ) |
85 |
84
|
biimpar |
⊢ ( ( 𝐴 ∈ LMod ∧ ( 𝑈 ∈ ( SubGrp ‘ 𝐴 ) ∧ ∀ 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∀ 𝑦 ∈ 𝑈 ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑦 ) ∈ 𝑈 ) ) → 𝑈 ∈ ( LSubSp ‘ 𝐴 ) ) |
86 |
24 49 78 85
|
syl12anc |
⊢ ( 𝜑 → 𝑈 ∈ ( LSubSp ‘ 𝐴 ) ) |
87 |
28 39
|
sseqtrrd |
⊢ ( 𝜑 → 𝑋 ⊆ 𝑈 ) |
88 |
26
|
lbslinds |
⊢ ( LBasis ‘ 𝐶 ) ⊆ ( LIndS ‘ 𝐶 ) |
89 |
88 13
|
sselid |
⊢ ( 𝜑 → 𝑋 ∈ ( LIndS ‘ 𝐶 ) ) |
90 |
31 44
|
sseqtrd |
⊢ ( 𝜑 → 𝑈 ⊆ ( Base ‘ 𝐴 ) ) |
91 |
|
eqid |
⊢ ( 𝐴 ↾s 𝑈 ) = ( 𝐴 ↾s 𝑈 ) |
92 |
91 81
|
ressbas2 |
⊢ ( 𝑈 ⊆ ( Base ‘ 𝐴 ) → 𝑈 = ( Base ‘ ( 𝐴 ↾s 𝑈 ) ) ) |
93 |
90 92
|
syl |
⊢ ( 𝜑 → 𝑈 = ( Base ‘ ( 𝐴 ↾s 𝑈 ) ) ) |
94 |
33 93 38
|
3eqtr3rd |
⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ ( 𝐴 ↾s 𝑈 ) ) ) |
95 |
91 79
|
resssca |
⊢ ( 𝑈 ∈ ( SubRing ‘ 𝐸 ) → ( Scalar ‘ 𝐴 ) = ( Scalar ‘ ( 𝐴 ↾s 𝑈 ) ) ) |
96 |
9 95
|
syl |
⊢ ( 𝜑 → ( Scalar ‘ 𝐴 ) = ( Scalar ‘ ( 𝐴 ↾s 𝑈 ) ) ) |
97 |
67 96
|
eqtr3d |
⊢ ( 𝜑 → ( Scalar ‘ 𝐶 ) = ( Scalar ‘ ( 𝐴 ↾s 𝑈 ) ) ) |
98 |
97
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐶 ) ) = ( Base ‘ ( Scalar ‘ ( 𝐴 ↾s 𝑈 ) ) ) ) |
99 |
97
|
fveq2d |
⊢ ( 𝜑 → ( 0g ‘ ( Scalar ‘ 𝐶 ) ) = ( 0g ‘ ( Scalar ‘ ( 𝐴 ↾s 𝑈 ) ) ) ) |
100 |
|
eqid |
⊢ ( +g ‘ 𝐸 ) = ( +g ‘ 𝐸 ) |
101 |
4 100
|
ressplusg |
⊢ ( 𝑈 ∈ ( SubRing ‘ 𝐸 ) → ( +g ‘ 𝐸 ) = ( +g ‘ 𝐹 ) ) |
102 |
9 101
|
syl |
⊢ ( 𝜑 → ( +g ‘ 𝐸 ) = ( +g ‘ 𝐹 ) ) |
103 |
41 43
|
sraaddg |
⊢ ( 𝜑 → ( +g ‘ 𝐸 ) = ( +g ‘ 𝐴 ) ) |
104 |
34 37
|
sraaddg |
⊢ ( 𝜑 → ( +g ‘ 𝐹 ) = ( +g ‘ 𝐶 ) ) |
105 |
102 103 104
|
3eqtr3rd |
⊢ ( 𝜑 → ( +g ‘ 𝐶 ) = ( +g ‘ 𝐴 ) ) |
106 |
|
eqid |
⊢ ( +g ‘ 𝐴 ) = ( +g ‘ 𝐴 ) |
107 |
91 106
|
ressplusg |
⊢ ( 𝑈 ∈ ( SubRing ‘ 𝐸 ) → ( +g ‘ 𝐴 ) = ( +g ‘ ( 𝐴 ↾s 𝑈 ) ) ) |
108 |
9 107
|
syl |
⊢ ( 𝜑 → ( +g ‘ 𝐴 ) = ( +g ‘ ( 𝐴 ↾s 𝑈 ) ) ) |
109 |
105 108
|
eqtrd |
⊢ ( 𝜑 → ( +g ‘ 𝐶 ) = ( +g ‘ ( 𝐴 ↾s 𝑈 ) ) ) |
110 |
109
|
oveqdr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝑥 ( +g ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( +g ‘ ( 𝐴 ↾s 𝑈 ) ) 𝑦 ) ) |
111 |
59 8
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐹 ↾s 𝑉 ) ∈ DivRing ) |
112 |
|
eqid |
⊢ ( 𝐹 ↾s 𝑉 ) = ( 𝐹 ↾s 𝑉 ) |
113 |
3 112
|
sralvec |
⊢ ( ( 𝐹 ∈ DivRing ∧ ( 𝐹 ↾s 𝑉 ) ∈ DivRing ∧ 𝑉 ∈ ( SubRing ‘ 𝐹 ) ) → 𝐶 ∈ LVec ) |
114 |
7 111 10 113
|
syl3anc |
⊢ ( 𝜑 → 𝐶 ∈ LVec ) |
115 |
|
lveclmod |
⊢ ( 𝐶 ∈ LVec → 𝐶 ∈ LMod ) |
116 |
114 115
|
syl |
⊢ ( 𝜑 → 𝐶 ∈ LMod ) |
117 |
|
eqid |
⊢ ( Scalar ‘ 𝐶 ) = ( Scalar ‘ 𝐶 ) |
118 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝐶 ) = ( ·𝑠 ‘ 𝐶 ) |
119 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝐶 ) ) = ( Base ‘ ( Scalar ‘ 𝐶 ) ) |
120 |
25 117 118 119
|
lmodvscl |
⊢ ( ( 𝐶 ∈ LMod ∧ 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐶 ) 𝑦 ) ∈ ( Base ‘ 𝐶 ) ) |
121 |
120
|
3expb |
⊢ ( ( 𝐶 ∈ LMod ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐶 ) 𝑦 ) ∈ ( Base ‘ 𝐶 ) ) |
122 |
116 121
|
sylan |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐶 ) 𝑦 ) ∈ ( Base ‘ 𝐶 ) ) |
123 |
2 6 9
|
drgextvsca |
⊢ ( 𝜑 → ( .r ‘ 𝐸 ) = ( ·𝑠 ‘ 𝐵 ) ) |
124 |
50 123
|
eqtr3d |
⊢ ( 𝜑 → ( ·𝑠 ‘ 𝐴 ) = ( ·𝑠 ‘ 𝐵 ) ) |
125 |
91 82
|
ressvsca |
⊢ ( 𝑈 ∈ ( SubRing ‘ 𝐸 ) → ( ·𝑠 ‘ 𝐴 ) = ( ·𝑠 ‘ ( 𝐴 ↾s 𝑈 ) ) ) |
126 |
9 125
|
syl |
⊢ ( 𝜑 → ( ·𝑠 ‘ 𝐴 ) = ( ·𝑠 ‘ ( 𝐴 ↾s 𝑈 ) ) ) |
127 |
4 74
|
ressmulr |
⊢ ( 𝑈 ∈ ( SubRing ‘ 𝐸 ) → ( .r ‘ 𝐸 ) = ( .r ‘ 𝐹 ) ) |
128 |
9 127
|
syl |
⊢ ( 𝜑 → ( .r ‘ 𝐸 ) = ( .r ‘ 𝐹 ) ) |
129 |
3 7 10
|
drgextvsca |
⊢ ( 𝜑 → ( .r ‘ 𝐹 ) = ( ·𝑠 ‘ 𝐶 ) ) |
130 |
128 123 129
|
3eqtr3d |
⊢ ( 𝜑 → ( ·𝑠 ‘ 𝐵 ) = ( ·𝑠 ‘ 𝐶 ) ) |
131 |
124 126 130
|
3eqtr3rd |
⊢ ( 𝜑 → ( ·𝑠 ‘ 𝐶 ) = ( ·𝑠 ‘ ( 𝐴 ↾s 𝑈 ) ) ) |
132 |
131
|
oveqdr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( ·𝑠 ‘ ( 𝐴 ↾s 𝑈 ) ) 𝑦 ) ) |
133 |
|
ovexd |
⊢ ( 𝜑 → ( 𝐴 ↾s 𝑈 ) ∈ V ) |
134 |
94 98 99 110 122 132 114 133
|
lindspropd |
⊢ ( 𝜑 → ( LIndS ‘ 𝐶 ) = ( LIndS ‘ ( 𝐴 ↾s 𝑈 ) ) ) |
135 |
89 134
|
eleqtrd |
⊢ ( 𝜑 → 𝑋 ∈ ( LIndS ‘ ( 𝐴 ↾s 𝑈 ) ) ) |
136 |
83 91
|
lsslinds |
⊢ ( ( 𝐴 ∈ LMod ∧ 𝑈 ∈ ( LSubSp ‘ 𝐴 ) ∧ 𝑋 ⊆ 𝑈 ) → ( 𝑋 ∈ ( LIndS ‘ ( 𝐴 ↾s 𝑈 ) ) ↔ 𝑋 ∈ ( LIndS ‘ 𝐴 ) ) ) |
137 |
136
|
biimpa |
⊢ ( ( ( 𝐴 ∈ LMod ∧ 𝑈 ∈ ( LSubSp ‘ 𝐴 ) ∧ 𝑋 ⊆ 𝑈 ) ∧ 𝑋 ∈ ( LIndS ‘ ( 𝐴 ↾s 𝑈 ) ) ) → 𝑋 ∈ ( LIndS ‘ 𝐴 ) ) |
138 |
24 86 87 135 137
|
syl31anc |
⊢ ( 𝜑 → 𝑋 ∈ ( LIndS ‘ 𝐴 ) ) |
139 |
|
eqid |
⊢ ( 0g ‘ 𝐴 ) = ( 0g ‘ 𝐴 ) |
140 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) |
141 |
81 80 79 82 139 140
|
islinds5 |
⊢ ( ( 𝐴 ∈ LMod ∧ 𝑋 ⊆ ( Base ‘ 𝐴 ) ) → ( 𝑋 ∈ ( LIndS ‘ 𝐴 ) ↔ ∀ 𝑤 ∈ ( ( Base ‘ ( Scalar ‘ 𝐴 ) ) ↑m 𝑋 ) ( ( 𝑤 finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∧ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑤 ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 0g ‘ 𝐴 ) ) → 𝑤 = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) ) ) |
142 |
141
|
biimpa |
⊢ ( ( ( 𝐴 ∈ LMod ∧ 𝑋 ⊆ ( Base ‘ 𝐴 ) ) ∧ 𝑋 ∈ ( LIndS ‘ 𝐴 ) ) → ∀ 𝑤 ∈ ( ( Base ‘ ( Scalar ‘ 𝐴 ) ) ↑m 𝑋 ) ( ( 𝑤 finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∧ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑤 ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 0g ‘ 𝐴 ) ) → 𝑤 = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) ) |
143 |
24 46 138 142
|
syl21anc |
⊢ ( 𝜑 → ∀ 𝑤 ∈ ( ( Base ‘ ( Scalar ‘ 𝐴 ) ) ↑m 𝑋 ) ( ( 𝑤 finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∧ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑤 ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 0g ‘ 𝐴 ) ) → 𝑤 = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) ) |
144 |
143
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ∀ 𝑤 ∈ ( ( Base ‘ ( Scalar ‘ 𝐴 ) ) ↑m 𝑋 ) ( ( 𝑤 finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∧ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑤 ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 0g ‘ 𝐴 ) ) → 𝑤 = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) ) |
145 |
|
eqid |
⊢ ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) |
146 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 0g ‘ 𝐹 ) ∈ V ) |
147 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝑌 ∈ ( LBasis ‘ 𝐵 ) ) |
148 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝑋 ∈ ( LBasis ‘ 𝐶 ) ) |
149 |
|
fvexd |
⊢ ( 𝜑 → ( Scalar ‘ 𝐴 ) ∈ V ) |
150 |
14 13
|
xpexd |
⊢ ( 𝜑 → ( 𝑌 × 𝑋 ) ∈ V ) |
151 |
|
eqid |
⊢ ( ( Scalar ‘ 𝐴 ) freeLMod ( 𝑌 × 𝑋 ) ) = ( ( Scalar ‘ 𝐴 ) freeLMod ( 𝑌 × 𝑋 ) ) |
152 |
|
eqid |
⊢ ( Base ‘ ( ( Scalar ‘ 𝐴 ) freeLMod ( 𝑌 × 𝑋 ) ) ) = ( Base ‘ ( ( Scalar ‘ 𝐴 ) freeLMod ( 𝑌 × 𝑋 ) ) ) |
153 |
151 80 140 152
|
frlmelbas |
⊢ ( ( ( Scalar ‘ 𝐴 ) ∈ V ∧ ( 𝑌 × 𝑋 ) ∈ V ) → ( 𝑊 ∈ ( Base ‘ ( ( Scalar ‘ 𝐴 ) freeLMod ( 𝑌 × 𝑋 ) ) ) ↔ ( 𝑊 ∈ ( ( Base ‘ ( Scalar ‘ 𝐴 ) ) ↑m ( 𝑌 × 𝑋 ) ) ∧ 𝑊 finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) ) |
154 |
149 150 153
|
syl2anc |
⊢ ( 𝜑 → ( 𝑊 ∈ ( Base ‘ ( ( Scalar ‘ 𝐴 ) freeLMod ( 𝑌 × 𝑋 ) ) ) ↔ ( 𝑊 ∈ ( ( Base ‘ ( Scalar ‘ 𝐴 ) ) ↑m ( 𝑌 × 𝑋 ) ) ∧ 𝑊 finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) ) |
155 |
15 154
|
mpbid |
⊢ ( 𝜑 → ( 𝑊 ∈ ( ( Base ‘ ( Scalar ‘ 𝐴 ) ) ↑m ( 𝑌 × 𝑋 ) ) ∧ 𝑊 finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
156 |
155
|
simpld |
⊢ ( 𝜑 → 𝑊 ∈ ( ( Base ‘ ( Scalar ‘ 𝐴 ) ) ↑m ( 𝑌 × 𝑋 ) ) ) |
157 |
|
fvexd |
⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∈ V ) |
158 |
157 150
|
elmapd |
⊢ ( 𝜑 → ( 𝑊 ∈ ( ( Base ‘ ( Scalar ‘ 𝐴 ) ) ↑m ( 𝑌 × 𝑋 ) ) ↔ 𝑊 : ( 𝑌 × 𝑋 ) ⟶ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
159 |
156 158
|
mpbid |
⊢ ( 𝜑 → 𝑊 : ( 𝑌 × 𝑋 ) ⟶ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) |
160 |
159
|
ffnd |
⊢ ( 𝜑 → 𝑊 Fn ( 𝑌 × 𝑋 ) ) |
161 |
160
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝑊 Fn ( 𝑌 × 𝑋 ) ) |
162 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝑗 ∈ 𝑌 ) |
163 |
155
|
simprd |
⊢ ( 𝜑 → 𝑊 finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
164 |
|
drngring |
⊢ ( 𝐸 ∈ DivRing → 𝐸 ∈ Ring ) |
165 |
6 164
|
syl |
⊢ ( 𝜑 → 𝐸 ∈ Ring ) |
166 |
|
ringmnd |
⊢ ( 𝐸 ∈ Ring → 𝐸 ∈ Mnd ) |
167 |
165 166
|
syl |
⊢ ( 𝜑 → 𝐸 ∈ Mnd ) |
168 |
|
subrgsubg |
⊢ ( 𝑉 ∈ ( SubRing ‘ 𝐸 ) → 𝑉 ∈ ( SubGrp ‘ 𝐸 ) ) |
169 |
20 168
|
syl |
⊢ ( 𝜑 → 𝑉 ∈ ( SubGrp ‘ 𝐸 ) ) |
170 |
|
eqid |
⊢ ( 0g ‘ 𝐸 ) = ( 0g ‘ 𝐸 ) |
171 |
170
|
subg0cl |
⊢ ( 𝑉 ∈ ( SubGrp ‘ 𝐸 ) → ( 0g ‘ 𝐸 ) ∈ 𝑉 ) |
172 |
169 171
|
syl |
⊢ ( 𝜑 → ( 0g ‘ 𝐸 ) ∈ 𝑉 ) |
173 |
53 172
|
sseldd |
⊢ ( 𝜑 → ( 0g ‘ 𝐸 ) ∈ 𝑈 ) |
174 |
4 29 170
|
ress0g |
⊢ ( ( 𝐸 ∈ Mnd ∧ ( 0g ‘ 𝐸 ) ∈ 𝑈 ∧ 𝑈 ⊆ ( Base ‘ 𝐸 ) ) → ( 0g ‘ 𝐸 ) = ( 0g ‘ 𝐹 ) ) |
175 |
167 173 31 174
|
syl3anc |
⊢ ( 𝜑 → ( 0g ‘ 𝐸 ) = ( 0g ‘ 𝐹 ) ) |
176 |
61
|
fveq2d |
⊢ ( 𝜑 → ( 0g ‘ 𝐾 ) = ( 0g ‘ ( Scalar ‘ 𝐶 ) ) ) |
177 |
5 170
|
subrg0 |
⊢ ( 𝑉 ∈ ( SubRing ‘ 𝐸 ) → ( 0g ‘ 𝐸 ) = ( 0g ‘ 𝐾 ) ) |
178 |
20 177
|
syl |
⊢ ( 𝜑 → ( 0g ‘ 𝐸 ) = ( 0g ‘ 𝐾 ) ) |
179 |
67
|
fveq2d |
⊢ ( 𝜑 → ( 0g ‘ ( Scalar ‘ 𝐴 ) ) = ( 0g ‘ ( Scalar ‘ 𝐶 ) ) ) |
180 |
176 178 179
|
3eqtr4d |
⊢ ( 𝜑 → ( 0g ‘ 𝐸 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
181 |
175 180
|
eqtr3d |
⊢ ( 𝜑 → ( 0g ‘ 𝐹 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
182 |
163 181
|
breqtrrd |
⊢ ( 𝜑 → 𝑊 finSupp ( 0g ‘ 𝐹 ) ) |
183 |
182
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝑊 finSupp ( 0g ‘ 𝐹 ) ) |
184 |
145 146 147 148 161 162 183
|
fsuppcurry1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) finSupp ( 0g ‘ 𝐹 ) ) |
185 |
181
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 0g ‘ 𝐹 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
186 |
184 185
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
187 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) |
188 |
159
|
fovrnda |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) ) → ( 𝑗 𝑊 𝑖 ) ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) |
189 |
188
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( 𝑗 𝑊 𝑖 ) ∈ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) |
190 |
187 189
|
fvmpt2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) = ( 𝑗 𝑊 𝑖 ) ) |
191 |
190
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) = ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) |
192 |
124
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( ·𝑠 ‘ 𝐴 ) = ( ·𝑠 ‘ 𝐵 ) ) |
193 |
192
|
oveqd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) = ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) |
194 |
191 193
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) = ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) |
195 |
194
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) |
196 |
195
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) |
197 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝐸 ∈ DivRing ) |
198 |
20
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝑉 ∈ ( SubRing ‘ 𝐸 ) ) |
199 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝐾 ∈ DivRing ) |
200 |
1 197 198 5 199 148
|
drgextgsum |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐸 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) = ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) |
201 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝑈 ∈ ( SubRing ‘ 𝐸 ) ) |
202 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝐹 ∈ DivRing ) |
203 |
2 197 201 4 202 148
|
drgextgsum |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐸 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) = ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) |
204 |
200 203
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) = ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) |
205 |
196 204
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) |
206 |
14
|
mptexd |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ∈ V ) |
207 |
|
eqid |
⊢ ( 0g ‘ 𝐵 ) = ( 0g ‘ 𝐵 ) |
208 |
2 4
|
sralvec |
⊢ ( ( 𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ ( SubRing ‘ 𝐸 ) ) → 𝐵 ∈ LVec ) |
209 |
6 7 9 208
|
syl3anc |
⊢ ( 𝜑 → 𝐵 ∈ LVec ) |
210 |
|
lveclmod |
⊢ ( 𝐵 ∈ LVec → 𝐵 ∈ LMod ) |
211 |
209 210
|
syl |
⊢ ( 𝜑 → 𝐵 ∈ LMod ) |
212 |
211
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝐵 ∈ LMod ) |
213 |
|
lmodabl |
⊢ ( 𝐵 ∈ LMod → 𝐵 ∈ Abel ) |
214 |
212 213
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝐵 ∈ Abel ) |
215 |
2
|
a1i |
⊢ ( 𝜑 → 𝐵 = ( ( subringAlg ‘ 𝐸 ) ‘ 𝑈 ) ) |
216 |
215 9 31
|
srasubrg |
⊢ ( 𝜑 → 𝑈 ∈ ( SubRing ‘ 𝐵 ) ) |
217 |
|
subrgsubg |
⊢ ( 𝑈 ∈ ( SubRing ‘ 𝐵 ) → 𝑈 ∈ ( SubGrp ‘ 𝐵 ) ) |
218 |
216 217
|
syl |
⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝐵 ) ) |
219 |
218
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝑈 ∈ ( SubGrp ‘ 𝐵 ) ) |
220 |
116
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → 𝐶 ∈ LMod ) |
221 |
68
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( Base ‘ ( Scalar ‘ 𝐴 ) ) = ( Base ‘ ( Scalar ‘ 𝐶 ) ) ) |
222 |
189 221
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( 𝑗 𝑊 𝑖 ) ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) ) |
223 |
28
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → 𝑋 ⊆ ( Base ‘ 𝐶 ) ) |
224 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → 𝑖 ∈ 𝑋 ) |
225 |
223 224
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → 𝑖 ∈ ( Base ‘ 𝐶 ) ) |
226 |
25 117 118 119
|
lmodvscl |
⊢ ( ( 𝐶 ∈ LMod ∧ ( 𝑗 𝑊 𝑖 ) ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) ∧ 𝑖 ∈ ( Base ‘ 𝐶 ) ) → ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∈ ( Base ‘ 𝐶 ) ) |
227 |
220 222 225 226
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∈ ( Base ‘ 𝐶 ) ) |
228 |
130
|
oveqd |
⊢ ( 𝜑 → ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) = ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ) |
229 |
228
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) = ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ) |
230 |
39
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → 𝑈 = ( Base ‘ 𝐶 ) ) |
231 |
227 229 230
|
3eltr4d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ∈ 𝑈 ) |
232 |
231
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) : 𝑋 ⟶ 𝑈 ) |
233 |
215 31
|
srasca |
⊢ ( 𝜑 → ( 𝐸 ↾s 𝑈 ) = ( Scalar ‘ 𝐵 ) ) |
234 |
4 233
|
syl5eq |
⊢ ( 𝜑 → 𝐹 = ( Scalar ‘ 𝐵 ) ) |
235 |
234
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝐹 = ( Scalar ‘ 𝐵 ) ) |
236 |
|
eqid |
⊢ ( Base ‘ 𝐵 ) = ( Base ‘ 𝐵 ) |
237 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( 𝑗 𝑊 𝑖 ) ∈ V ) |
238 |
28 40
|
sstrd |
⊢ ( 𝜑 → 𝑋 ⊆ ( Base ‘ 𝐸 ) ) |
239 |
238
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) ) → 𝑋 ⊆ ( Base ‘ 𝐸 ) ) |
240 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) ) → 𝑖 ∈ 𝑋 ) |
241 |
239 240
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) ) → 𝑖 ∈ ( Base ‘ 𝐸 ) ) |
242 |
241
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → 𝑖 ∈ ( Base ‘ 𝐸 ) ) |
243 |
215 31
|
srabase |
⊢ ( 𝜑 → ( Base ‘ 𝐸 ) = ( Base ‘ 𝐵 ) ) |
244 |
243
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( Base ‘ 𝐸 ) = ( Base ‘ 𝐵 ) ) |
245 |
242 244
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → 𝑖 ∈ ( Base ‘ 𝐵 ) ) |
246 |
|
eqid |
⊢ ( 0g ‘ 𝐹 ) = ( 0g ‘ 𝐹 ) |
247 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝐵 ) = ( ·𝑠 ‘ 𝐵 ) |
248 |
148 212 235 236 237 245 207 246 247 184
|
mptscmfsupp0 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) finSupp ( 0g ‘ 𝐵 ) ) |
249 |
207 214 148 219 232 248
|
gsumsubgcl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ∈ 𝑈 ) |
250 |
234
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ 𝐹 ) = ( Base ‘ ( Scalar ‘ 𝐵 ) ) ) |
251 |
33 250
|
eqtrd |
⊢ ( 𝜑 → 𝑈 = ( Base ‘ ( Scalar ‘ 𝐵 ) ) ) |
252 |
251
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝑈 = ( Base ‘ ( Scalar ‘ 𝐵 ) ) ) |
253 |
249 252
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ∈ ( Base ‘ ( Scalar ‘ 𝐵 ) ) ) |
254 |
253
|
fmpttd |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) : 𝑌 ⟶ ( Base ‘ ( Scalar ‘ 𝐵 ) ) ) |
255 |
254
|
ffund |
⊢ ( 𝜑 → Fun ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ) |
256 |
|
fvexd |
⊢ ( 𝜑 → ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ∈ V ) |
257 |
|
fconstmpt |
⊢ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) = ( 𝑖 ∈ 𝑋 ↦ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
258 |
257
|
eqeq2i |
⊢ ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ↔ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
259 |
|
ovex |
⊢ ( 𝑘 𝑊 𝑖 ) ∈ V |
260 |
259
|
rgenw |
⊢ ∀ 𝑖 ∈ 𝑋 ( 𝑘 𝑊 𝑖 ) ∈ V |
261 |
|
mpteqb |
⊢ ( ∀ 𝑖 ∈ 𝑋 ( 𝑘 𝑊 𝑖 ) ∈ V → ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ↔ ∀ 𝑖 ∈ 𝑋 ( 𝑘 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
262 |
260 261
|
ax-mp |
⊢ ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ↔ ∀ 𝑖 ∈ 𝑋 ( 𝑘 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
263 |
258 262
|
bitri |
⊢ ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ↔ ∀ 𝑖 ∈ 𝑋 ( 𝑘 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
264 |
263
|
necon3abii |
⊢ ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ↔ ¬ ∀ 𝑖 ∈ 𝑋 ( 𝑘 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
265 |
|
df-ov |
⊢ ( 𝑘 𝑊 𝑖 ) = ( 𝑊 ‘ 〈 𝑘 , 𝑖 〉 ) |
266 |
265
|
eqcomi |
⊢ ( 𝑊 ‘ 〈 𝑘 , 𝑖 〉 ) = ( 𝑘 𝑊 𝑖 ) |
267 |
266
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( 𝑊 ‘ 〈 𝑘 , 𝑖 〉 ) = ( 𝑘 𝑊 𝑖 ) ) |
268 |
267
|
eqeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝑊 ‘ 〈 𝑘 , 𝑖 〉 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ↔ ( 𝑘 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
269 |
268
|
necon3abid |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝑊 ‘ 〈 𝑘 , 𝑖 〉 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ↔ ¬ ( 𝑘 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
270 |
269
|
rexbidva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( ∃ 𝑖 ∈ 𝑋 ( 𝑊 ‘ 〈 𝑘 , 𝑖 〉 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ↔ ∃ 𝑖 ∈ 𝑋 ¬ ( 𝑘 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
271 |
|
rexnal |
⊢ ( ∃ 𝑖 ∈ 𝑋 ¬ ( 𝑘 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ↔ ¬ ∀ 𝑖 ∈ 𝑋 ( 𝑘 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
272 |
270 271
|
bitr2di |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( ¬ ∀ 𝑖 ∈ 𝑋 ( 𝑘 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ↔ ∃ 𝑖 ∈ 𝑋 ( 𝑊 ‘ 〈 𝑘 , 𝑖 〉 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
273 |
264 272
|
syl5bb |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ) → ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ↔ ∃ 𝑖 ∈ 𝑋 ( 𝑊 ‘ 〈 𝑘 , 𝑖 〉 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
274 |
273
|
rabbidva |
⊢ ( 𝜑 → { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } = { 𝑘 ∈ 𝑌 ∣ ∃ 𝑖 ∈ 𝑋 ( 𝑊 ‘ 〈 𝑘 , 𝑖 〉 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) |
275 |
|
fveq2 |
⊢ ( 𝑧 = 〈 𝑘 , 𝑖 〉 → ( 𝑊 ‘ 𝑧 ) = ( 𝑊 ‘ 〈 𝑘 , 𝑖 〉 ) ) |
276 |
275
|
neeq1d |
⊢ ( 𝑧 = 〈 𝑘 , 𝑖 〉 → ( ( 𝑊 ‘ 𝑧 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ↔ ( 𝑊 ‘ 〈 𝑘 , 𝑖 〉 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
277 |
276
|
dmrab |
⊢ dom { 𝑧 ∈ ( 𝑌 × 𝑋 ) ∣ ( 𝑊 ‘ 𝑧 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } = { 𝑘 ∈ 𝑌 ∣ ∃ 𝑖 ∈ 𝑋 ( 𝑊 ‘ 〈 𝑘 , 𝑖 〉 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } |
278 |
274 277
|
eqtr4di |
⊢ ( 𝜑 → { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } = dom { 𝑧 ∈ ( 𝑌 × 𝑋 ) ∣ ( 𝑊 ‘ 𝑧 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) |
279 |
|
fvexd |
⊢ ( 𝜑 → ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∈ V ) |
280 |
|
suppvalfn |
⊢ ( ( 𝑊 Fn ( 𝑌 × 𝑋 ) ∧ ( 𝑌 × 𝑋 ) ∈ V ∧ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∈ V ) → ( 𝑊 supp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) = { 𝑧 ∈ ( 𝑌 × 𝑋 ) ∣ ( 𝑊 ‘ 𝑧 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) |
281 |
160 150 279 280
|
syl3anc |
⊢ ( 𝜑 → ( 𝑊 supp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) = { 𝑧 ∈ ( 𝑌 × 𝑋 ) ∣ ( 𝑊 ‘ 𝑧 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) |
282 |
163
|
fsuppimpd |
⊢ ( 𝜑 → ( 𝑊 supp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ∈ Fin ) |
283 |
281 282
|
eqeltrrd |
⊢ ( 𝜑 → { 𝑧 ∈ ( 𝑌 × 𝑋 ) ∣ ( 𝑊 ‘ 𝑧 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ∈ Fin ) |
284 |
|
dmfi |
⊢ ( { 𝑧 ∈ ( 𝑌 × 𝑋 ) ∣ ( 𝑊 ‘ 𝑧 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ∈ Fin → dom { 𝑧 ∈ ( 𝑌 × 𝑋 ) ∣ ( 𝑊 ‘ 𝑧 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ∈ Fin ) |
285 |
283 284
|
syl |
⊢ ( 𝜑 → dom { 𝑧 ∈ ( 𝑌 × 𝑋 ) ∣ ( 𝑊 ‘ 𝑧 ) ≠ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ∈ Fin ) |
286 |
278 285
|
eqeltrd |
⊢ ( 𝜑 → { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ∈ Fin ) |
287 |
|
nfv |
⊢ Ⅎ 𝑖 𝜑 |
288 |
|
nfcv |
⊢ Ⅎ 𝑖 𝑌 |
289 |
|
nfmpt1 |
⊢ Ⅎ 𝑖 ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) |
290 |
|
nfcv |
⊢ Ⅎ 𝑖 ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) |
291 |
289 290
|
nfne |
⊢ Ⅎ 𝑖 ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) |
292 |
291 288
|
nfrabw |
⊢ Ⅎ 𝑖 { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } |
293 |
288 292
|
nfdif |
⊢ Ⅎ 𝑖 ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) |
294 |
293
|
nfcri |
⊢ Ⅎ 𝑖 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) |
295 |
287 294
|
nfan |
⊢ Ⅎ 𝑖 ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) |
296 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) → 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) |
297 |
296
|
eldifad |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) → 𝑗 ∈ 𝑌 ) |
298 |
296
|
eldifbd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) → ¬ 𝑗 ∈ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) |
299 |
|
oveq1 |
⊢ ( 𝑘 = 𝑗 → ( 𝑘 𝑊 𝑖 ) = ( 𝑗 𝑊 𝑖 ) ) |
300 |
299
|
mpteq2dv |
⊢ ( 𝑘 = 𝑗 → ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) |
301 |
300
|
neeq1d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ↔ ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) ) |
302 |
301
|
elrab |
⊢ ( 𝑗 ∈ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ↔ ( 𝑗 ∈ 𝑌 ∧ ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) ) |
303 |
298 302
|
sylnib |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) → ¬ ( 𝑗 ∈ 𝑌 ∧ ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) ) |
304 |
297 303
|
mpnanrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) → ¬ ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) |
305 |
|
nne |
⊢ ( ¬ ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ↔ ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) |
306 |
304 305
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) → ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) |
307 |
306 257
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) → ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
308 |
|
ovex |
⊢ ( 𝑗 𝑊 𝑖 ) ∈ V |
309 |
308
|
rgenw |
⊢ ∀ 𝑖 ∈ 𝑋 ( 𝑗 𝑊 𝑖 ) ∈ V |
310 |
|
mpteqb |
⊢ ( ∀ 𝑖 ∈ 𝑋 ( 𝑗 𝑊 𝑖 ) ∈ V → ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ↔ ∀ 𝑖 ∈ 𝑋 ( 𝑗 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
311 |
309 310
|
ax-mp |
⊢ ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ↔ ∀ 𝑖 ∈ 𝑋 ( 𝑗 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
312 |
307 311
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) → ∀ 𝑖 ∈ 𝑋 ( 𝑗 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
313 |
312
|
r19.21bi |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) ∧ 𝑖 ∈ 𝑋 ) → ( 𝑗 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
314 |
313
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) = ( ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) |
315 |
2 6 9
|
drgext0g |
⊢ ( 𝜑 → ( 0g ‘ 𝐸 ) = ( 0g ‘ 𝐵 ) ) |
316 |
2 6 9
|
drgext0gsca |
⊢ ( 𝜑 → ( 0g ‘ 𝐵 ) = ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) |
317 |
315 180 316
|
3eqtr3d |
⊢ ( 𝜑 → ( 0g ‘ ( Scalar ‘ 𝐴 ) ) = ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) |
318 |
317
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) ∧ 𝑖 ∈ 𝑋 ) → ( 0g ‘ ( Scalar ‘ 𝐴 ) ) = ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) |
319 |
318
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) ∧ 𝑖 ∈ 𝑋 ) → ( ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) = ( ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) |
320 |
211
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) ∧ 𝑖 ∈ 𝑋 ) → 𝐵 ∈ LMod ) |
321 |
297 245
|
syldanl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) ∧ 𝑖 ∈ 𝑋 ) → 𝑖 ∈ ( Base ‘ 𝐵 ) ) |
322 |
|
eqid |
⊢ ( Scalar ‘ 𝐵 ) = ( Scalar ‘ 𝐵 ) |
323 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝐵 ) ) = ( 0g ‘ ( Scalar ‘ 𝐵 ) ) |
324 |
236 322 247 323 207
|
lmod0vs |
⊢ ( ( 𝐵 ∈ LMod ∧ 𝑖 ∈ ( Base ‘ 𝐵 ) ) → ( ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) = ( 0g ‘ 𝐵 ) ) |
325 |
320 321 324
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) ∧ 𝑖 ∈ 𝑋 ) → ( ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) = ( 0g ‘ 𝐵 ) ) |
326 |
314 319 325
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) = ( 0g ‘ 𝐵 ) ) |
327 |
295 326
|
mpteq2da |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) → ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( 0g ‘ 𝐵 ) ) ) |
328 |
327
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) → ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) = ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( 0g ‘ 𝐵 ) ) ) ) |
329 |
211 213
|
syl |
⊢ ( 𝜑 → 𝐵 ∈ Abel ) |
330 |
|
ablgrp |
⊢ ( 𝐵 ∈ Abel → 𝐵 ∈ Grp ) |
331 |
|
grpmnd |
⊢ ( 𝐵 ∈ Grp → 𝐵 ∈ Mnd ) |
332 |
329 330 331
|
3syl |
⊢ ( 𝜑 → 𝐵 ∈ Mnd ) |
333 |
207
|
gsumz |
⊢ ( ( 𝐵 ∈ Mnd ∧ 𝑋 ∈ ( LBasis ‘ 𝐶 ) ) → ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( 0g ‘ 𝐵 ) ) ) = ( 0g ‘ 𝐵 ) ) |
334 |
332 13 333
|
syl2anc |
⊢ ( 𝜑 → ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( 0g ‘ 𝐵 ) ) ) = ( 0g ‘ 𝐵 ) ) |
335 |
334
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) → ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( 0g ‘ 𝐵 ) ) ) = ( 0g ‘ 𝐵 ) ) |
336 |
316
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) → ( 0g ‘ 𝐵 ) = ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) |
337 |
328 335 336
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑌 ∖ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) → ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) = ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) |
338 |
337 14
|
suppss2 |
⊢ ( 𝜑 → ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) supp ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) ⊆ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) |
339 |
|
suppssfifsupp |
⊢ ( ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ∈ V ∧ Fun ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ∧ ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ∈ V ) ∧ ( { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ∈ Fin ∧ ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) supp ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) ⊆ { 𝑘 ∈ 𝑌 ∣ ( 𝑖 ∈ 𝑋 ↦ ( 𝑘 𝑊 𝑖 ) ) ≠ ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) } ) ) → ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) finSupp ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) |
340 |
206 255 256 286 338 339
|
syl32anc |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) finSupp ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) |
341 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) = ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ) |
342 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ∈ V ) |
343 |
341 342
|
fvmpt2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ‘ 𝑗 ) = ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) |
344 |
343
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) = ( ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) |
345 |
344
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑌 ↦ ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) = ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) |
346 |
345
|
oveq2d |
⊢ ( 𝜑 → ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) ) |
347 |
124
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( ·𝑠 ‘ 𝐴 ) = ( ·𝑠 ‘ 𝐵 ) ) |
348 |
50
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( .r ‘ 𝐸 ) = ( ·𝑠 ‘ 𝐴 ) ) |
349 |
348
|
oveqd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) = ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) |
350 |
349
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) |
351 |
123
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( .r ‘ 𝐸 ) = ( ·𝑠 ‘ 𝐵 ) ) |
352 |
351
|
oveqd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) = ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) |
353 |
352
|
mpteq2dv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) |
354 |
350 353
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) |
355 |
354
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) |
356 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝑗 = 𝑗 ) |
357 |
347 355 356
|
oveq123d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐴 ) 𝑗 ) = ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) |
358 |
204
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) = ( ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) |
359 |
357 358
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐴 ) 𝑗 ) = ( ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) |
360 |
359
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐴 ) 𝑗 ) ) = ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) |
361 |
360
|
oveq2d |
⊢ ( 𝜑 → ( 𝐴 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐴 ) 𝑗 ) ) ) = ( 𝐴 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) ) |
362 |
1 29
|
sraring |
⊢ ( ( 𝐸 ∈ Ring ∧ 𝑉 ⊆ ( Base ‘ 𝐸 ) ) → 𝐴 ∈ Ring ) |
363 |
165 43 362
|
syl2anc |
⊢ ( 𝜑 → 𝐴 ∈ Ring ) |
364 |
|
ringcmn |
⊢ ( 𝐴 ∈ Ring → 𝐴 ∈ CMnd ) |
365 |
363 364
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ CMnd ) |
366 |
165
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) ) → 𝐸 ∈ Ring ) |
367 |
|
eqid |
⊢ ( LBasis ‘ 𝐵 ) = ( LBasis ‘ 𝐵 ) |
368 |
236 367
|
lbsss |
⊢ ( 𝑌 ∈ ( LBasis ‘ 𝐵 ) → 𝑌 ⊆ ( Base ‘ 𝐵 ) ) |
369 |
14 368
|
syl |
⊢ ( 𝜑 → 𝑌 ⊆ ( Base ‘ 𝐵 ) ) |
370 |
369 243
|
sseqtrrd |
⊢ ( 𝜑 → 𝑌 ⊆ ( Base ‘ 𝐸 ) ) |
371 |
370
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) ) → 𝑌 ⊆ ( Base ‘ 𝐸 ) ) |
372 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) ) → 𝑗 ∈ 𝑌 ) |
373 |
371 372
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) ) → 𝑗 ∈ ( Base ‘ 𝐸 ) ) |
374 |
29 74
|
ringcl |
⊢ ( ( 𝐸 ∈ Ring ∧ 𝑖 ∈ ( Base ‘ 𝐸 ) ∧ 𝑗 ∈ ( Base ‘ 𝐸 ) ) → ( 𝑖 ( .r ‘ 𝐸 ) 𝑗 ) ∈ ( Base ‘ 𝐸 ) ) |
375 |
366 241 373 374
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) ) → ( 𝑖 ( .r ‘ 𝐸 ) 𝑗 ) ∈ ( Base ‘ 𝐸 ) ) |
376 |
44
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) ) → ( Base ‘ 𝐸 ) = ( Base ‘ 𝐴 ) ) |
377 |
375 376
|
eleqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) ) → ( 𝑖 ( .r ‘ 𝐸 ) 𝑗 ) ∈ ( Base ‘ 𝐴 ) ) |
378 |
377
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑗 ∈ 𝑌 ∀ 𝑖 ∈ 𝑋 ( 𝑖 ( .r ‘ 𝐸 ) 𝑗 ) ∈ ( Base ‘ 𝐴 ) ) |
379 |
11
|
fmpo |
⊢ ( ∀ 𝑗 ∈ 𝑌 ∀ 𝑖 ∈ 𝑋 ( 𝑖 ( .r ‘ 𝐸 ) 𝑗 ) ∈ ( Base ‘ 𝐴 ) ↔ 𝐷 : ( 𝑌 × 𝑋 ) ⟶ ( Base ‘ 𝐴 ) ) |
380 |
378 379
|
sylib |
⊢ ( 𝜑 → 𝐷 : ( 𝑌 × 𝑋 ) ⟶ ( Base ‘ 𝐴 ) ) |
381 |
79 80 82 81 24 159 380 150
|
lcomf |
⊢ ( 𝜑 → ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) : ( 𝑌 × 𝑋 ) ⟶ ( Base ‘ 𝐴 ) ) |
382 |
79 80 82 81 24 159 380 150 139 140 163
|
lcomfsupp |
⊢ ( 𝜑 → ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) finSupp ( 0g ‘ 𝐴 ) ) |
383 |
81 139 365 14 13 381 382
|
gsumxp |
⊢ ( 𝜑 → ( 𝐴 Σg ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) ) = ( 𝐴 Σg ( 𝑗 ∈ 𝑌 ↦ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) 𝑖 ) ) ) ) ) ) |
384 |
165
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → 𝐸 ∈ Ring ) |
385 |
159
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → 𝑊 : ( 𝑌 × 𝑋 ) ⟶ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) |
386 |
64 62
|
eqtrd |
⊢ ( 𝜑 → 𝑉 = ( Base ‘ ( Scalar ‘ 𝐶 ) ) ) |
387 |
386 43
|
eqsstrrd |
⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐶 ) ) ⊆ ( Base ‘ 𝐸 ) ) |
388 |
68 387
|
eqsstrd |
⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐴 ) ) ⊆ ( Base ‘ 𝐸 ) ) |
389 |
388 44
|
sseqtrd |
⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐴 ) ) ⊆ ( Base ‘ 𝐴 ) ) |
390 |
389
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → ( Base ‘ ( Scalar ‘ 𝐴 ) ) ⊆ ( Base ‘ 𝐴 ) ) |
391 |
385 390
|
fssd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → 𝑊 : ( 𝑌 × 𝑋 ) ⟶ ( Base ‘ 𝐴 ) ) |
392 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → 𝑗 ∈ 𝑌 ) |
393 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → 𝑖 ∈ 𝑋 ) |
394 |
391 392 393
|
fovrnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → ( 𝑗 𝑊 𝑖 ) ∈ ( Base ‘ 𝐴 ) ) |
395 |
44
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → ( Base ‘ 𝐸 ) = ( Base ‘ 𝐴 ) ) |
396 |
394 395
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → ( 𝑗 𝑊 𝑖 ) ∈ ( Base ‘ 𝐸 ) ) |
397 |
241
|
3impb |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → 𝑖 ∈ ( Base ‘ 𝐸 ) ) |
398 |
373
|
3impb |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → 𝑗 ∈ ( Base ‘ 𝐸 ) ) |
399 |
29 74
|
ringass |
⊢ ( ( 𝐸 ∈ Ring ∧ ( ( 𝑗 𝑊 𝑖 ) ∈ ( Base ‘ 𝐸 ) ∧ 𝑖 ∈ ( Base ‘ 𝐸 ) ∧ 𝑗 ∈ ( Base ‘ 𝐸 ) ) ) → ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) = ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) ( 𝑖 ( .r ‘ 𝐸 ) 𝑗 ) ) ) |
400 |
384 396 397 398 399
|
syl13anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) = ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) ( 𝑖 ( .r ‘ 𝐸 ) 𝑗 ) ) ) |
401 |
400
|
mpoeq3dva |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ) = ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) ( 𝑖 ( .r ‘ 𝐸 ) 𝑗 ) ) ) ) |
402 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → ( 𝑗 𝑊 𝑖 ) ∈ V ) |
403 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → ( 𝑖 ( .r ‘ 𝐸 ) 𝑗 ) ∈ V ) |
404 |
|
fnov |
⊢ ( 𝑊 Fn ( 𝑌 × 𝑋 ) ↔ 𝑊 = ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) |
405 |
160 404
|
sylib |
⊢ ( 𝜑 → 𝑊 = ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) |
406 |
11
|
a1i |
⊢ ( 𝜑 → 𝐷 = ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( 𝑖 ( .r ‘ 𝐸 ) 𝑗 ) ) ) |
407 |
14 13 402 403 405 406
|
offval22 |
⊢ ( 𝜑 → ( 𝑊 ∘f ( .r ‘ 𝐸 ) 𝐷 ) = ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) ( 𝑖 ( .r ‘ 𝐸 ) 𝑗 ) ) ) ) |
408 |
50
|
ofeqd |
⊢ ( 𝜑 → ∘f ( .r ‘ 𝐸 ) = ∘f ( ·𝑠 ‘ 𝐴 ) ) |
409 |
408
|
oveqd |
⊢ ( 𝜑 → ( 𝑊 ∘f ( .r ‘ 𝐸 ) 𝐷 ) = ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) ) |
410 |
401 407 409
|
3eqtr2rd |
⊢ ( 𝜑 → ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) = ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ) ) |
411 |
410
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) = ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ) ) |
412 |
411
|
oveqd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( 𝑗 ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) 𝑖 ) = ( 𝑗 ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ) 𝑖 ) ) |
413 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → 𝑗 ∈ 𝑌 ) |
414 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ∈ V ) |
415 |
|
eqid |
⊢ ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ) = ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ) |
416 |
415
|
ovmpt4g |
⊢ ( ( 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ∧ ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ∈ V ) → ( 𝑗 ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ) 𝑖 ) = ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ) |
417 |
413 224 414 416
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( 𝑗 ( 𝑗 ∈ 𝑌 , 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ) 𝑖 ) = ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ) |
418 |
412 417
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( 𝑗 ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) 𝑖 ) = ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ) |
419 |
418
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ) ) |
420 |
419
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐸 Σg ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) 𝑖 ) ) ) = ( 𝐸 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ) ) ) |
421 |
165
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝐸 ∈ Ring ) |
422 |
370
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝑗 ∈ ( Base ‘ 𝐸 ) ) |
423 |
165
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → 𝐸 ∈ Ring ) |
424 |
387
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( Base ‘ ( Scalar ‘ 𝐶 ) ) ⊆ ( Base ‘ 𝐸 ) ) |
425 |
424 222
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( 𝑗 𝑊 𝑖 ) ∈ ( Base ‘ 𝐸 ) ) |
426 |
29 74
|
ringcl |
⊢ ( ( 𝐸 ∈ Ring ∧ ( 𝑗 𝑊 𝑖 ) ∈ ( Base ‘ 𝐸 ) ∧ 𝑖 ∈ ( Base ‘ 𝐸 ) ) → ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ∈ ( Base ‘ 𝐸 ) ) |
427 |
423 425 242 426
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ∈ ( Base ‘ 𝐸 ) ) |
428 |
315
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 0g ‘ 𝐸 ) = ( 0g ‘ 𝐵 ) ) |
429 |
248 353 428
|
3brtr4d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ) finSupp ( 0g ‘ 𝐸 ) ) |
430 |
29 170 100 74 421 148 422 427 429
|
gsummulc1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐸 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ( .r ‘ 𝐸 ) 𝑗 ) ) ) = ( ( 𝐸 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ) ) ( .r ‘ 𝐸 ) 𝑗 ) ) |
431 |
420 430
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐸 Σg ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) 𝑖 ) ) ) = ( ( 𝐸 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ) ) ( .r ‘ 𝐸 ) 𝑗 ) ) |
432 |
148
|
mptexd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) 𝑖 ) ) ∈ V ) |
433 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝐴 ∈ LMod ) |
434 |
43
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → 𝑉 ⊆ ( Base ‘ 𝐸 ) ) |
435 |
1 432 197 433 434
|
gsumsra |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐸 Σg ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) 𝑖 ) ) ) = ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) 𝑖 ) ) ) ) |
436 |
148
|
mptexd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ) ∈ V ) |
437 |
1 436 197 433 434
|
gsumsra |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐸 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ) ) = ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ) ) ) |
438 |
437
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( ( 𝐸 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ) ) ( .r ‘ 𝐸 ) 𝑗 ) = ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ) ) ( .r ‘ 𝐸 ) 𝑗 ) ) |
439 |
50
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( .r ‘ 𝐸 ) = ( ·𝑠 ‘ 𝐴 ) ) |
440 |
350
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ) ) = ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) ) |
441 |
439 440 356
|
oveq123d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ) ) ( .r ‘ 𝐸 ) 𝑗 ) = ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐴 ) 𝑗 ) ) |
442 |
438 441
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( ( 𝐸 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( .r ‘ 𝐸 ) 𝑖 ) ) ) ( .r ‘ 𝐸 ) 𝑗 ) = ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐴 ) 𝑗 ) ) |
443 |
431 435 442
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) 𝑖 ) ) ) = ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐴 ) 𝑗 ) ) |
444 |
443
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑌 ↦ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) 𝑖 ) ) ) ) = ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐴 ) 𝑗 ) ) ) |
445 |
444
|
oveq2d |
⊢ ( 𝜑 → ( 𝐴 Σg ( 𝑗 ∈ 𝑌 ↦ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 ( 𝑊 ∘f ( ·𝑠 ‘ 𝐴 ) 𝐷 ) 𝑖 ) ) ) ) ) = ( 𝐴 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐴 ) 𝑗 ) ) ) ) |
446 |
383 16 445
|
3eqtr3rd |
⊢ ( 𝜑 → ( 𝐴 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐴 ) 𝑗 ) ) ) = ( 0g ‘ 𝐴 ) ) |
447 |
1 6 20
|
drgext0g |
⊢ ( 𝜑 → ( 0g ‘ 𝐸 ) = ( 0g ‘ 𝐴 ) ) |
448 |
446 447 315
|
3eqtr2d |
⊢ ( 𝜑 → ( 𝐴 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐴 ) 𝑗 ) ) ) = ( 0g ‘ 𝐵 ) ) |
449 |
1 6 20 5 8 14
|
drgextgsum |
⊢ ( 𝜑 → ( 𝐸 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 𝐴 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) ) |
450 |
2 6 9 4 7 14
|
drgextgsum |
⊢ ( 𝜑 → ( 𝐸 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) ) |
451 |
449 450
|
eqtr3d |
⊢ ( 𝜑 → ( 𝐴 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) ) |
452 |
361 448 451
|
3eqtr3rd |
⊢ ( 𝜑 → ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 0g ‘ 𝐵 ) ) |
453 |
346 452
|
eqtrd |
⊢ ( 𝜑 → ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 0g ‘ 𝐵 ) ) |
454 |
|
breq1 |
⊢ ( 𝑏 = ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) → ( 𝑏 finSupp ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ↔ ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) finSupp ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) ) |
455 |
|
nfmpt1 |
⊢ Ⅎ 𝑗 ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) |
456 |
455
|
nfeq2 |
⊢ Ⅎ 𝑗 𝑏 = ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) |
457 |
|
fveq1 |
⊢ ( 𝑏 = ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) → ( 𝑏 ‘ 𝑗 ) = ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ‘ 𝑗 ) ) |
458 |
457
|
oveq1d |
⊢ ( 𝑏 = ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) → ( ( 𝑏 ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) = ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) |
459 |
458
|
adantr |
⊢ ( ( 𝑏 = ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ∧ 𝑗 ∈ 𝑌 ) → ( ( 𝑏 ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) = ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) |
460 |
456 459
|
mpteq2da |
⊢ ( 𝑏 = ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) → ( 𝑗 ∈ 𝑌 ↦ ( ( 𝑏 ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) = ( 𝑗 ∈ 𝑌 ↦ ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) |
461 |
460
|
oveq2d |
⊢ ( 𝑏 = ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) → ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝑏 ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) ) |
462 |
461
|
eqeq1d |
⊢ ( 𝑏 = ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) → ( ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝑏 ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 0g ‘ 𝐵 ) ↔ ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 0g ‘ 𝐵 ) ) ) |
463 |
454 462
|
anbi12d |
⊢ ( 𝑏 = ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) → ( ( 𝑏 finSupp ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ∧ ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝑏 ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 0g ‘ 𝐵 ) ) ↔ ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) finSupp ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ∧ ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 0g ‘ 𝐵 ) ) ) ) |
464 |
|
eqeq1 |
⊢ ( 𝑏 = ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) → ( 𝑏 = ( 𝑌 × { ( 0g ‘ ( Scalar ‘ 𝐵 ) ) } ) ↔ ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) = ( 𝑌 × { ( 0g ‘ ( Scalar ‘ 𝐵 ) ) } ) ) ) |
465 |
463 464
|
imbi12d |
⊢ ( 𝑏 = ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) → ( ( ( 𝑏 finSupp ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ∧ ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝑏 ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 0g ‘ 𝐵 ) ) → 𝑏 = ( 𝑌 × { ( 0g ‘ ( Scalar ‘ 𝐵 ) ) } ) ) ↔ ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) finSupp ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ∧ ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 0g ‘ 𝐵 ) ) → ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) = ( 𝑌 × { ( 0g ‘ ( Scalar ‘ 𝐵 ) ) } ) ) ) ) |
466 |
367
|
lbslinds |
⊢ ( LBasis ‘ 𝐵 ) ⊆ ( LIndS ‘ 𝐵 ) |
467 |
466 14
|
sselid |
⊢ ( 𝜑 → 𝑌 ∈ ( LIndS ‘ 𝐵 ) ) |
468 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝐵 ) ) = ( Base ‘ ( Scalar ‘ 𝐵 ) ) |
469 |
236 468 322 247 207 323
|
islinds5 |
⊢ ( ( 𝐵 ∈ LMod ∧ 𝑌 ⊆ ( Base ‘ 𝐵 ) ) → ( 𝑌 ∈ ( LIndS ‘ 𝐵 ) ↔ ∀ 𝑏 ∈ ( ( Base ‘ ( Scalar ‘ 𝐵 ) ) ↑m 𝑌 ) ( ( 𝑏 finSupp ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ∧ ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝑏 ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 0g ‘ 𝐵 ) ) → 𝑏 = ( 𝑌 × { ( 0g ‘ ( Scalar ‘ 𝐵 ) ) } ) ) ) ) |
470 |
469
|
biimpa |
⊢ ( ( ( 𝐵 ∈ LMod ∧ 𝑌 ⊆ ( Base ‘ 𝐵 ) ) ∧ 𝑌 ∈ ( LIndS ‘ 𝐵 ) ) → ∀ 𝑏 ∈ ( ( Base ‘ ( Scalar ‘ 𝐵 ) ) ↑m 𝑌 ) ( ( 𝑏 finSupp ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ∧ ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝑏 ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 0g ‘ 𝐵 ) ) → 𝑏 = ( 𝑌 × { ( 0g ‘ ( Scalar ‘ 𝐵 ) ) } ) ) ) |
471 |
211 369 467 470
|
syl21anc |
⊢ ( 𝜑 → ∀ 𝑏 ∈ ( ( Base ‘ ( Scalar ‘ 𝐵 ) ) ↑m 𝑌 ) ( ( 𝑏 finSupp ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ∧ ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( 𝑏 ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 0g ‘ 𝐵 ) ) → 𝑏 = ( 𝑌 × { ( 0g ‘ ( Scalar ‘ 𝐵 ) ) } ) ) ) |
472 |
|
fvexd |
⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐵 ) ) ∈ V ) |
473 |
|
elmapg |
⊢ ( ( ( Base ‘ ( Scalar ‘ 𝐵 ) ) ∈ V ∧ 𝑌 ∈ ( LBasis ‘ 𝐵 ) ) → ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ∈ ( ( Base ‘ ( Scalar ‘ 𝐵 ) ) ↑m 𝑌 ) ↔ ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) : 𝑌 ⟶ ( Base ‘ ( Scalar ‘ 𝐵 ) ) ) ) |
474 |
473
|
biimpar |
⊢ ( ( ( ( Base ‘ ( Scalar ‘ 𝐵 ) ) ∈ V ∧ 𝑌 ∈ ( LBasis ‘ 𝐵 ) ) ∧ ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) : 𝑌 ⟶ ( Base ‘ ( Scalar ‘ 𝐵 ) ) ) → ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ∈ ( ( Base ‘ ( Scalar ‘ 𝐵 ) ) ↑m 𝑌 ) ) |
475 |
472 14 254 474
|
syl21anc |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ∈ ( ( Base ‘ ( Scalar ‘ 𝐵 ) ) ↑m 𝑌 ) ) |
476 |
465 471 475
|
rspcdva |
⊢ ( 𝜑 → ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) finSupp ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ∧ ( 𝐵 Σg ( 𝑗 ∈ 𝑌 ↦ ( ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) ‘ 𝑗 ) ( ·𝑠 ‘ 𝐵 ) 𝑗 ) ) ) = ( 0g ‘ 𝐵 ) ) → ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) = ( 𝑌 × { ( 0g ‘ ( Scalar ‘ 𝐵 ) ) } ) ) ) |
477 |
340 453 476
|
mp2and |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) = ( 𝑌 × { ( 0g ‘ ( Scalar ‘ 𝐵 ) ) } ) ) |
478 |
|
fconstmpt |
⊢ ( 𝑌 × { ( 0g ‘ ( Scalar ‘ 𝐵 ) ) } ) = ( 𝑗 ∈ 𝑌 ↦ ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) |
479 |
477 478
|
eqtrdi |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) = ( 𝑗 ∈ 𝑌 ↦ ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) ) |
480 |
|
ovex |
⊢ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ∈ V |
481 |
480
|
rgenw |
⊢ ∀ 𝑗 ∈ 𝑌 ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ∈ V |
482 |
|
mpteqb |
⊢ ( ∀ 𝑗 ∈ 𝑌 ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ∈ V → ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) = ( 𝑗 ∈ 𝑌 ↦ ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) ↔ ∀ 𝑗 ∈ 𝑌 ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) = ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) ) |
483 |
481 482
|
ax-mp |
⊢ ( ( 𝑗 ∈ 𝑌 ↦ ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) ) = ( 𝑗 ∈ 𝑌 ↦ ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) ↔ ∀ 𝑗 ∈ 𝑌 ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) = ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) |
484 |
479 483
|
sylib |
⊢ ( 𝜑 → ∀ 𝑗 ∈ 𝑌 ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) = ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) |
485 |
484
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐵 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑗 𝑊 𝑖 ) ( ·𝑠 ‘ 𝐵 ) 𝑖 ) ) ) = ( 0g ‘ ( Scalar ‘ 𝐵 ) ) ) |
486 |
315 447 316
|
3eqtr3rd |
⊢ ( 𝜑 → ( 0g ‘ ( Scalar ‘ 𝐵 ) ) = ( 0g ‘ 𝐴 ) ) |
487 |
486
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 0g ‘ ( Scalar ‘ 𝐵 ) ) = ( 0g ‘ 𝐴 ) ) |
488 |
205 485 487
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 0g ‘ 𝐴 ) ) |
489 |
186 488
|
jca |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∧ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 0g ‘ 𝐴 ) ) ) |
490 |
189
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) : 𝑋 ⟶ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) |
491 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( Base ‘ ( Scalar ‘ 𝐴 ) ) ∈ V ) |
492 |
491 148
|
elmapd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ∈ ( ( Base ‘ ( Scalar ‘ 𝐴 ) ) ↑m 𝑋 ) ↔ ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) : 𝑋 ⟶ ( Base ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
493 |
490 492
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ∈ ( ( Base ‘ ( Scalar ‘ 𝐴 ) ) ↑m 𝑋 ) ) |
494 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) → 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) |
495 |
494
|
breq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) → ( 𝑤 finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ↔ ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
496 |
|
nfv |
⊢ Ⅎ 𝑖 ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) |
497 |
|
nfmpt1 |
⊢ Ⅎ 𝑖 ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) |
498 |
497
|
nfeq2 |
⊢ Ⅎ 𝑖 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) |
499 |
496 498
|
nfan |
⊢ Ⅎ 𝑖 ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) |
500 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) ∧ 𝑖 ∈ 𝑋 ) → 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) |
501 |
500
|
fveq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) ∧ 𝑖 ∈ 𝑋 ) → ( 𝑤 ‘ 𝑖 ) = ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ) |
502 |
501
|
oveq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) ∧ 𝑖 ∈ 𝑋 ) → ( ( 𝑤 ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) = ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) |
503 |
499 502
|
mpteq2da |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) → ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑤 ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) |
504 |
503
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) → ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑤 ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) ) |
505 |
504
|
eqeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) → ( ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑤 ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 0g ‘ 𝐴 ) ↔ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 0g ‘ 𝐴 ) ) ) |
506 |
495 505
|
anbi12d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) → ( ( 𝑤 finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∧ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑤 ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 0g ‘ 𝐴 ) ) ↔ ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∧ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 0g ‘ 𝐴 ) ) ) ) |
507 |
494
|
eqeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) → ( 𝑤 = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ↔ ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) ) |
508 |
506 507
|
imbi12d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) ∧ 𝑤 = ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ) → ( ( ( 𝑤 finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∧ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑤 ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 0g ‘ 𝐴 ) ) → 𝑤 = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) ↔ ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∧ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 0g ‘ 𝐴 ) ) → ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) ) ) |
509 |
493 508
|
rspcdv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( ∀ 𝑤 ∈ ( ( Base ‘ ( Scalar ‘ 𝐴 ) ) ↑m 𝑋 ) ( ( 𝑤 finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∧ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( 𝑤 ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 0g ‘ 𝐴 ) ) → 𝑤 = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) → ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) finSupp ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∧ ( 𝐴 Σg ( 𝑖 ∈ 𝑋 ↦ ( ( ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) ‘ 𝑖 ) ( ·𝑠 ‘ 𝐴 ) 𝑖 ) ) ) = ( 0g ‘ 𝐴 ) ) → ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) ) ) |
510 |
144 489 509
|
mp2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) = ( 𝑋 × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) |
511 |
510 257
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ∈ 𝑋 ↦ ( 𝑗 𝑊 𝑖 ) ) = ( 𝑖 ∈ 𝑋 ↦ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
512 |
511 311
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ) → ∀ 𝑖 ∈ 𝑋 ( 𝑗 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
513 |
512
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑗 ∈ 𝑌 ∀ 𝑖 ∈ 𝑋 ( 𝑗 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
514 |
|
eqidd |
⊢ ( ( 𝑗 = 𝑘 ∧ 𝑖 = 𝑙 ) → ( 0g ‘ ( Scalar ‘ 𝐴 ) ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
515 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑌 ∧ 𝑖 ∈ 𝑋 ) → ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∈ V ) |
516 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑌 ∧ 𝑙 ∈ 𝑋 ) → ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ∈ V ) |
517 |
160 514 515 516
|
fnmpoovd |
⊢ ( 𝜑 → ( 𝑊 = ( 𝑘 ∈ 𝑌 , 𝑙 ∈ 𝑋 ↦ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ↔ ∀ 𝑗 ∈ 𝑌 ∀ 𝑖 ∈ 𝑋 ( 𝑗 𝑊 𝑖 ) = ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
518 |
513 517
|
mpbird |
⊢ ( 𝜑 → 𝑊 = ( 𝑘 ∈ 𝑌 , 𝑙 ∈ 𝑋 ↦ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) ) |
519 |
|
fconstmpo |
⊢ ( ( 𝑌 × 𝑋 ) × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) = ( 𝑘 ∈ 𝑌 , 𝑙 ∈ 𝑋 ↦ ( 0g ‘ ( Scalar ‘ 𝐴 ) ) ) |
520 |
518 519
|
eqtr4di |
⊢ ( 𝜑 → 𝑊 = ( ( 𝑌 × 𝑋 ) × { ( 0g ‘ ( Scalar ‘ 𝐴 ) ) } ) ) |