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 |
|- ( ph -> E e. DivRing ) |
7 |
|
fedgmul.2 |
|- ( ph -> F e. DivRing ) |
8 |
|
fedgmul.3 |
|- ( ph -> K e. DivRing ) |
9 |
|
fedgmul.4 |
|- ( ph -> U e. ( SubRing ` E ) ) |
10 |
|
fedgmul.5 |
|- ( ph -> V e. ( SubRing ` F ) ) |
11 |
|
fedgmullem.d |
|- D = ( j e. Y , i e. X |-> ( i ( .r ` E ) j ) ) |
12 |
|
fedgmullem.h |
|- H = ( j e. Y , i e. X |-> ( ( G ` j ) ` i ) ) |
13 |
|
fedgmullem.x |
|- ( ph -> X e. ( LBasis ` C ) ) |
14 |
|
fedgmullem.y |
|- ( ph -> Y e. ( LBasis ` B ) ) |
15 |
|
fedgmullem2.1 |
|- ( ph -> W e. ( Base ` ( ( Scalar ` A ) freeLMod ( Y X. X ) ) ) ) |
16 |
|
fedgmullem2.2 |
|- ( ph -> ( A gsum ( W oF ( .s ` A ) D ) ) = ( 0g ` A ) ) |
17 |
4
|
subsubrg |
|- ( U e. ( SubRing ` E ) -> ( V e. ( SubRing ` F ) <-> ( V e. ( SubRing ` E ) /\ V C_ U ) ) ) |
18 |
17
|
biimpa |
|- ( ( U e. ( SubRing ` E ) /\ V e. ( SubRing ` F ) ) -> ( V e. ( SubRing ` E ) /\ V C_ U ) ) |
19 |
9 10 18
|
syl2anc |
|- ( ph -> ( V e. ( SubRing ` E ) /\ V C_ U ) ) |
20 |
19
|
simpld |
|- ( ph -> V e. ( SubRing ` E ) ) |
21 |
1 5
|
sralvec |
|- ( ( E e. DivRing /\ K e. DivRing /\ V e. ( SubRing ` E ) ) -> A e. LVec ) |
22 |
6 8 20 21
|
syl3anc |
|- ( ph -> A e. LVec ) |
23 |
|
lveclmod |
|- ( A e. LVec -> A e. LMod ) |
24 |
22 23
|
syl |
|- ( ph -> A e. LMod ) |
25 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
26 |
|
eqid |
|- ( LBasis ` C ) = ( LBasis ` C ) |
27 |
25 26
|
lbsss |
|- ( X e. ( LBasis ` C ) -> X C_ ( Base ` C ) ) |
28 |
13 27
|
syl |
|- ( ph -> X C_ ( Base ` C ) ) |
29 |
|
eqid |
|- ( Base ` E ) = ( Base ` E ) |
30 |
29
|
subrgss |
|- ( U e. ( SubRing ` E ) -> U C_ ( Base ` E ) ) |
31 |
9 30
|
syl |
|- ( ph -> U C_ ( Base ` E ) ) |
32 |
4 29
|
ressbas2 |
|- ( U C_ ( Base ` E ) -> U = ( Base ` F ) ) |
33 |
31 32
|
syl |
|- ( ph -> U = ( Base ` F ) ) |
34 |
3
|
a1i |
|- ( ph -> C = ( ( subringAlg ` F ) ` V ) ) |
35 |
|
eqid |
|- ( Base ` F ) = ( Base ` F ) |
36 |
35
|
subrgss |
|- ( V e. ( SubRing ` F ) -> V C_ ( Base ` F ) ) |
37 |
10 36
|
syl |
|- ( ph -> V C_ ( Base ` F ) ) |
38 |
34 37
|
srabase |
|- ( ph -> ( Base ` F ) = ( Base ` C ) ) |
39 |
33 38
|
eqtrd |
|- ( ph -> U = ( Base ` C ) ) |
40 |
39 31
|
eqsstrrd |
|- ( ph -> ( Base ` C ) C_ ( Base ` E ) ) |
41 |
1
|
a1i |
|- ( ph -> A = ( ( subringAlg ` E ) ` V ) ) |
42 |
29
|
subrgss |
|- ( V e. ( SubRing ` E ) -> V C_ ( Base ` E ) ) |
43 |
20 42
|
syl |
|- ( ph -> V C_ ( Base ` E ) ) |
44 |
41 43
|
srabase |
|- ( ph -> ( Base ` E ) = ( Base ` A ) ) |
45 |
40 44
|
sseqtrd |
|- ( ph -> ( Base ` C ) C_ ( Base ` A ) ) |
46 |
28 45
|
sstrd |
|- ( ph -> X C_ ( Base ` A ) ) |
47 |
41 9 43
|
srasubrg |
|- ( ph -> U e. ( SubRing ` A ) ) |
48 |
|
subrgsubg |
|- ( U e. ( SubRing ` A ) -> U e. ( SubGrp ` A ) ) |
49 |
47 48
|
syl |
|- ( ph -> U e. ( SubGrp ` A ) ) |
50 |
1 6 20
|
drgextvsca |
|- ( ph -> ( .r ` E ) = ( .s ` A ) ) |
51 |
50
|
oveqdr |
|- ( ( ph /\ ( x e. ( Base ` ( Scalar ` A ) ) /\ y e. U ) ) -> ( x ( .r ` E ) y ) = ( x ( .s ` A ) y ) ) |
52 |
9
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` ( Scalar ` A ) ) /\ y e. U ) ) -> U e. ( SubRing ` E ) ) |
53 |
19
|
simprd |
|- ( ph -> V C_ U ) |
54 |
53
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` ( Scalar ` A ) ) /\ y e. U ) ) -> V C_ U ) |
55 |
|
simprl |
|- ( ( ph /\ ( x e. ( Base ` ( Scalar ` A ) ) /\ y e. U ) ) -> x e. ( Base ` ( Scalar ` A ) ) ) |
56 |
|
ressabs |
|- ( ( U e. ( SubRing ` E ) /\ V C_ U ) -> ( ( E |`s U ) |`s V ) = ( E |`s V ) ) |
57 |
9 53 56
|
syl2anc |
|- ( ph -> ( ( E |`s U ) |`s V ) = ( E |`s V ) ) |
58 |
4
|
oveq1i |
|- ( F |`s V ) = ( ( E |`s U ) |`s V ) |
59 |
57 58 5
|
3eqtr4g |
|- ( ph -> ( F |`s V ) = K ) |
60 |
34 37
|
srasca |
|- ( ph -> ( F |`s V ) = ( Scalar ` C ) ) |
61 |
59 60
|
eqtr3d |
|- ( ph -> K = ( Scalar ` C ) ) |
62 |
61
|
fveq2d |
|- ( ph -> ( Base ` K ) = ( Base ` ( Scalar ` C ) ) ) |
63 |
5 29
|
ressbas2 |
|- ( V C_ ( Base ` E ) -> V = ( Base ` K ) ) |
64 |
43 63
|
syl |
|- ( ph -> V = ( Base ` K ) ) |
65 |
41 43
|
srasca |
|- ( ph -> ( E |`s V ) = ( Scalar ` A ) ) |
66 |
5 65
|
syl5eq |
|- ( ph -> K = ( Scalar ` A ) ) |
67 |
59 60 66
|
3eqtr3rd |
|- ( ph -> ( Scalar ` A ) = ( Scalar ` C ) ) |
68 |
67
|
fveq2d |
|- ( ph -> ( Base ` ( Scalar ` A ) ) = ( Base ` ( Scalar ` C ) ) ) |
69 |
62 64 68
|
3eqtr4d |
|- ( ph -> V = ( Base ` ( Scalar ` A ) ) ) |
70 |
69
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` ( Scalar ` A ) ) /\ y e. U ) ) -> V = ( Base ` ( Scalar ` A ) ) ) |
71 |
55 70
|
eleqtrrd |
|- ( ( ph /\ ( x e. ( Base ` ( Scalar ` A ) ) /\ y e. U ) ) -> x e. V ) |
72 |
54 71
|
sseldd |
|- ( ( ph /\ ( x e. ( Base ` ( Scalar ` A ) ) /\ y e. U ) ) -> x e. U ) |
73 |
|
simprr |
|- ( ( ph /\ ( x e. ( Base ` ( Scalar ` A ) ) /\ y e. U ) ) -> y e. U ) |
74 |
|
eqid |
|- ( .r ` E ) = ( .r ` E ) |
75 |
74
|
subrgmcl |
|- ( ( U e. ( SubRing ` E ) /\ x e. U /\ y e. U ) -> ( x ( .r ` E ) y ) e. U ) |
76 |
52 72 73 75
|
syl3anc |
|- ( ( ph /\ ( x e. ( Base ` ( Scalar ` A ) ) /\ y e. U ) ) -> ( x ( .r ` E ) y ) e. U ) |
77 |
51 76
|
eqeltrrd |
|- ( ( ph /\ ( x e. ( Base ` ( Scalar ` A ) ) /\ y e. U ) ) -> ( x ( .s ` A ) y ) e. U ) |
78 |
77
|
ralrimivva |
|- ( ph -> A. x e. ( Base ` ( Scalar ` A ) ) A. y e. U ( x ( .s ` A ) y ) e. 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 e. LMod -> ( U e. ( LSubSp ` A ) <-> ( U e. ( SubGrp ` A ) /\ A. x e. ( Base ` ( Scalar ` A ) ) A. y e. U ( x ( .s ` A ) y ) e. U ) ) ) |
85 |
84
|
biimpar |
|- ( ( A e. LMod /\ ( U e. ( SubGrp ` A ) /\ A. x e. ( Base ` ( Scalar ` A ) ) A. y e. U ( x ( .s ` A ) y ) e. U ) ) -> U e. ( LSubSp ` A ) ) |
86 |
24 49 78 85
|
syl12anc |
|- ( ph -> U e. ( LSubSp ` A ) ) |
87 |
28 39
|
sseqtrrd |
|- ( ph -> X C_ U ) |
88 |
26
|
lbslinds |
|- ( LBasis ` C ) C_ ( LIndS ` C ) |
89 |
88 13
|
sselid |
|- ( ph -> X e. ( LIndS ` C ) ) |
90 |
31 44
|
sseqtrd |
|- ( ph -> U C_ ( Base ` A ) ) |
91 |
|
eqid |
|- ( A |`s U ) = ( A |`s U ) |
92 |
91 81
|
ressbas2 |
|- ( U C_ ( Base ` A ) -> U = ( Base ` ( A |`s U ) ) ) |
93 |
90 92
|
syl |
|- ( ph -> U = ( Base ` ( A |`s U ) ) ) |
94 |
33 93 38
|
3eqtr3rd |
|- ( ph -> ( Base ` C ) = ( Base ` ( A |`s U ) ) ) |
95 |
91 79
|
resssca |
|- ( U e. ( SubRing ` E ) -> ( Scalar ` A ) = ( Scalar ` ( A |`s U ) ) ) |
96 |
9 95
|
syl |
|- ( ph -> ( Scalar ` A ) = ( Scalar ` ( A |`s U ) ) ) |
97 |
67 96
|
eqtr3d |
|- ( ph -> ( Scalar ` C ) = ( Scalar ` ( A |`s U ) ) ) |
98 |
97
|
fveq2d |
|- ( ph -> ( Base ` ( Scalar ` C ) ) = ( Base ` ( Scalar ` ( A |`s U ) ) ) ) |
99 |
97
|
fveq2d |
|- ( ph -> ( 0g ` ( Scalar ` C ) ) = ( 0g ` ( Scalar ` ( A |`s U ) ) ) ) |
100 |
|
eqid |
|- ( +g ` E ) = ( +g ` E ) |
101 |
4 100
|
ressplusg |
|- ( U e. ( SubRing ` E ) -> ( +g ` E ) = ( +g ` F ) ) |
102 |
9 101
|
syl |
|- ( ph -> ( +g ` E ) = ( +g ` F ) ) |
103 |
41 43
|
sraaddg |
|- ( ph -> ( +g ` E ) = ( +g ` A ) ) |
104 |
34 37
|
sraaddg |
|- ( ph -> ( +g ` F ) = ( +g ` C ) ) |
105 |
102 103 104
|
3eqtr3rd |
|- ( ph -> ( +g ` C ) = ( +g ` A ) ) |
106 |
|
eqid |
|- ( +g ` A ) = ( +g ` A ) |
107 |
91 106
|
ressplusg |
|- ( U e. ( SubRing ` E ) -> ( +g ` A ) = ( +g ` ( A |`s U ) ) ) |
108 |
9 107
|
syl |
|- ( ph -> ( +g ` A ) = ( +g ` ( A |`s U ) ) ) |
109 |
105 108
|
eqtrd |
|- ( ph -> ( +g ` C ) = ( +g ` ( A |`s U ) ) ) |
110 |
109
|
oveqdr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( +g ` C ) y ) = ( x ( +g ` ( A |`s U ) ) y ) ) |
111 |
59 8
|
eqeltrd |
|- ( ph -> ( F |`s V ) e. DivRing ) |
112 |
|
eqid |
|- ( F |`s V ) = ( F |`s V ) |
113 |
3 112
|
sralvec |
|- ( ( F e. DivRing /\ ( F |`s V ) e. DivRing /\ V e. ( SubRing ` F ) ) -> C e. LVec ) |
114 |
7 111 10 113
|
syl3anc |
|- ( ph -> C e. LVec ) |
115 |
|
lveclmod |
|- ( C e. LVec -> C e. LMod ) |
116 |
114 115
|
syl |
|- ( ph -> C e. 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 e. LMod /\ x e. ( Base ` ( Scalar ` C ) ) /\ y e. ( Base ` C ) ) -> ( x ( .s ` C ) y ) e. ( Base ` C ) ) |
121 |
120
|
3expb |
|- ( ( C e. LMod /\ ( x e. ( Base ` ( Scalar ` C ) ) /\ y e. ( Base ` C ) ) ) -> ( x ( .s ` C ) y ) e. ( Base ` C ) ) |
122 |
116 121
|
sylan |
|- ( ( ph /\ ( x e. ( Base ` ( Scalar ` C ) ) /\ y e. ( Base ` C ) ) ) -> ( x ( .s ` C ) y ) e. ( Base ` C ) ) |
123 |
2 6 9
|
drgextvsca |
|- ( ph -> ( .r ` E ) = ( .s ` B ) ) |
124 |
50 123
|
eqtr3d |
|- ( ph -> ( .s ` A ) = ( .s ` B ) ) |
125 |
91 82
|
ressvsca |
|- ( U e. ( SubRing ` E ) -> ( .s ` A ) = ( .s ` ( A |`s U ) ) ) |
126 |
9 125
|
syl |
|- ( ph -> ( .s ` A ) = ( .s ` ( A |`s U ) ) ) |
127 |
4 74
|
ressmulr |
|- ( U e. ( SubRing ` E ) -> ( .r ` E ) = ( .r ` F ) ) |
128 |
9 127
|
syl |
|- ( ph -> ( .r ` E ) = ( .r ` F ) ) |
129 |
3 7 10
|
drgextvsca |
|- ( ph -> ( .r ` F ) = ( .s ` C ) ) |
130 |
128 123 129
|
3eqtr3d |
|- ( ph -> ( .s ` B ) = ( .s ` C ) ) |
131 |
124 126 130
|
3eqtr3rd |
|- ( ph -> ( .s ` C ) = ( .s ` ( A |`s U ) ) ) |
132 |
131
|
oveqdr |
|- ( ( ph /\ ( x e. ( Base ` ( Scalar ` C ) ) /\ y e. ( Base ` C ) ) ) -> ( x ( .s ` C ) y ) = ( x ( .s ` ( A |`s U ) ) y ) ) |
133 |
|
ovexd |
|- ( ph -> ( A |`s U ) e. _V ) |
134 |
94 98 99 110 122 132 114 133
|
lindspropd |
|- ( ph -> ( LIndS ` C ) = ( LIndS ` ( A |`s U ) ) ) |
135 |
89 134
|
eleqtrd |
|- ( ph -> X e. ( LIndS ` ( A |`s U ) ) ) |
136 |
83 91
|
lsslinds |
|- ( ( A e. LMod /\ U e. ( LSubSp ` A ) /\ X C_ U ) -> ( X e. ( LIndS ` ( A |`s U ) ) <-> X e. ( LIndS ` A ) ) ) |
137 |
136
|
biimpa |
|- ( ( ( A e. LMod /\ U e. ( LSubSp ` A ) /\ X C_ U ) /\ X e. ( LIndS ` ( A |`s U ) ) ) -> X e. ( LIndS ` A ) ) |
138 |
24 86 87 135 137
|
syl31anc |
|- ( ph -> X e. ( 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 e. LMod /\ X C_ ( Base ` A ) ) -> ( X e. ( LIndS ` A ) <-> A. w e. ( ( Base ` ( Scalar ` A ) ) ^m X ) ( ( w finSupp ( 0g ` ( Scalar ` A ) ) /\ ( A gsum ( i e. X |-> ( ( w ` i ) ( .s ` A ) i ) ) ) = ( 0g ` A ) ) -> w = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) ) ) |
142 |
141
|
biimpa |
|- ( ( ( A e. LMod /\ X C_ ( Base ` A ) ) /\ X e. ( LIndS ` A ) ) -> A. w e. ( ( Base ` ( Scalar ` A ) ) ^m X ) ( ( w finSupp ( 0g ` ( Scalar ` A ) ) /\ ( A gsum ( i e. X |-> ( ( w ` i ) ( .s ` A ) i ) ) ) = ( 0g ` A ) ) -> w = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) ) |
143 |
24 46 138 142
|
syl21anc |
|- ( ph -> A. w e. ( ( Base ` ( Scalar ` A ) ) ^m X ) ( ( w finSupp ( 0g ` ( Scalar ` A ) ) /\ ( A gsum ( i e. X |-> ( ( w ` i ) ( .s ` A ) i ) ) ) = ( 0g ` A ) ) -> w = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) ) |
144 |
143
|
adantr |
|- ( ( ph /\ j e. Y ) -> A. w e. ( ( Base ` ( Scalar ` A ) ) ^m X ) ( ( w finSupp ( 0g ` ( Scalar ` A ) ) /\ ( A gsum ( i e. X |-> ( ( w ` i ) ( .s ` A ) i ) ) ) = ( 0g ` A ) ) -> w = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) ) |
145 |
|
eqid |
|- ( i e. X |-> ( j W i ) ) = ( i e. X |-> ( j W i ) ) |
146 |
|
fvexd |
|- ( ( ph /\ j e. Y ) -> ( 0g ` F ) e. _V ) |
147 |
14
|
adantr |
|- ( ( ph /\ j e. Y ) -> Y e. ( LBasis ` B ) ) |
148 |
13
|
adantr |
|- ( ( ph /\ j e. Y ) -> X e. ( LBasis ` C ) ) |
149 |
|
fvexd |
|- ( ph -> ( Scalar ` A ) e. _V ) |
150 |
14 13
|
xpexd |
|- ( ph -> ( Y X. X ) e. _V ) |
151 |
|
eqid |
|- ( ( Scalar ` A ) freeLMod ( Y X. X ) ) = ( ( Scalar ` A ) freeLMod ( Y X. X ) ) |
152 |
|
eqid |
|- ( Base ` ( ( Scalar ` A ) freeLMod ( Y X. X ) ) ) = ( Base ` ( ( Scalar ` A ) freeLMod ( Y X. X ) ) ) |
153 |
151 80 140 152
|
frlmelbas |
|- ( ( ( Scalar ` A ) e. _V /\ ( Y X. X ) e. _V ) -> ( W e. ( Base ` ( ( Scalar ` A ) freeLMod ( Y X. X ) ) ) <-> ( W e. ( ( Base ` ( Scalar ` A ) ) ^m ( Y X. X ) ) /\ W finSupp ( 0g ` ( Scalar ` A ) ) ) ) ) |
154 |
149 150 153
|
syl2anc |
|- ( ph -> ( W e. ( Base ` ( ( Scalar ` A ) freeLMod ( Y X. X ) ) ) <-> ( W e. ( ( Base ` ( Scalar ` A ) ) ^m ( Y X. X ) ) /\ W finSupp ( 0g ` ( Scalar ` A ) ) ) ) ) |
155 |
15 154
|
mpbid |
|- ( ph -> ( W e. ( ( Base ` ( Scalar ` A ) ) ^m ( Y X. X ) ) /\ W finSupp ( 0g ` ( Scalar ` A ) ) ) ) |
156 |
155
|
simpld |
|- ( ph -> W e. ( ( Base ` ( Scalar ` A ) ) ^m ( Y X. X ) ) ) |
157 |
|
fvexd |
|- ( ph -> ( Base ` ( Scalar ` A ) ) e. _V ) |
158 |
157 150
|
elmapd |
|- ( ph -> ( W e. ( ( Base ` ( Scalar ` A ) ) ^m ( Y X. X ) ) <-> W : ( Y X. X ) --> ( Base ` ( Scalar ` A ) ) ) ) |
159 |
156 158
|
mpbid |
|- ( ph -> W : ( Y X. X ) --> ( Base ` ( Scalar ` A ) ) ) |
160 |
159
|
ffnd |
|- ( ph -> W Fn ( Y X. X ) ) |
161 |
160
|
adantr |
|- ( ( ph /\ j e. Y ) -> W Fn ( Y X. X ) ) |
162 |
|
simpr |
|- ( ( ph /\ j e. Y ) -> j e. Y ) |
163 |
155
|
simprd |
|- ( ph -> W finSupp ( 0g ` ( Scalar ` A ) ) ) |
164 |
|
drngring |
|- ( E e. DivRing -> E e. Ring ) |
165 |
6 164
|
syl |
|- ( ph -> E e. Ring ) |
166 |
|
ringmnd |
|- ( E e. Ring -> E e. Mnd ) |
167 |
165 166
|
syl |
|- ( ph -> E e. Mnd ) |
168 |
|
subrgsubg |
|- ( V e. ( SubRing ` E ) -> V e. ( SubGrp ` E ) ) |
169 |
20 168
|
syl |
|- ( ph -> V e. ( SubGrp ` E ) ) |
170 |
|
eqid |
|- ( 0g ` E ) = ( 0g ` E ) |
171 |
170
|
subg0cl |
|- ( V e. ( SubGrp ` E ) -> ( 0g ` E ) e. V ) |
172 |
169 171
|
syl |
|- ( ph -> ( 0g ` E ) e. V ) |
173 |
53 172
|
sseldd |
|- ( ph -> ( 0g ` E ) e. U ) |
174 |
4 29 170
|
ress0g |
|- ( ( E e. Mnd /\ ( 0g ` E ) e. U /\ U C_ ( Base ` E ) ) -> ( 0g ` E ) = ( 0g ` F ) ) |
175 |
167 173 31 174
|
syl3anc |
|- ( ph -> ( 0g ` E ) = ( 0g ` F ) ) |
176 |
61
|
fveq2d |
|- ( ph -> ( 0g ` K ) = ( 0g ` ( Scalar ` C ) ) ) |
177 |
5 170
|
subrg0 |
|- ( V e. ( SubRing ` E ) -> ( 0g ` E ) = ( 0g ` K ) ) |
178 |
20 177
|
syl |
|- ( ph -> ( 0g ` E ) = ( 0g ` K ) ) |
179 |
67
|
fveq2d |
|- ( ph -> ( 0g ` ( Scalar ` A ) ) = ( 0g ` ( Scalar ` C ) ) ) |
180 |
176 178 179
|
3eqtr4d |
|- ( ph -> ( 0g ` E ) = ( 0g ` ( Scalar ` A ) ) ) |
181 |
175 180
|
eqtr3d |
|- ( ph -> ( 0g ` F ) = ( 0g ` ( Scalar ` A ) ) ) |
182 |
163 181
|
breqtrrd |
|- ( ph -> W finSupp ( 0g ` F ) ) |
183 |
182
|
adantr |
|- ( ( ph /\ j e. Y ) -> W finSupp ( 0g ` F ) ) |
184 |
145 146 147 148 161 162 183
|
fsuppcurry1 |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( j W i ) ) finSupp ( 0g ` F ) ) |
185 |
181
|
adantr |
|- ( ( ph /\ j e. Y ) -> ( 0g ` F ) = ( 0g ` ( Scalar ` A ) ) ) |
186 |
184 185
|
breqtrd |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( j W i ) ) finSupp ( 0g ` ( Scalar ` A ) ) ) |
187 |
|
eqidd |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( j W i ) ) = ( i e. X |-> ( j W i ) ) ) |
188 |
159
|
fovrnda |
|- ( ( ph /\ ( j e. Y /\ i e. X ) ) -> ( j W i ) e. ( Base ` ( Scalar ` A ) ) ) |
189 |
188
|
anassrs |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( j W i ) e. ( Base ` ( Scalar ` A ) ) ) |
190 |
187 189
|
fvmpt2d |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( ( i e. X |-> ( j W i ) ) ` i ) = ( j W i ) ) |
191 |
190
|
oveq1d |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( ( ( i e. X |-> ( j W i ) ) ` i ) ( .s ` A ) i ) = ( ( j W i ) ( .s ` A ) i ) ) |
192 |
124
|
ad2antrr |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( .s ` A ) = ( .s ` B ) ) |
193 |
192
|
oveqd |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( ( j W i ) ( .s ` A ) i ) = ( ( j W i ) ( .s ` B ) i ) ) |
194 |
191 193
|
eqtrd |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( ( ( i e. X |-> ( j W i ) ) ` i ) ( .s ` A ) i ) = ( ( j W i ) ( .s ` B ) i ) ) |
195 |
194
|
mpteq2dva |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( ( ( i e. X |-> ( j W i ) ) ` i ) ( .s ` A ) i ) ) = ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) |
196 |
195
|
oveq2d |
|- ( ( ph /\ j e. Y ) -> ( A gsum ( i e. X |-> ( ( ( i e. X |-> ( j W i ) ) ` i ) ( .s ` A ) i ) ) ) = ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) |
197 |
6
|
adantr |
|- ( ( ph /\ j e. Y ) -> E e. DivRing ) |
198 |
20
|
adantr |
|- ( ( ph /\ j e. Y ) -> V e. ( SubRing ` E ) ) |
199 |
8
|
adantr |
|- ( ( ph /\ j e. Y ) -> K e. DivRing ) |
200 |
1 197 198 5 199 148
|
drgextgsum |
|- ( ( ph /\ j e. Y ) -> ( E gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) = ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) |
201 |
9
|
adantr |
|- ( ( ph /\ j e. Y ) -> U e. ( SubRing ` E ) ) |
202 |
7
|
adantr |
|- ( ( ph /\ j e. Y ) -> F e. DivRing ) |
203 |
2 197 201 4 202 148
|
drgextgsum |
|- ( ( ph /\ j e. Y ) -> ( E gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) = ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) |
204 |
200 203
|
eqtr3d |
|- ( ( ph /\ j e. Y ) -> ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) = ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) |
205 |
196 204
|
eqtrd |
|- ( ( ph /\ j e. Y ) -> ( A gsum ( i e. X |-> ( ( ( i e. X |-> ( j W i ) ) ` i ) ( .s ` A ) i ) ) ) = ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) |
206 |
14
|
mptexd |
|- ( ph -> ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) e. _V ) |
207 |
|
eqid |
|- ( 0g ` B ) = ( 0g ` B ) |
208 |
2 4
|
sralvec |
|- ( ( E e. DivRing /\ F e. DivRing /\ U e. ( SubRing ` E ) ) -> B e. LVec ) |
209 |
6 7 9 208
|
syl3anc |
|- ( ph -> B e. LVec ) |
210 |
|
lveclmod |
|- ( B e. LVec -> B e. LMod ) |
211 |
209 210
|
syl |
|- ( ph -> B e. LMod ) |
212 |
211
|
adantr |
|- ( ( ph /\ j e. Y ) -> B e. LMod ) |
213 |
|
lmodabl |
|- ( B e. LMod -> B e. Abel ) |
214 |
212 213
|
syl |
|- ( ( ph /\ j e. Y ) -> B e. Abel ) |
215 |
2
|
a1i |
|- ( ph -> B = ( ( subringAlg ` E ) ` U ) ) |
216 |
215 9 31
|
srasubrg |
|- ( ph -> U e. ( SubRing ` B ) ) |
217 |
|
subrgsubg |
|- ( U e. ( SubRing ` B ) -> U e. ( SubGrp ` B ) ) |
218 |
216 217
|
syl |
|- ( ph -> U e. ( SubGrp ` B ) ) |
219 |
218
|
adantr |
|- ( ( ph /\ j e. Y ) -> U e. ( SubGrp ` B ) ) |
220 |
116
|
ad2antrr |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> C e. LMod ) |
221 |
68
|
ad2antrr |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( Base ` ( Scalar ` A ) ) = ( Base ` ( Scalar ` C ) ) ) |
222 |
189 221
|
eleqtrd |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( j W i ) e. ( Base ` ( Scalar ` C ) ) ) |
223 |
28
|
ad2antrr |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> X C_ ( Base ` C ) ) |
224 |
|
simpr |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> i e. X ) |
225 |
223 224
|
sseldd |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> i e. ( Base ` C ) ) |
226 |
25 117 118 119
|
lmodvscl |
|- ( ( C e. LMod /\ ( j W i ) e. ( Base ` ( Scalar ` C ) ) /\ i e. ( Base ` C ) ) -> ( ( j W i ) ( .s ` C ) i ) e. ( Base ` C ) ) |
227 |
220 222 225 226
|
syl3anc |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( ( j W i ) ( .s ` C ) i ) e. ( Base ` C ) ) |
228 |
130
|
oveqd |
|- ( ph -> ( ( j W i ) ( .s ` B ) i ) = ( ( j W i ) ( .s ` C ) i ) ) |
229 |
228
|
ad2antrr |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( ( j W i ) ( .s ` B ) i ) = ( ( j W i ) ( .s ` C ) i ) ) |
230 |
39
|
ad2antrr |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> U = ( Base ` C ) ) |
231 |
227 229 230
|
3eltr4d |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( ( j W i ) ( .s ` B ) i ) e. U ) |
232 |
231
|
fmpttd |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) : X --> U ) |
233 |
215 31
|
srasca |
|- ( ph -> ( E |`s U ) = ( Scalar ` B ) ) |
234 |
4 233
|
syl5eq |
|- ( ph -> F = ( Scalar ` B ) ) |
235 |
234
|
adantr |
|- ( ( ph /\ j e. Y ) -> F = ( Scalar ` B ) ) |
236 |
|
eqid |
|- ( Base ` B ) = ( Base ` B ) |
237 |
|
ovexd |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( j W i ) e. _V ) |
238 |
28 40
|
sstrd |
|- ( ph -> X C_ ( Base ` E ) ) |
239 |
238
|
adantr |
|- ( ( ph /\ ( j e. Y /\ i e. X ) ) -> X C_ ( Base ` E ) ) |
240 |
|
simprr |
|- ( ( ph /\ ( j e. Y /\ i e. X ) ) -> i e. X ) |
241 |
239 240
|
sseldd |
|- ( ( ph /\ ( j e. Y /\ i e. X ) ) -> i e. ( Base ` E ) ) |
242 |
241
|
anassrs |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> i e. ( Base ` E ) ) |
243 |
215 31
|
srabase |
|- ( ph -> ( Base ` E ) = ( Base ` B ) ) |
244 |
243
|
ad2antrr |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( Base ` E ) = ( Base ` B ) ) |
245 |
242 244
|
eleqtrd |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> i e. ( 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 |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) finSupp ( 0g ` B ) ) |
249 |
207 214 148 219 232 248
|
gsumsubgcl |
|- ( ( ph /\ j e. Y ) -> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) e. U ) |
250 |
234
|
fveq2d |
|- ( ph -> ( Base ` F ) = ( Base ` ( Scalar ` B ) ) ) |
251 |
33 250
|
eqtrd |
|- ( ph -> U = ( Base ` ( Scalar ` B ) ) ) |
252 |
251
|
adantr |
|- ( ( ph /\ j e. Y ) -> U = ( Base ` ( Scalar ` B ) ) ) |
253 |
249 252
|
eleqtrd |
|- ( ( ph /\ j e. Y ) -> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) e. ( Base ` ( Scalar ` B ) ) ) |
254 |
253
|
fmpttd |
|- ( ph -> ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) : Y --> ( Base ` ( Scalar ` B ) ) ) |
255 |
254
|
ffund |
|- ( ph -> Fun ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ) |
256 |
|
fvexd |
|- ( ph -> ( 0g ` ( Scalar ` B ) ) e. _V ) |
257 |
|
fconstmpt |
|- ( X X. { ( 0g ` ( Scalar ` A ) ) } ) = ( i e. X |-> ( 0g ` ( Scalar ` A ) ) ) |
258 |
257
|
eqeq2i |
|- ( ( i e. X |-> ( k W i ) ) = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) <-> ( i e. X |-> ( k W i ) ) = ( i e. X |-> ( 0g ` ( Scalar ` A ) ) ) ) |
259 |
|
ovex |
|- ( k W i ) e. _V |
260 |
259
|
rgenw |
|- A. i e. X ( k W i ) e. _V |
261 |
|
mpteqb |
|- ( A. i e. X ( k W i ) e. _V -> ( ( i e. X |-> ( k W i ) ) = ( i e. X |-> ( 0g ` ( Scalar ` A ) ) ) <-> A. i e. X ( k W i ) = ( 0g ` ( Scalar ` A ) ) ) ) |
262 |
260 261
|
ax-mp |
|- ( ( i e. X |-> ( k W i ) ) = ( i e. X |-> ( 0g ` ( Scalar ` A ) ) ) <-> A. i e. X ( k W i ) = ( 0g ` ( Scalar ` A ) ) ) |
263 |
258 262
|
bitri |
|- ( ( i e. X |-> ( k W i ) ) = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) <-> A. i e. X ( k W i ) = ( 0g ` ( Scalar ` A ) ) ) |
264 |
263
|
necon3abii |
|- ( ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) <-> -. A. i e. 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 |
|- ( ( ( ph /\ k e. Y ) /\ i e. X ) -> ( W ` <. k , i >. ) = ( k W i ) ) |
268 |
267
|
eqeq1d |
|- ( ( ( ph /\ k e. Y ) /\ i e. X ) -> ( ( W ` <. k , i >. ) = ( 0g ` ( Scalar ` A ) ) <-> ( k W i ) = ( 0g ` ( Scalar ` A ) ) ) ) |
269 |
268
|
necon3abid |
|- ( ( ( ph /\ k e. Y ) /\ i e. X ) -> ( ( W ` <. k , i >. ) =/= ( 0g ` ( Scalar ` A ) ) <-> -. ( k W i ) = ( 0g ` ( Scalar ` A ) ) ) ) |
270 |
269
|
rexbidva |
|- ( ( ph /\ k e. Y ) -> ( E. i e. X ( W ` <. k , i >. ) =/= ( 0g ` ( Scalar ` A ) ) <-> E. i e. X -. ( k W i ) = ( 0g ` ( Scalar ` A ) ) ) ) |
271 |
|
rexnal |
|- ( E. i e. X -. ( k W i ) = ( 0g ` ( Scalar ` A ) ) <-> -. A. i e. X ( k W i ) = ( 0g ` ( Scalar ` A ) ) ) |
272 |
270 271
|
bitr2di |
|- ( ( ph /\ k e. Y ) -> ( -. A. i e. X ( k W i ) = ( 0g ` ( Scalar ` A ) ) <-> E. i e. X ( W ` <. k , i >. ) =/= ( 0g ` ( Scalar ` A ) ) ) ) |
273 |
264 272
|
syl5bb |
|- ( ( ph /\ k e. Y ) -> ( ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) <-> E. i e. X ( W ` <. k , i >. ) =/= ( 0g ` ( Scalar ` A ) ) ) ) |
274 |
273
|
rabbidva |
|- ( ph -> { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } = { k e. Y | E. i e. 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 e. ( Y X. X ) | ( W ` z ) =/= ( 0g ` ( Scalar ` A ) ) } = { k e. Y | E. i e. X ( W ` <. k , i >. ) =/= ( 0g ` ( Scalar ` A ) ) } |
278 |
274 277
|
eqtr4di |
|- ( ph -> { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } = dom { z e. ( Y X. X ) | ( W ` z ) =/= ( 0g ` ( Scalar ` A ) ) } ) |
279 |
|
fvexd |
|- ( ph -> ( 0g ` ( Scalar ` A ) ) e. _V ) |
280 |
|
suppvalfn |
|- ( ( W Fn ( Y X. X ) /\ ( Y X. X ) e. _V /\ ( 0g ` ( Scalar ` A ) ) e. _V ) -> ( W supp ( 0g ` ( Scalar ` A ) ) ) = { z e. ( Y X. X ) | ( W ` z ) =/= ( 0g ` ( Scalar ` A ) ) } ) |
281 |
160 150 279 280
|
syl3anc |
|- ( ph -> ( W supp ( 0g ` ( Scalar ` A ) ) ) = { z e. ( Y X. X ) | ( W ` z ) =/= ( 0g ` ( Scalar ` A ) ) } ) |
282 |
163
|
fsuppimpd |
|- ( ph -> ( W supp ( 0g ` ( Scalar ` A ) ) ) e. Fin ) |
283 |
281 282
|
eqeltrrd |
|- ( ph -> { z e. ( Y X. X ) | ( W ` z ) =/= ( 0g ` ( Scalar ` A ) ) } e. Fin ) |
284 |
|
dmfi |
|- ( { z e. ( Y X. X ) | ( W ` z ) =/= ( 0g ` ( Scalar ` A ) ) } e. Fin -> dom { z e. ( Y X. X ) | ( W ` z ) =/= ( 0g ` ( Scalar ` A ) ) } e. Fin ) |
285 |
283 284
|
syl |
|- ( ph -> dom { z e. ( Y X. X ) | ( W ` z ) =/= ( 0g ` ( Scalar ` A ) ) } e. Fin ) |
286 |
278 285
|
eqeltrd |
|- ( ph -> { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } e. Fin ) |
287 |
|
nfv |
|- F/ i ph |
288 |
|
nfcv |
|- F/_ i Y |
289 |
|
nfmpt1 |
|- F/_ i ( i e. X |-> ( k W i ) ) |
290 |
|
nfcv |
|- F/_ i ( X X. { ( 0g ` ( Scalar ` A ) ) } ) |
291 |
289 290
|
nfne |
|- F/ i ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) |
292 |
291 288
|
nfrabw |
|- F/_ i { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } |
293 |
288 292
|
nfdif |
|- F/_ i ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) |
294 |
293
|
nfcri |
|- F/ i j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) |
295 |
287 294
|
nfan |
|- F/ i ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) |
296 |
|
simpr |
|- ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) -> j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) |
297 |
296
|
eldifad |
|- ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) -> j e. Y ) |
298 |
296
|
eldifbd |
|- ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) -> -. j e. { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) |
299 |
|
oveq1 |
|- ( k = j -> ( k W i ) = ( j W i ) ) |
300 |
299
|
mpteq2dv |
|- ( k = j -> ( i e. X |-> ( k W i ) ) = ( i e. X |-> ( j W i ) ) ) |
301 |
300
|
neeq1d |
|- ( k = j -> ( ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) <-> ( i e. X |-> ( j W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) ) |
302 |
301
|
elrab |
|- ( j e. { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } <-> ( j e. Y /\ ( i e. X |-> ( j W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) ) |
303 |
298 302
|
sylnib |
|- ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) -> -. ( j e. Y /\ ( i e. X |-> ( j W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) ) |
304 |
297 303
|
mpnanrd |
|- ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) -> -. ( i e. X |-> ( j W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) |
305 |
|
nne |
|- ( -. ( i e. X |-> ( j W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) <-> ( i e. X |-> ( j W i ) ) = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) |
306 |
304 305
|
sylib |
|- ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) -> ( i e. X |-> ( j W i ) ) = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) |
307 |
306 257
|
eqtrdi |
|- ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) -> ( i e. X |-> ( j W i ) ) = ( i e. X |-> ( 0g ` ( Scalar ` A ) ) ) ) |
308 |
|
ovex |
|- ( j W i ) e. _V |
309 |
308
|
rgenw |
|- A. i e. X ( j W i ) e. _V |
310 |
|
mpteqb |
|- ( A. i e. X ( j W i ) e. _V -> ( ( i e. X |-> ( j W i ) ) = ( i e. X |-> ( 0g ` ( Scalar ` A ) ) ) <-> A. i e. X ( j W i ) = ( 0g ` ( Scalar ` A ) ) ) ) |
311 |
309 310
|
ax-mp |
|- ( ( i e. X |-> ( j W i ) ) = ( i e. X |-> ( 0g ` ( Scalar ` A ) ) ) <-> A. i e. X ( j W i ) = ( 0g ` ( Scalar ` A ) ) ) |
312 |
307 311
|
sylib |
|- ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) -> A. i e. X ( j W i ) = ( 0g ` ( Scalar ` A ) ) ) |
313 |
312
|
r19.21bi |
|- ( ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) /\ i e. X ) -> ( j W i ) = ( 0g ` ( Scalar ` A ) ) ) |
314 |
313
|
oveq1d |
|- ( ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) /\ i e. X ) -> ( ( j W i ) ( .s ` B ) i ) = ( ( 0g ` ( Scalar ` A ) ) ( .s ` B ) i ) ) |
315 |
2 6 9
|
drgext0g |
|- ( ph -> ( 0g ` E ) = ( 0g ` B ) ) |
316 |
2 6 9
|
drgext0gsca |
|- ( ph -> ( 0g ` B ) = ( 0g ` ( Scalar ` B ) ) ) |
317 |
315 180 316
|
3eqtr3d |
|- ( ph -> ( 0g ` ( Scalar ` A ) ) = ( 0g ` ( Scalar ` B ) ) ) |
318 |
317
|
ad2antrr |
|- ( ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) /\ i e. X ) -> ( 0g ` ( Scalar ` A ) ) = ( 0g ` ( Scalar ` B ) ) ) |
319 |
318
|
oveq1d |
|- ( ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) /\ i e. X ) -> ( ( 0g ` ( Scalar ` A ) ) ( .s ` B ) i ) = ( ( 0g ` ( Scalar ` B ) ) ( .s ` B ) i ) ) |
320 |
211
|
ad2antrr |
|- ( ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) /\ i e. X ) -> B e. LMod ) |
321 |
297 245
|
syldanl |
|- ( ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) /\ i e. X ) -> i e. ( Base ` B ) ) |
322 |
|
eqid |
|- ( Scalar ` B ) = ( Scalar ` B ) |
323 |
|
eqid |
|- ( 0g ` ( Scalar ` B ) ) = ( 0g ` ( Scalar ` B ) ) |
324 |
236 322 247 323 207
|
lmod0vs |
|- ( ( B e. LMod /\ i e. ( Base ` B ) ) -> ( ( 0g ` ( Scalar ` B ) ) ( .s ` B ) i ) = ( 0g ` B ) ) |
325 |
320 321 324
|
syl2anc |
|- ( ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) /\ i e. X ) -> ( ( 0g ` ( Scalar ` B ) ) ( .s ` B ) i ) = ( 0g ` B ) ) |
326 |
314 319 325
|
3eqtrd |
|- ( ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) /\ i e. X ) -> ( ( j W i ) ( .s ` B ) i ) = ( 0g ` B ) ) |
327 |
295 326
|
mpteq2da |
|- ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) -> ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) = ( i e. X |-> ( 0g ` B ) ) ) |
328 |
327
|
oveq2d |
|- ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) -> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) = ( B gsum ( i e. X |-> ( 0g ` B ) ) ) ) |
329 |
211 213
|
syl |
|- ( ph -> B e. Abel ) |
330 |
|
ablgrp |
|- ( B e. Abel -> B e. Grp ) |
331 |
|
grpmnd |
|- ( B e. Grp -> B e. Mnd ) |
332 |
329 330 331
|
3syl |
|- ( ph -> B e. Mnd ) |
333 |
207
|
gsumz |
|- ( ( B e. Mnd /\ X e. ( LBasis ` C ) ) -> ( B gsum ( i e. X |-> ( 0g ` B ) ) ) = ( 0g ` B ) ) |
334 |
332 13 333
|
syl2anc |
|- ( ph -> ( B gsum ( i e. X |-> ( 0g ` B ) ) ) = ( 0g ` B ) ) |
335 |
334
|
adantr |
|- ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) -> ( B gsum ( i e. X |-> ( 0g ` B ) ) ) = ( 0g ` B ) ) |
336 |
316
|
adantr |
|- ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) -> ( 0g ` B ) = ( 0g ` ( Scalar ` B ) ) ) |
337 |
328 335 336
|
3eqtrd |
|- ( ( ph /\ j e. ( Y \ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) -> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) = ( 0g ` ( Scalar ` B ) ) ) |
338 |
337 14
|
suppss2 |
|- ( ph -> ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) supp ( 0g ` ( Scalar ` B ) ) ) C_ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) |
339 |
|
suppssfifsupp |
|- ( ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) e. _V /\ Fun ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) /\ ( 0g ` ( Scalar ` B ) ) e. _V ) /\ ( { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } e. Fin /\ ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) supp ( 0g ` ( Scalar ` B ) ) ) C_ { k e. Y | ( i e. X |-> ( k W i ) ) =/= ( X X. { ( 0g ` ( Scalar ` A ) ) } ) } ) ) -> ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) finSupp ( 0g ` ( Scalar ` B ) ) ) |
340 |
206 255 256 286 338 339
|
syl32anc |
|- ( ph -> ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) finSupp ( 0g ` ( Scalar ` B ) ) ) |
341 |
|
eqidd |
|- ( ph -> ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) = ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ) |
342 |
|
ovexd |
|- ( ( ph /\ j e. Y ) -> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) e. _V ) |
343 |
341 342
|
fvmpt2d |
|- ( ( ph /\ j e. Y ) -> ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ` j ) = ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) |
344 |
343
|
oveq1d |
|- ( ( ph /\ j e. Y ) -> ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ` j ) ( .s ` B ) j ) = ( ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) |
345 |
344
|
mpteq2dva |
|- ( ph -> ( j e. Y |-> ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ` j ) ( .s ` B ) j ) ) = ( j e. Y |-> ( ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) ) |
346 |
345
|
oveq2d |
|- ( ph -> ( B gsum ( j e. Y |-> ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ` j ) ( .s ` B ) j ) ) ) = ( B gsum ( j e. Y |-> ( ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) ) ) |
347 |
124
|
adantr |
|- ( ( ph /\ j e. Y ) -> ( .s ` A ) = ( .s ` B ) ) |
348 |
50
|
ad2antrr |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( .r ` E ) = ( .s ` A ) ) |
349 |
348
|
oveqd |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( ( j W i ) ( .r ` E ) i ) = ( ( j W i ) ( .s ` A ) i ) ) |
350 |
349
|
mpteq2dva |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( ( j W i ) ( .r ` E ) i ) ) = ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) ) |
351 |
123
|
adantr |
|- ( ( ph /\ j e. Y ) -> ( .r ` E ) = ( .s ` B ) ) |
352 |
351
|
oveqd |
|- ( ( ph /\ j e. Y ) -> ( ( j W i ) ( .r ` E ) i ) = ( ( j W i ) ( .s ` B ) i ) ) |
353 |
352
|
mpteq2dv |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( ( j W i ) ( .r ` E ) i ) ) = ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) |
354 |
350 353
|
eqtr3d |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) = ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) |
355 |
354
|
oveq2d |
|- ( ( ph /\ j e. Y ) -> ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) ) = ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) |
356 |
|
eqidd |
|- ( ( ph /\ j e. Y ) -> j = j ) |
357 |
347 355 356
|
oveq123d |
|- ( ( ph /\ j e. Y ) -> ( ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) ) ( .s ` A ) j ) = ( ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) |
358 |
204
|
oveq1d |
|- ( ( ph /\ j e. Y ) -> ( ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) = ( ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) |
359 |
357 358
|
eqtrd |
|- ( ( ph /\ j e. Y ) -> ( ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) ) ( .s ` A ) j ) = ( ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) |
360 |
359
|
mpteq2dva |
|- ( ph -> ( j e. Y |-> ( ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) ) ( .s ` A ) j ) ) = ( j e. Y |-> ( ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) ) |
361 |
360
|
oveq2d |
|- ( ph -> ( A gsum ( j e. Y |-> ( ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) ) ( .s ` A ) j ) ) ) = ( A gsum ( j e. Y |-> ( ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) ) ) |
362 |
1 29
|
sraring |
|- ( ( E e. Ring /\ V C_ ( Base ` E ) ) -> A e. Ring ) |
363 |
165 43 362
|
syl2anc |
|- ( ph -> A e. Ring ) |
364 |
|
ringcmn |
|- ( A e. Ring -> A e. CMnd ) |
365 |
363 364
|
syl |
|- ( ph -> A e. CMnd ) |
366 |
165
|
adantr |
|- ( ( ph /\ ( j e. Y /\ i e. X ) ) -> E e. Ring ) |
367 |
|
eqid |
|- ( LBasis ` B ) = ( LBasis ` B ) |
368 |
236 367
|
lbsss |
|- ( Y e. ( LBasis ` B ) -> Y C_ ( Base ` B ) ) |
369 |
14 368
|
syl |
|- ( ph -> Y C_ ( Base ` B ) ) |
370 |
369 243
|
sseqtrrd |
|- ( ph -> Y C_ ( Base ` E ) ) |
371 |
370
|
adantr |
|- ( ( ph /\ ( j e. Y /\ i e. X ) ) -> Y C_ ( Base ` E ) ) |
372 |
|
simprl |
|- ( ( ph /\ ( j e. Y /\ i e. X ) ) -> j e. Y ) |
373 |
371 372
|
sseldd |
|- ( ( ph /\ ( j e. Y /\ i e. X ) ) -> j e. ( Base ` E ) ) |
374 |
29 74
|
ringcl |
|- ( ( E e. Ring /\ i e. ( Base ` E ) /\ j e. ( Base ` E ) ) -> ( i ( .r ` E ) j ) e. ( Base ` E ) ) |
375 |
366 241 373 374
|
syl3anc |
|- ( ( ph /\ ( j e. Y /\ i e. X ) ) -> ( i ( .r ` E ) j ) e. ( Base ` E ) ) |
376 |
44
|
adantr |
|- ( ( ph /\ ( j e. Y /\ i e. X ) ) -> ( Base ` E ) = ( Base ` A ) ) |
377 |
375 376
|
eleqtrd |
|- ( ( ph /\ ( j e. Y /\ i e. X ) ) -> ( i ( .r ` E ) j ) e. ( Base ` A ) ) |
378 |
377
|
ralrimivva |
|- ( ph -> A. j e. Y A. i e. X ( i ( .r ` E ) j ) e. ( Base ` A ) ) |
379 |
11
|
fmpo |
|- ( A. j e. Y A. i e. X ( i ( .r ` E ) j ) e. ( Base ` A ) <-> D : ( Y X. X ) --> ( Base ` A ) ) |
380 |
378 379
|
sylib |
|- ( ph -> D : ( Y X. X ) --> ( Base ` A ) ) |
381 |
79 80 82 81 24 159 380 150
|
lcomf |
|- ( ph -> ( W oF ( .s ` A ) D ) : ( Y X. X ) --> ( Base ` A ) ) |
382 |
79 80 82 81 24 159 380 150 139 140 163
|
lcomfsupp |
|- ( ph -> ( W oF ( .s ` A ) D ) finSupp ( 0g ` A ) ) |
383 |
81 139 365 14 13 381 382
|
gsumxp |
|- ( ph -> ( A gsum ( W oF ( .s ` A ) D ) ) = ( A gsum ( j e. Y |-> ( A gsum ( i e. X |-> ( j ( W oF ( .s ` A ) D ) i ) ) ) ) ) ) |
384 |
165
|
3ad2ant1 |
|- ( ( ph /\ j e. Y /\ i e. X ) -> E e. Ring ) |
385 |
159
|
3ad2ant1 |
|- ( ( ph /\ j e. Y /\ i e. X ) -> W : ( Y X. X ) --> ( Base ` ( Scalar ` A ) ) ) |
386 |
64 62
|
eqtrd |
|- ( ph -> V = ( Base ` ( Scalar ` C ) ) ) |
387 |
386 43
|
eqsstrrd |
|- ( ph -> ( Base ` ( Scalar ` C ) ) C_ ( Base ` E ) ) |
388 |
68 387
|
eqsstrd |
|- ( ph -> ( Base ` ( Scalar ` A ) ) C_ ( Base ` E ) ) |
389 |
388 44
|
sseqtrd |
|- ( ph -> ( Base ` ( Scalar ` A ) ) C_ ( Base ` A ) ) |
390 |
389
|
3ad2ant1 |
|- ( ( ph /\ j e. Y /\ i e. X ) -> ( Base ` ( Scalar ` A ) ) C_ ( Base ` A ) ) |
391 |
385 390
|
fssd |
|- ( ( ph /\ j e. Y /\ i e. X ) -> W : ( Y X. X ) --> ( Base ` A ) ) |
392 |
|
simp2 |
|- ( ( ph /\ j e. Y /\ i e. X ) -> j e. Y ) |
393 |
|
simp3 |
|- ( ( ph /\ j e. Y /\ i e. X ) -> i e. X ) |
394 |
391 392 393
|
fovrnd |
|- ( ( ph /\ j e. Y /\ i e. X ) -> ( j W i ) e. ( Base ` A ) ) |
395 |
44
|
3ad2ant1 |
|- ( ( ph /\ j e. Y /\ i e. X ) -> ( Base ` E ) = ( Base ` A ) ) |
396 |
394 395
|
eleqtrrd |
|- ( ( ph /\ j e. Y /\ i e. X ) -> ( j W i ) e. ( Base ` E ) ) |
397 |
241
|
3impb |
|- ( ( ph /\ j e. Y /\ i e. X ) -> i e. ( Base ` E ) ) |
398 |
373
|
3impb |
|- ( ( ph /\ j e. Y /\ i e. X ) -> j e. ( Base ` E ) ) |
399 |
29 74
|
ringass |
|- ( ( E e. Ring /\ ( ( j W i ) e. ( Base ` E ) /\ i e. ( Base ` E ) /\ j e. ( 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 |
|- ( ( ph /\ j e. Y /\ i e. X ) -> ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) = ( ( j W i ) ( .r ` E ) ( i ( .r ` E ) j ) ) ) |
401 |
400
|
mpoeq3dva |
|- ( ph -> ( j e. Y , i e. X |-> ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) ) = ( j e. Y , i e. X |-> ( ( j W i ) ( .r ` E ) ( i ( .r ` E ) j ) ) ) ) |
402 |
|
ovexd |
|- ( ( ph /\ j e. Y /\ i e. X ) -> ( j W i ) e. _V ) |
403 |
|
ovexd |
|- ( ( ph /\ j e. Y /\ i e. X ) -> ( i ( .r ` E ) j ) e. _V ) |
404 |
|
fnov |
|- ( W Fn ( Y X. X ) <-> W = ( j e. Y , i e. X |-> ( j W i ) ) ) |
405 |
160 404
|
sylib |
|- ( ph -> W = ( j e. Y , i e. X |-> ( j W i ) ) ) |
406 |
11
|
a1i |
|- ( ph -> D = ( j e. Y , i e. X |-> ( i ( .r ` E ) j ) ) ) |
407 |
14 13 402 403 405 406
|
offval22 |
|- ( ph -> ( W oF ( .r ` E ) D ) = ( j e. Y , i e. X |-> ( ( j W i ) ( .r ` E ) ( i ( .r ` E ) j ) ) ) ) |
408 |
|
ofeq |
|- ( ( .r ` E ) = ( .s ` A ) -> oF ( .r ` E ) = oF ( .s ` A ) ) |
409 |
50 408
|
syl |
|- ( ph -> oF ( .r ` E ) = oF ( .s ` A ) ) |
410 |
409
|
oveqd |
|- ( ph -> ( W oF ( .r ` E ) D ) = ( W oF ( .s ` A ) D ) ) |
411 |
401 407 410
|
3eqtr2rd |
|- ( ph -> ( W oF ( .s ` A ) D ) = ( j e. Y , i e. X |-> ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) ) ) |
412 |
411
|
ad2antrr |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( W oF ( .s ` A ) D ) = ( j e. Y , i e. X |-> ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) ) ) |
413 |
412
|
oveqd |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( j ( W oF ( .s ` A ) D ) i ) = ( j ( j e. Y , i e. X |-> ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) ) i ) ) |
414 |
|
simplr |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> j e. Y ) |
415 |
|
ovexd |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) e. _V ) |
416 |
|
eqid |
|- ( j e. Y , i e. X |-> ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) ) = ( j e. Y , i e. X |-> ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) ) |
417 |
416
|
ovmpt4g |
|- ( ( j e. Y /\ i e. X /\ ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) e. _V ) -> ( j ( j e. Y , i e. X |-> ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) ) i ) = ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) ) |
418 |
414 224 415 417
|
syl3anc |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( j ( j e. Y , i e. X |-> ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) ) i ) = ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) ) |
419 |
413 418
|
eqtrd |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( j ( W oF ( .s ` A ) D ) i ) = ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) ) |
420 |
419
|
mpteq2dva |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( j ( W oF ( .s ` A ) D ) i ) ) = ( i e. X |-> ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) ) ) |
421 |
420
|
oveq2d |
|- ( ( ph /\ j e. Y ) -> ( E gsum ( i e. X |-> ( j ( W oF ( .s ` A ) D ) i ) ) ) = ( E gsum ( i e. X |-> ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) ) ) ) |
422 |
165
|
adantr |
|- ( ( ph /\ j e. Y ) -> E e. Ring ) |
423 |
370
|
sselda |
|- ( ( ph /\ j e. Y ) -> j e. ( Base ` E ) ) |
424 |
165
|
ad2antrr |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> E e. Ring ) |
425 |
387
|
ad2antrr |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( Base ` ( Scalar ` C ) ) C_ ( Base ` E ) ) |
426 |
425 222
|
sseldd |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( j W i ) e. ( Base ` E ) ) |
427 |
29 74
|
ringcl |
|- ( ( E e. Ring /\ ( j W i ) e. ( Base ` E ) /\ i e. ( Base ` E ) ) -> ( ( j W i ) ( .r ` E ) i ) e. ( Base ` E ) ) |
428 |
424 426 242 427
|
syl3anc |
|- ( ( ( ph /\ j e. Y ) /\ i e. X ) -> ( ( j W i ) ( .r ` E ) i ) e. ( Base ` E ) ) |
429 |
315
|
adantr |
|- ( ( ph /\ j e. Y ) -> ( 0g ` E ) = ( 0g ` B ) ) |
430 |
248 353 429
|
3brtr4d |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( ( j W i ) ( .r ` E ) i ) ) finSupp ( 0g ` E ) ) |
431 |
29 170 100 74 422 148 423 428 430
|
gsummulc1 |
|- ( ( ph /\ j e. Y ) -> ( E gsum ( i e. X |-> ( ( ( j W i ) ( .r ` E ) i ) ( .r ` E ) j ) ) ) = ( ( E gsum ( i e. X |-> ( ( j W i ) ( .r ` E ) i ) ) ) ( .r ` E ) j ) ) |
432 |
421 431
|
eqtrd |
|- ( ( ph /\ j e. Y ) -> ( E gsum ( i e. X |-> ( j ( W oF ( .s ` A ) D ) i ) ) ) = ( ( E gsum ( i e. X |-> ( ( j W i ) ( .r ` E ) i ) ) ) ( .r ` E ) j ) ) |
433 |
148
|
mptexd |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( j ( W oF ( .s ` A ) D ) i ) ) e. _V ) |
434 |
24
|
adantr |
|- ( ( ph /\ j e. Y ) -> A e. LMod ) |
435 |
43
|
adantr |
|- ( ( ph /\ j e. Y ) -> V C_ ( Base ` E ) ) |
436 |
1 433 197 434 435
|
gsumsra |
|- ( ( ph /\ j e. Y ) -> ( E gsum ( i e. X |-> ( j ( W oF ( .s ` A ) D ) i ) ) ) = ( A gsum ( i e. X |-> ( j ( W oF ( .s ` A ) D ) i ) ) ) ) |
437 |
148
|
mptexd |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( ( j W i ) ( .r ` E ) i ) ) e. _V ) |
438 |
1 437 197 434 435
|
gsumsra |
|- ( ( ph /\ j e. Y ) -> ( E gsum ( i e. X |-> ( ( j W i ) ( .r ` E ) i ) ) ) = ( A gsum ( i e. X |-> ( ( j W i ) ( .r ` E ) i ) ) ) ) |
439 |
438
|
oveq1d |
|- ( ( ph /\ j e. Y ) -> ( ( E gsum ( i e. X |-> ( ( j W i ) ( .r ` E ) i ) ) ) ( .r ` E ) j ) = ( ( A gsum ( i e. X |-> ( ( j W i ) ( .r ` E ) i ) ) ) ( .r ` E ) j ) ) |
440 |
50
|
adantr |
|- ( ( ph /\ j e. Y ) -> ( .r ` E ) = ( .s ` A ) ) |
441 |
350
|
oveq2d |
|- ( ( ph /\ j e. Y ) -> ( A gsum ( i e. X |-> ( ( j W i ) ( .r ` E ) i ) ) ) = ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) ) ) |
442 |
440 441 356
|
oveq123d |
|- ( ( ph /\ j e. Y ) -> ( ( A gsum ( i e. X |-> ( ( j W i ) ( .r ` E ) i ) ) ) ( .r ` E ) j ) = ( ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) ) ( .s ` A ) j ) ) |
443 |
439 442
|
eqtrd |
|- ( ( ph /\ j e. Y ) -> ( ( E gsum ( i e. X |-> ( ( j W i ) ( .r ` E ) i ) ) ) ( .r ` E ) j ) = ( ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) ) ( .s ` A ) j ) ) |
444 |
432 436 443
|
3eqtr3d |
|- ( ( ph /\ j e. Y ) -> ( A gsum ( i e. X |-> ( j ( W oF ( .s ` A ) D ) i ) ) ) = ( ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) ) ( .s ` A ) j ) ) |
445 |
444
|
mpteq2dva |
|- ( ph -> ( j e. Y |-> ( A gsum ( i e. X |-> ( j ( W oF ( .s ` A ) D ) i ) ) ) ) = ( j e. Y |-> ( ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) ) ( .s ` A ) j ) ) ) |
446 |
445
|
oveq2d |
|- ( ph -> ( A gsum ( j e. Y |-> ( A gsum ( i e. X |-> ( j ( W oF ( .s ` A ) D ) i ) ) ) ) ) = ( A gsum ( j e. Y |-> ( ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) ) ( .s ` A ) j ) ) ) ) |
447 |
383 16 446
|
3eqtr3rd |
|- ( ph -> ( A gsum ( j e. Y |-> ( ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) ) ( .s ` A ) j ) ) ) = ( 0g ` A ) ) |
448 |
1 6 20
|
drgext0g |
|- ( ph -> ( 0g ` E ) = ( 0g ` A ) ) |
449 |
447 448 315
|
3eqtr2d |
|- ( ph -> ( A gsum ( j e. Y |-> ( ( A gsum ( i e. X |-> ( ( j W i ) ( .s ` A ) i ) ) ) ( .s ` A ) j ) ) ) = ( 0g ` B ) ) |
450 |
1 6 20 5 8 14
|
drgextgsum |
|- ( ph -> ( E gsum ( j e. Y |-> ( ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) ) = ( A gsum ( j e. Y |-> ( ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) ) ) |
451 |
2 6 9 4 7 14
|
drgextgsum |
|- ( ph -> ( E gsum ( j e. Y |-> ( ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) ) = ( B gsum ( j e. Y |-> ( ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) ) ) |
452 |
450 451
|
eqtr3d |
|- ( ph -> ( A gsum ( j e. Y |-> ( ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) ) = ( B gsum ( j e. Y |-> ( ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) ) ) |
453 |
361 449 452
|
3eqtr3rd |
|- ( ph -> ( B gsum ( j e. Y |-> ( ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ( .s ` B ) j ) ) ) = ( 0g ` B ) ) |
454 |
346 453
|
eqtrd |
|- ( ph -> ( B gsum ( j e. Y |-> ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ` j ) ( .s ` B ) j ) ) ) = ( 0g ` B ) ) |
455 |
|
breq1 |
|- ( b = ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) -> ( b finSupp ( 0g ` ( Scalar ` B ) ) <-> ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) finSupp ( 0g ` ( Scalar ` B ) ) ) ) |
456 |
|
nfmpt1 |
|- F/_ j ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) |
457 |
456
|
nfeq2 |
|- F/ j b = ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) |
458 |
|
fveq1 |
|- ( b = ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) -> ( b ` j ) = ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ` j ) ) |
459 |
458
|
oveq1d |
|- ( b = ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) -> ( ( b ` j ) ( .s ` B ) j ) = ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ` j ) ( .s ` B ) j ) ) |
460 |
459
|
adantr |
|- ( ( b = ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) /\ j e. Y ) -> ( ( b ` j ) ( .s ` B ) j ) = ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ` j ) ( .s ` B ) j ) ) |
461 |
457 460
|
mpteq2da |
|- ( b = ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) -> ( j e. Y |-> ( ( b ` j ) ( .s ` B ) j ) ) = ( j e. Y |-> ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ` j ) ( .s ` B ) j ) ) ) |
462 |
461
|
oveq2d |
|- ( b = ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) -> ( B gsum ( j e. Y |-> ( ( b ` j ) ( .s ` B ) j ) ) ) = ( B gsum ( j e. Y |-> ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ` j ) ( .s ` B ) j ) ) ) ) |
463 |
462
|
eqeq1d |
|- ( b = ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) -> ( ( B gsum ( j e. Y |-> ( ( b ` j ) ( .s ` B ) j ) ) ) = ( 0g ` B ) <-> ( B gsum ( j e. Y |-> ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ` j ) ( .s ` B ) j ) ) ) = ( 0g ` B ) ) ) |
464 |
455 463
|
anbi12d |
|- ( b = ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) -> ( ( b finSupp ( 0g ` ( Scalar ` B ) ) /\ ( B gsum ( j e. Y |-> ( ( b ` j ) ( .s ` B ) j ) ) ) = ( 0g ` B ) ) <-> ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) finSupp ( 0g ` ( Scalar ` B ) ) /\ ( B gsum ( j e. Y |-> ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ` j ) ( .s ` B ) j ) ) ) = ( 0g ` B ) ) ) ) |
465 |
|
eqeq1 |
|- ( b = ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) -> ( b = ( Y X. { ( 0g ` ( Scalar ` B ) ) } ) <-> ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) = ( Y X. { ( 0g ` ( Scalar ` B ) ) } ) ) ) |
466 |
464 465
|
imbi12d |
|- ( b = ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) -> ( ( ( b finSupp ( 0g ` ( Scalar ` B ) ) /\ ( B gsum ( j e. Y |-> ( ( b ` j ) ( .s ` B ) j ) ) ) = ( 0g ` B ) ) -> b = ( Y X. { ( 0g ` ( Scalar ` B ) ) } ) ) <-> ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) finSupp ( 0g ` ( Scalar ` B ) ) /\ ( B gsum ( j e. Y |-> ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ` j ) ( .s ` B ) j ) ) ) = ( 0g ` B ) ) -> ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) = ( Y X. { ( 0g ` ( Scalar ` B ) ) } ) ) ) ) |
467 |
367
|
lbslinds |
|- ( LBasis ` B ) C_ ( LIndS ` B ) |
468 |
467 14
|
sselid |
|- ( ph -> Y e. ( LIndS ` B ) ) |
469 |
|
eqid |
|- ( Base ` ( Scalar ` B ) ) = ( Base ` ( Scalar ` B ) ) |
470 |
236 469 322 247 207 323
|
islinds5 |
|- ( ( B e. LMod /\ Y C_ ( Base ` B ) ) -> ( Y e. ( LIndS ` B ) <-> A. b e. ( ( Base ` ( Scalar ` B ) ) ^m Y ) ( ( b finSupp ( 0g ` ( Scalar ` B ) ) /\ ( B gsum ( j e. Y |-> ( ( b ` j ) ( .s ` B ) j ) ) ) = ( 0g ` B ) ) -> b = ( Y X. { ( 0g ` ( Scalar ` B ) ) } ) ) ) ) |
471 |
470
|
biimpa |
|- ( ( ( B e. LMod /\ Y C_ ( Base ` B ) ) /\ Y e. ( LIndS ` B ) ) -> A. b e. ( ( Base ` ( Scalar ` B ) ) ^m Y ) ( ( b finSupp ( 0g ` ( Scalar ` B ) ) /\ ( B gsum ( j e. Y |-> ( ( b ` j ) ( .s ` B ) j ) ) ) = ( 0g ` B ) ) -> b = ( Y X. { ( 0g ` ( Scalar ` B ) ) } ) ) ) |
472 |
211 369 468 471
|
syl21anc |
|- ( ph -> A. b e. ( ( Base ` ( Scalar ` B ) ) ^m Y ) ( ( b finSupp ( 0g ` ( Scalar ` B ) ) /\ ( B gsum ( j e. Y |-> ( ( b ` j ) ( .s ` B ) j ) ) ) = ( 0g ` B ) ) -> b = ( Y X. { ( 0g ` ( Scalar ` B ) ) } ) ) ) |
473 |
|
fvexd |
|- ( ph -> ( Base ` ( Scalar ` B ) ) e. _V ) |
474 |
|
elmapg |
|- ( ( ( Base ` ( Scalar ` B ) ) e. _V /\ Y e. ( LBasis ` B ) ) -> ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) e. ( ( Base ` ( Scalar ` B ) ) ^m Y ) <-> ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) : Y --> ( Base ` ( Scalar ` B ) ) ) ) |
475 |
474
|
biimpar |
|- ( ( ( ( Base ` ( Scalar ` B ) ) e. _V /\ Y e. ( LBasis ` B ) ) /\ ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) : Y --> ( Base ` ( Scalar ` B ) ) ) -> ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) e. ( ( Base ` ( Scalar ` B ) ) ^m Y ) ) |
476 |
473 14 254 475
|
syl21anc |
|- ( ph -> ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) e. ( ( Base ` ( Scalar ` B ) ) ^m Y ) ) |
477 |
466 472 476
|
rspcdva |
|- ( ph -> ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) finSupp ( 0g ` ( Scalar ` B ) ) /\ ( B gsum ( j e. Y |-> ( ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) ` j ) ( .s ` B ) j ) ) ) = ( 0g ` B ) ) -> ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) = ( Y X. { ( 0g ` ( Scalar ` B ) ) } ) ) ) |
478 |
340 454 477
|
mp2and |
|- ( ph -> ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) = ( Y X. { ( 0g ` ( Scalar ` B ) ) } ) ) |
479 |
|
fconstmpt |
|- ( Y X. { ( 0g ` ( Scalar ` B ) ) } ) = ( j e. Y |-> ( 0g ` ( Scalar ` B ) ) ) |
480 |
478 479
|
eqtrdi |
|- ( ph -> ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) = ( j e. Y |-> ( 0g ` ( Scalar ` B ) ) ) ) |
481 |
|
ovex |
|- ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) e. _V |
482 |
481
|
rgenw |
|- A. j e. Y ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) e. _V |
483 |
|
mpteqb |
|- ( A. j e. Y ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) e. _V -> ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) = ( j e. Y |-> ( 0g ` ( Scalar ` B ) ) ) <-> A. j e. Y ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) = ( 0g ` ( Scalar ` B ) ) ) ) |
484 |
482 483
|
ax-mp |
|- ( ( j e. Y |-> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) ) = ( j e. Y |-> ( 0g ` ( Scalar ` B ) ) ) <-> A. j e. Y ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) = ( 0g ` ( Scalar ` B ) ) ) |
485 |
480 484
|
sylib |
|- ( ph -> A. j e. Y ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) = ( 0g ` ( Scalar ` B ) ) ) |
486 |
485
|
r19.21bi |
|- ( ( ph /\ j e. Y ) -> ( B gsum ( i e. X |-> ( ( j W i ) ( .s ` B ) i ) ) ) = ( 0g ` ( Scalar ` B ) ) ) |
487 |
315 448 316
|
3eqtr3rd |
|- ( ph -> ( 0g ` ( Scalar ` B ) ) = ( 0g ` A ) ) |
488 |
487
|
adantr |
|- ( ( ph /\ j e. Y ) -> ( 0g ` ( Scalar ` B ) ) = ( 0g ` A ) ) |
489 |
205 486 488
|
3eqtrd |
|- ( ( ph /\ j e. Y ) -> ( A gsum ( i e. X |-> ( ( ( i e. X |-> ( j W i ) ) ` i ) ( .s ` A ) i ) ) ) = ( 0g ` A ) ) |
490 |
186 489
|
jca |
|- ( ( ph /\ j e. Y ) -> ( ( i e. X |-> ( j W i ) ) finSupp ( 0g ` ( Scalar ` A ) ) /\ ( A gsum ( i e. X |-> ( ( ( i e. X |-> ( j W i ) ) ` i ) ( .s ` A ) i ) ) ) = ( 0g ` A ) ) ) |
491 |
189
|
fmpttd |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( j W i ) ) : X --> ( Base ` ( Scalar ` A ) ) ) |
492 |
|
fvexd |
|- ( ( ph /\ j e. Y ) -> ( Base ` ( Scalar ` A ) ) e. _V ) |
493 |
492 148
|
elmapd |
|- ( ( ph /\ j e. Y ) -> ( ( i e. X |-> ( j W i ) ) e. ( ( Base ` ( Scalar ` A ) ) ^m X ) <-> ( i e. X |-> ( j W i ) ) : X --> ( Base ` ( Scalar ` A ) ) ) ) |
494 |
491 493
|
mpbird |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( j W i ) ) e. ( ( Base ` ( Scalar ` A ) ) ^m X ) ) |
495 |
|
simpr |
|- ( ( ( ph /\ j e. Y ) /\ w = ( i e. X |-> ( j W i ) ) ) -> w = ( i e. X |-> ( j W i ) ) ) |
496 |
495
|
breq1d |
|- ( ( ( ph /\ j e. Y ) /\ w = ( i e. X |-> ( j W i ) ) ) -> ( w finSupp ( 0g ` ( Scalar ` A ) ) <-> ( i e. X |-> ( j W i ) ) finSupp ( 0g ` ( Scalar ` A ) ) ) ) |
497 |
|
nfv |
|- F/ i ( ph /\ j e. Y ) |
498 |
|
nfmpt1 |
|- F/_ i ( i e. X |-> ( j W i ) ) |
499 |
498
|
nfeq2 |
|- F/ i w = ( i e. X |-> ( j W i ) ) |
500 |
497 499
|
nfan |
|- F/ i ( ( ph /\ j e. Y ) /\ w = ( i e. X |-> ( j W i ) ) ) |
501 |
|
simplr |
|- ( ( ( ( ph /\ j e. Y ) /\ w = ( i e. X |-> ( j W i ) ) ) /\ i e. X ) -> w = ( i e. X |-> ( j W i ) ) ) |
502 |
501
|
fveq1d |
|- ( ( ( ( ph /\ j e. Y ) /\ w = ( i e. X |-> ( j W i ) ) ) /\ i e. X ) -> ( w ` i ) = ( ( i e. X |-> ( j W i ) ) ` i ) ) |
503 |
502
|
oveq1d |
|- ( ( ( ( ph /\ j e. Y ) /\ w = ( i e. X |-> ( j W i ) ) ) /\ i e. X ) -> ( ( w ` i ) ( .s ` A ) i ) = ( ( ( i e. X |-> ( j W i ) ) ` i ) ( .s ` A ) i ) ) |
504 |
500 503
|
mpteq2da |
|- ( ( ( ph /\ j e. Y ) /\ w = ( i e. X |-> ( j W i ) ) ) -> ( i e. X |-> ( ( w ` i ) ( .s ` A ) i ) ) = ( i e. X |-> ( ( ( i e. X |-> ( j W i ) ) ` i ) ( .s ` A ) i ) ) ) |
505 |
504
|
oveq2d |
|- ( ( ( ph /\ j e. Y ) /\ w = ( i e. X |-> ( j W i ) ) ) -> ( A gsum ( i e. X |-> ( ( w ` i ) ( .s ` A ) i ) ) ) = ( A gsum ( i e. X |-> ( ( ( i e. X |-> ( j W i ) ) ` i ) ( .s ` A ) i ) ) ) ) |
506 |
505
|
eqeq1d |
|- ( ( ( ph /\ j e. Y ) /\ w = ( i e. X |-> ( j W i ) ) ) -> ( ( A gsum ( i e. X |-> ( ( w ` i ) ( .s ` A ) i ) ) ) = ( 0g ` A ) <-> ( A gsum ( i e. X |-> ( ( ( i e. X |-> ( j W i ) ) ` i ) ( .s ` A ) i ) ) ) = ( 0g ` A ) ) ) |
507 |
496 506
|
anbi12d |
|- ( ( ( ph /\ j e. Y ) /\ w = ( i e. X |-> ( j W i ) ) ) -> ( ( w finSupp ( 0g ` ( Scalar ` A ) ) /\ ( A gsum ( i e. X |-> ( ( w ` i ) ( .s ` A ) i ) ) ) = ( 0g ` A ) ) <-> ( ( i e. X |-> ( j W i ) ) finSupp ( 0g ` ( Scalar ` A ) ) /\ ( A gsum ( i e. X |-> ( ( ( i e. X |-> ( j W i ) ) ` i ) ( .s ` A ) i ) ) ) = ( 0g ` A ) ) ) ) |
508 |
495
|
eqeq1d |
|- ( ( ( ph /\ j e. Y ) /\ w = ( i e. X |-> ( j W i ) ) ) -> ( w = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) <-> ( i e. X |-> ( j W i ) ) = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) ) |
509 |
507 508
|
imbi12d |
|- ( ( ( ph /\ j e. Y ) /\ w = ( i e. X |-> ( j W i ) ) ) -> ( ( ( w finSupp ( 0g ` ( Scalar ` A ) ) /\ ( A gsum ( i e. X |-> ( ( w ` i ) ( .s ` A ) i ) ) ) = ( 0g ` A ) ) -> w = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) <-> ( ( ( i e. X |-> ( j W i ) ) finSupp ( 0g ` ( Scalar ` A ) ) /\ ( A gsum ( i e. X |-> ( ( ( i e. X |-> ( j W i ) ) ` i ) ( .s ` A ) i ) ) ) = ( 0g ` A ) ) -> ( i e. X |-> ( j W i ) ) = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) ) ) |
510 |
494 509
|
rspcdv |
|- ( ( ph /\ j e. Y ) -> ( A. w e. ( ( Base ` ( Scalar ` A ) ) ^m X ) ( ( w finSupp ( 0g ` ( Scalar ` A ) ) /\ ( A gsum ( i e. X |-> ( ( w ` i ) ( .s ` A ) i ) ) ) = ( 0g ` A ) ) -> w = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) -> ( ( ( i e. X |-> ( j W i ) ) finSupp ( 0g ` ( Scalar ` A ) ) /\ ( A gsum ( i e. X |-> ( ( ( i e. X |-> ( j W i ) ) ` i ) ( .s ` A ) i ) ) ) = ( 0g ` A ) ) -> ( i e. X |-> ( j W i ) ) = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) ) ) |
511 |
144 490 510
|
mp2d |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( j W i ) ) = ( X X. { ( 0g ` ( Scalar ` A ) ) } ) ) |
512 |
511 257
|
eqtrdi |
|- ( ( ph /\ j e. Y ) -> ( i e. X |-> ( j W i ) ) = ( i e. X |-> ( 0g ` ( Scalar ` A ) ) ) ) |
513 |
512 311
|
sylib |
|- ( ( ph /\ j e. Y ) -> A. i e. X ( j W i ) = ( 0g ` ( Scalar ` A ) ) ) |
514 |
513
|
ralrimiva |
|- ( ph -> A. j e. Y A. i e. X ( j W i ) = ( 0g ` ( Scalar ` A ) ) ) |
515 |
|
eqidd |
|- ( ( j = k /\ i = l ) -> ( 0g ` ( Scalar ` A ) ) = ( 0g ` ( Scalar ` A ) ) ) |
516 |
|
fvexd |
|- ( ( ph /\ j e. Y /\ i e. X ) -> ( 0g ` ( Scalar ` A ) ) e. _V ) |
517 |
|
fvexd |
|- ( ( ph /\ k e. Y /\ l e. X ) -> ( 0g ` ( Scalar ` A ) ) e. _V ) |
518 |
160 515 516 517
|
fnmpoovd |
|- ( ph -> ( W = ( k e. Y , l e. X |-> ( 0g ` ( Scalar ` A ) ) ) <-> A. j e. Y A. i e. X ( j W i ) = ( 0g ` ( Scalar ` A ) ) ) ) |
519 |
514 518
|
mpbird |
|- ( ph -> W = ( k e. Y , l e. X |-> ( 0g ` ( Scalar ` A ) ) ) ) |
520 |
|
fconstmpo |
|- ( ( Y X. X ) X. { ( 0g ` ( Scalar ` A ) ) } ) = ( k e. Y , l e. X |-> ( 0g ` ( Scalar ` A ) ) ) |
521 |
519 520
|
eqtr4di |
|- ( ph -> W = ( ( Y X. X ) X. { ( 0g ` ( Scalar ` A ) ) } ) ) |