Step |
Hyp |
Ref |
Expression |
1 |
|
elrgspn.b |
|- B = ( Base ` R ) |
2 |
|
elrgspn.m |
|- M = ( mulGrp ` R ) |
3 |
|
elrgspn.x |
|- .x. = ( .g ` R ) |
4 |
|
elrgspn.n |
|- N = ( RingSpan ` R ) |
5 |
|
elrgspn.f |
|- F = { f e. ( ZZ ^m Word A ) | f finSupp 0 } |
6 |
|
elrgspn.r |
|- ( ph -> R e. Ring ) |
7 |
|
elrgspn.a |
|- ( ph -> A C_ B ) |
8 |
|
elrgspnlem1.1 |
|- S = ran ( g e. F |-> ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) |
9 |
|
eqid |
|- ( g e. F |-> ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) = ( g e. F |-> ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) |
10 |
|
fveq1 |
|- ( g = ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) -> ( g ` w ) = ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) ) |
11 |
10
|
oveq1d |
|- ( g = ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) -> ( ( g ` w ) .x. ( M gsum w ) ) = ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) ) |
12 |
11
|
mpteq2dv |
|- ( g = ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) -> ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) = ( w e. Word A |-> ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) ) ) |
13 |
12
|
oveq2d |
|- ( g = ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) -> ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) = ( R gsum ( w e. Word A |-> ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) ) ) ) |
14 |
13
|
eqeq2d |
|- ( g = ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) -> ( x = ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) <-> x = ( R gsum ( w e. Word A |-> ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) ) ) ) ) |
15 |
|
breq1 |
|- ( f = ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) -> ( f finSupp 0 <-> ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) finSupp 0 ) ) |
16 |
|
zex |
|- ZZ e. _V |
17 |
16
|
a1i |
|- ( ( ph /\ x e. A ) -> ZZ e. _V ) |
18 |
1
|
fvexi |
|- B e. _V |
19 |
18
|
a1i |
|- ( ph -> B e. _V ) |
20 |
19 7
|
ssexd |
|- ( ph -> A e. _V ) |
21 |
|
wrdexg |
|- ( A e. _V -> Word A e. _V ) |
22 |
20 21
|
syl |
|- ( ph -> Word A e. _V ) |
23 |
22
|
adantr |
|- ( ( ph /\ x e. A ) -> Word A e. _V ) |
24 |
|
1zzd |
|- ( ( ( ( ph /\ x e. A ) /\ v e. Word A ) /\ v = <" x "> ) -> 1 e. ZZ ) |
25 |
|
0zd |
|- ( ( ( ( ph /\ x e. A ) /\ v e. Word A ) /\ -. v = <" x "> ) -> 0 e. ZZ ) |
26 |
24 25
|
ifclda |
|- ( ( ( ph /\ x e. A ) /\ v e. Word A ) -> if ( v = <" x "> , 1 , 0 ) e. ZZ ) |
27 |
26
|
fmpttd |
|- ( ( ph /\ x e. A ) -> ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) : Word A --> ZZ ) |
28 |
17 23 27
|
elmapdd |
|- ( ( ph /\ x e. A ) -> ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) e. ( ZZ ^m Word A ) ) |
29 |
28
|
elexd |
|- ( ( ph /\ x e. A ) -> ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) e. _V ) |
30 |
27
|
ffund |
|- ( ( ph /\ x e. A ) -> Fun ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ) |
31 |
|
0zd |
|- ( ( ph /\ x e. A ) -> 0 e. ZZ ) |
32 |
|
snfi |
|- { <" x "> } e. Fin |
33 |
32
|
a1i |
|- ( ( ph /\ x e. A ) -> { <" x "> } e. Fin ) |
34 |
|
eldifsni |
|- ( v e. ( Word A \ { <" x "> } ) -> v =/= <" x "> ) |
35 |
34
|
adantl |
|- ( ( ( ph /\ x e. A ) /\ v e. ( Word A \ { <" x "> } ) ) -> v =/= <" x "> ) |
36 |
35
|
neneqd |
|- ( ( ( ph /\ x e. A ) /\ v e. ( Word A \ { <" x "> } ) ) -> -. v = <" x "> ) |
37 |
36
|
iffalsed |
|- ( ( ( ph /\ x e. A ) /\ v e. ( Word A \ { <" x "> } ) ) -> if ( v = <" x "> , 1 , 0 ) = 0 ) |
38 |
37 23
|
suppss2 |
|- ( ( ph /\ x e. A ) -> ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) supp 0 ) C_ { <" x "> } ) |
39 |
|
suppssfifsupp |
|- ( ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) e. _V /\ Fun ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) /\ 0 e. ZZ ) /\ ( { <" x "> } e. Fin /\ ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) supp 0 ) C_ { <" x "> } ) ) -> ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) finSupp 0 ) |
40 |
29 30 31 33 38 39
|
syl32anc |
|- ( ( ph /\ x e. A ) -> ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) finSupp 0 ) |
41 |
15 28 40
|
elrabd |
|- ( ( ph /\ x e. A ) -> ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) e. { f e. ( ZZ ^m Word A ) | f finSupp 0 } ) |
42 |
41 5
|
eleqtrrdi |
|- ( ( ph /\ x e. A ) -> ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) e. F ) |
43 |
|
eqeq2 |
|- ( x = if ( w = <" x "> , x , ( 0g ` R ) ) -> ( ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) = x <-> ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) = if ( w = <" x "> , x , ( 0g ` R ) ) ) ) |
44 |
|
eqeq2 |
|- ( ( 0g ` R ) = if ( w = <" x "> , x , ( 0g ` R ) ) -> ( ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) = ( 0g ` R ) <-> ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) = if ( w = <" x "> , x , ( 0g ` R ) ) ) ) |
45 |
|
eqid |
|- ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) = ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) |
46 |
|
simpr |
|- ( ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) /\ v = w ) -> v = w ) |
47 |
|
simplr |
|- ( ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) /\ v = w ) -> w = <" x "> ) |
48 |
46 47
|
eqtrd |
|- ( ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) /\ v = w ) -> v = <" x "> ) |
49 |
48
|
iftrued |
|- ( ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) /\ v = w ) -> if ( v = <" x "> , 1 , 0 ) = 1 ) |
50 |
|
simplr |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) -> w e. Word A ) |
51 |
|
1zzd |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) -> 1 e. ZZ ) |
52 |
45 49 50 51
|
fvmptd2 |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) -> ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) = 1 ) |
53 |
|
simpr |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) -> w = <" x "> ) |
54 |
53
|
oveq2d |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) -> ( M gsum w ) = ( M gsum <" x "> ) ) |
55 |
7
|
sselda |
|- ( ( ph /\ x e. A ) -> x e. B ) |
56 |
55
|
ad2antrr |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) -> x e. B ) |
57 |
2 1
|
mgpbas |
|- B = ( Base ` M ) |
58 |
57
|
gsumws1 |
|- ( x e. B -> ( M gsum <" x "> ) = x ) |
59 |
56 58
|
syl |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) -> ( M gsum <" x "> ) = x ) |
60 |
54 59
|
eqtrd |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) -> ( M gsum w ) = x ) |
61 |
52 60
|
oveq12d |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) -> ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) = ( 1 .x. x ) ) |
62 |
1 3
|
mulg1 |
|- ( x e. B -> ( 1 .x. x ) = x ) |
63 |
56 62
|
syl |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) -> ( 1 .x. x ) = x ) |
64 |
61 63
|
eqtrd |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ w = <" x "> ) -> ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) = x ) |
65 |
|
eqeq1 |
|- ( v = w -> ( v = <" x "> <-> w = <" x "> ) ) |
66 |
65
|
notbid |
|- ( v = w -> ( -. v = <" x "> <-> -. w = <" x "> ) ) |
67 |
66
|
biimparc |
|- ( ( -. w = <" x "> /\ v = w ) -> -. v = <" x "> ) |
68 |
67
|
adantll |
|- ( ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ -. w = <" x "> ) /\ v = w ) -> -. v = <" x "> ) |
69 |
68
|
iffalsed |
|- ( ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ -. w = <" x "> ) /\ v = w ) -> if ( v = <" x "> , 1 , 0 ) = 0 ) |
70 |
|
simplr |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ -. w = <" x "> ) -> w e. Word A ) |
71 |
|
0zd |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ -. w = <" x "> ) -> 0 e. ZZ ) |
72 |
45 69 70 71
|
fvmptd2 |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ -. w = <" x "> ) -> ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) = 0 ) |
73 |
72
|
oveq1d |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ -. w = <" x "> ) -> ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) = ( 0 .x. ( M gsum w ) ) ) |
74 |
2
|
ringmgp |
|- ( R e. Ring -> M e. Mnd ) |
75 |
6 74
|
syl |
|- ( ph -> M e. Mnd ) |
76 |
75
|
ad3antrrr |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ -. w = <" x "> ) -> M e. Mnd ) |
77 |
|
sswrd |
|- ( A C_ B -> Word A C_ Word B ) |
78 |
7 77
|
syl |
|- ( ph -> Word A C_ Word B ) |
79 |
78
|
adantr |
|- ( ( ph /\ x e. A ) -> Word A C_ Word B ) |
80 |
79
|
sselda |
|- ( ( ( ph /\ x e. A ) /\ w e. Word A ) -> w e. Word B ) |
81 |
80
|
adantr |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ -. w = <" x "> ) -> w e. Word B ) |
82 |
57
|
gsumwcl |
|- ( ( M e. Mnd /\ w e. Word B ) -> ( M gsum w ) e. B ) |
83 |
76 81 82
|
syl2anc |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ -. w = <" x "> ) -> ( M gsum w ) e. B ) |
84 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
85 |
1 84 3
|
mulg0 |
|- ( ( M gsum w ) e. B -> ( 0 .x. ( M gsum w ) ) = ( 0g ` R ) ) |
86 |
83 85
|
syl |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ -. w = <" x "> ) -> ( 0 .x. ( M gsum w ) ) = ( 0g ` R ) ) |
87 |
73 86
|
eqtrd |
|- ( ( ( ( ph /\ x e. A ) /\ w e. Word A ) /\ -. w = <" x "> ) -> ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) = ( 0g ` R ) ) |
88 |
43 44 64 87
|
ifbothda |
|- ( ( ( ph /\ x e. A ) /\ w e. Word A ) -> ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) = if ( w = <" x "> , x , ( 0g ` R ) ) ) |
89 |
88
|
mpteq2dva |
|- ( ( ph /\ x e. A ) -> ( w e. Word A |-> ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) ) = ( w e. Word A |-> if ( w = <" x "> , x , ( 0g ` R ) ) ) ) |
90 |
89
|
oveq2d |
|- ( ( ph /\ x e. A ) -> ( R gsum ( w e. Word A |-> ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) ) ) = ( R gsum ( w e. Word A |-> if ( w = <" x "> , x , ( 0g ` R ) ) ) ) ) |
91 |
|
ringmnd |
|- ( R e. Ring -> R e. Mnd ) |
92 |
6 91
|
syl |
|- ( ph -> R e. Mnd ) |
93 |
92
|
adantr |
|- ( ( ph /\ x e. A ) -> R e. Mnd ) |
94 |
|
simpr |
|- ( ( ph /\ x e. A ) -> x e. A ) |
95 |
94
|
s1cld |
|- ( ( ph /\ x e. A ) -> <" x "> e. Word A ) |
96 |
|
eqid |
|- ( w e. Word A |-> if ( w = <" x "> , x , ( 0g ` R ) ) ) = ( w e. Word A |-> if ( w = <" x "> , x , ( 0g ` R ) ) ) |
97 |
7 1
|
sseqtrdi |
|- ( ph -> A C_ ( Base ` R ) ) |
98 |
97
|
sselda |
|- ( ( ph /\ x e. A ) -> x e. ( Base ` R ) ) |
99 |
84 93 23 95 96 98
|
gsummptif1n0 |
|- ( ( ph /\ x e. A ) -> ( R gsum ( w e. Word A |-> if ( w = <" x "> , x , ( 0g ` R ) ) ) ) = x ) |
100 |
90 99
|
eqtr2d |
|- ( ( ph /\ x e. A ) -> x = ( R gsum ( w e. Word A |-> ( ( ( v e. Word A |-> if ( v = <" x "> , 1 , 0 ) ) ` w ) .x. ( M gsum w ) ) ) ) ) |
101 |
14 42 100
|
rspcedvdw |
|- ( ( ph /\ x e. A ) -> E. g e. F x = ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) |
102 |
9 101 94
|
elrnmptd |
|- ( ( ph /\ x e. A ) -> x e. ran ( g e. F |-> ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) ) |
103 |
102 8
|
eleqtrrdi |
|- ( ( ph /\ x e. A ) -> x e. S ) |
104 |
103
|
ex |
|- ( ph -> ( x e. A -> x e. S ) ) |
105 |
104
|
ssrdv |
|- ( ph -> A C_ S ) |