| 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 |
1
|
a1i |
|- ( ph -> B = ( Base ` R ) ) |
| 9 |
4
|
a1i |
|- ( ph -> N = ( RingSpan ` R ) ) |
| 10 |
|
eqidd |
|- ( ph -> ( N ` A ) = ( N ` A ) ) |
| 11 |
6 8 7 9 10
|
rgspncl |
|- ( ph -> ( N ` A ) e. ( SubRing ` R ) ) |
| 12 |
1
|
subrgss |
|- ( ( N ` A ) e. ( SubRing ` R ) -> ( N ` A ) C_ B ) |
| 13 |
11 12
|
syl |
|- ( ph -> ( N ` A ) C_ B ) |
| 14 |
13
|
sselda |
|- ( ( ph /\ X e. ( N ` A ) ) -> X e. B ) |
| 15 |
|
simpr |
|- ( ( ( ph /\ g e. F ) /\ X = ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) -> X = ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) |
| 16 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
| 17 |
6
|
ringcmnd |
|- ( ph -> R e. CMnd ) |
| 18 |
17
|
adantr |
|- ( ( ph /\ g e. F ) -> R e. CMnd ) |
| 19 |
1
|
fvexi |
|- B e. _V |
| 20 |
19
|
a1i |
|- ( ph -> B e. _V ) |
| 21 |
20 7
|
ssexd |
|- ( ph -> A e. _V ) |
| 22 |
21
|
adantr |
|- ( ( ph /\ g e. F ) -> A e. _V ) |
| 23 |
|
wrdexg |
|- ( A e. _V -> Word A e. _V ) |
| 24 |
22 23
|
syl |
|- ( ( ph /\ g e. F ) -> Word A e. _V ) |
| 25 |
6
|
ringgrpd |
|- ( ph -> R e. Grp ) |
| 26 |
25
|
ad2antrr |
|- ( ( ( ph /\ g e. F ) /\ w e. Word A ) -> R e. Grp ) |
| 27 |
|
zex |
|- ZZ e. _V |
| 28 |
27
|
a1i |
|- ( ( ph /\ g e. F ) -> ZZ e. _V ) |
| 29 |
|
breq1 |
|- ( f = g -> ( f finSupp 0 <-> g finSupp 0 ) ) |
| 30 |
29 5
|
elrab2 |
|- ( g e. F <-> ( g e. ( ZZ ^m Word A ) /\ g finSupp 0 ) ) |
| 31 |
30
|
biimpi |
|- ( g e. F -> ( g e. ( ZZ ^m Word A ) /\ g finSupp 0 ) ) |
| 32 |
31
|
simpld |
|- ( g e. F -> g e. ( ZZ ^m Word A ) ) |
| 33 |
32
|
adantl |
|- ( ( ph /\ g e. F ) -> g e. ( ZZ ^m Word A ) ) |
| 34 |
24 28 33
|
elmaprd |
|- ( ( ph /\ g e. F ) -> g : Word A --> ZZ ) |
| 35 |
34
|
ffvelcdmda |
|- ( ( ( ph /\ g e. F ) /\ w e. Word A ) -> ( g ` w ) e. ZZ ) |
| 36 |
2
|
ringmgp |
|- ( R e. Ring -> M e. Mnd ) |
| 37 |
6 36
|
syl |
|- ( ph -> M e. Mnd ) |
| 38 |
37
|
ad2antrr |
|- ( ( ( ph /\ g e. F ) /\ w e. Word A ) -> M e. Mnd ) |
| 39 |
|
sswrd |
|- ( A C_ B -> Word A C_ Word B ) |
| 40 |
7 39
|
syl |
|- ( ph -> Word A C_ Word B ) |
| 41 |
40
|
adantr |
|- ( ( ph /\ g e. F ) -> Word A C_ Word B ) |
| 42 |
41
|
sselda |
|- ( ( ( ph /\ g e. F ) /\ w e. Word A ) -> w e. Word B ) |
| 43 |
2 1
|
mgpbas |
|- B = ( Base ` M ) |
| 44 |
43
|
gsumwcl |
|- ( ( M e. Mnd /\ w e. Word B ) -> ( M gsum w ) e. B ) |
| 45 |
38 42 44
|
syl2anc |
|- ( ( ( ph /\ g e. F ) /\ w e. Word A ) -> ( M gsum w ) e. B ) |
| 46 |
1 3 26 35 45
|
mulgcld |
|- ( ( ( ph /\ g e. F ) /\ w e. Word A ) -> ( ( g ` w ) .x. ( M gsum w ) ) e. B ) |
| 47 |
46
|
fmpttd |
|- ( ( ph /\ g e. F ) -> ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) : Word A --> B ) |
| 48 |
34
|
feqmptd |
|- ( ( ph /\ g e. F ) -> g = ( w e. Word A |-> ( g ` w ) ) ) |
| 49 |
31
|
simprd |
|- ( g e. F -> g finSupp 0 ) |
| 50 |
49
|
adantl |
|- ( ( ph /\ g e. F ) -> g finSupp 0 ) |
| 51 |
48 50
|
eqbrtrrd |
|- ( ( ph /\ g e. F ) -> ( w e. Word A |-> ( g ` w ) ) finSupp 0 ) |
| 52 |
1 16 3
|
mulg0 |
|- ( y e. B -> ( 0 .x. y ) = ( 0g ` R ) ) |
| 53 |
52
|
adantl |
|- ( ( ( ph /\ g e. F ) /\ y e. B ) -> ( 0 .x. y ) = ( 0g ` R ) ) |
| 54 |
|
fvexd |
|- ( ( ph /\ g e. F ) -> ( 0g ` R ) e. _V ) |
| 55 |
51 53 35 45 54
|
fsuppssov1 |
|- ( ( ph /\ g e. F ) -> ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) finSupp ( 0g ` R ) ) |
| 56 |
1 16 18 24 47 55
|
gsumcl |
|- ( ( ph /\ g e. F ) -> ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) e. B ) |
| 57 |
56
|
adantr |
|- ( ( ( ph /\ g e. F ) /\ X = ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) -> ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) e. B ) |
| 58 |
15 57
|
eqeltrd |
|- ( ( ( ph /\ g e. F ) /\ X = ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) -> X e. B ) |
| 59 |
58
|
r19.29an |
|- ( ( ph /\ E. g e. F X = ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) -> X e. B ) |
| 60 |
6
|
adantr |
|- ( ( ph /\ X e. B ) -> R e. Ring ) |
| 61 |
7
|
adantr |
|- ( ( ph /\ X e. B ) -> A C_ B ) |
| 62 |
|
fveq1 |
|- ( h = i -> ( h ` w ) = ( i ` w ) ) |
| 63 |
62
|
oveq1d |
|- ( h = i -> ( ( h ` w ) .x. ( M gsum w ) ) = ( ( i ` w ) .x. ( M gsum w ) ) ) |
| 64 |
63
|
mpteq2dv |
|- ( h = i -> ( w e. Word A |-> ( ( h ` w ) .x. ( M gsum w ) ) ) = ( w e. Word A |-> ( ( i ` w ) .x. ( M gsum w ) ) ) ) |
| 65 |
|
fveq2 |
|- ( w = v -> ( i ` w ) = ( i ` v ) ) |
| 66 |
|
oveq2 |
|- ( w = v -> ( M gsum w ) = ( M gsum v ) ) |
| 67 |
65 66
|
oveq12d |
|- ( w = v -> ( ( i ` w ) .x. ( M gsum w ) ) = ( ( i ` v ) .x. ( M gsum v ) ) ) |
| 68 |
67
|
cbvmptv |
|- ( w e. Word A |-> ( ( i ` w ) .x. ( M gsum w ) ) ) = ( v e. Word A |-> ( ( i ` v ) .x. ( M gsum v ) ) ) |
| 69 |
64 68
|
eqtrdi |
|- ( h = i -> ( w e. Word A |-> ( ( h ` w ) .x. ( M gsum w ) ) ) = ( v e. Word A |-> ( ( i ` v ) .x. ( M gsum v ) ) ) ) |
| 70 |
69
|
oveq2d |
|- ( h = i -> ( R gsum ( w e. Word A |-> ( ( h ` w ) .x. ( M gsum w ) ) ) ) = ( R gsum ( v e. Word A |-> ( ( i ` v ) .x. ( M gsum v ) ) ) ) ) |
| 71 |
70
|
cbvmptv |
|- ( h e. F |-> ( R gsum ( w e. Word A |-> ( ( h ` w ) .x. ( M gsum w ) ) ) ) ) = ( i e. F |-> ( R gsum ( v e. Word A |-> ( ( i ` v ) .x. ( M gsum v ) ) ) ) ) |
| 72 |
71
|
rneqi |
|- ran ( h e. F |-> ( R gsum ( w e. Word A |-> ( ( h ` w ) .x. ( M gsum w ) ) ) ) ) = ran ( i e. F |-> ( R gsum ( v e. Word A |-> ( ( i ` v ) .x. ( M gsum v ) ) ) ) ) |
| 73 |
1 2 3 4 5 60 61 72
|
elrgspnlem4 |
|- ( ( ph /\ X e. B ) -> ( N ` A ) = ran ( h e. F |-> ( R gsum ( w e. Word A |-> ( ( h ` w ) .x. ( M gsum w ) ) ) ) ) ) |
| 74 |
73
|
eleq2d |
|- ( ( ph /\ X e. B ) -> ( X e. ( N ` A ) <-> X e. ran ( h e. F |-> ( R gsum ( w e. Word A |-> ( ( h ` w ) .x. ( M gsum w ) ) ) ) ) ) ) |
| 75 |
|
fveq1 |
|- ( h = g -> ( h ` w ) = ( g ` w ) ) |
| 76 |
75
|
oveq1d |
|- ( h = g -> ( ( h ` w ) .x. ( M gsum w ) ) = ( ( g ` w ) .x. ( M gsum w ) ) ) |
| 77 |
76
|
mpteq2dv |
|- ( h = g -> ( w e. Word A |-> ( ( h ` w ) .x. ( M gsum w ) ) ) = ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) |
| 78 |
77
|
oveq2d |
|- ( h = g -> ( R gsum ( w e. Word A |-> ( ( h ` w ) .x. ( M gsum w ) ) ) ) = ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) |
| 79 |
78
|
cbvmptv |
|- ( h e. F |-> ( R gsum ( w e. Word A |-> ( ( h ` w ) .x. ( M gsum w ) ) ) ) ) = ( g e. F |-> ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) |
| 80 |
79
|
elrnmpt |
|- ( X e. B -> ( X e. ran ( h e. F |-> ( R gsum ( w e. Word A |-> ( ( h ` w ) .x. ( M gsum w ) ) ) ) ) <-> E. g e. F X = ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) ) |
| 81 |
80
|
adantl |
|- ( ( ph /\ X e. B ) -> ( X e. ran ( h e. F |-> ( R gsum ( w e. Word A |-> ( ( h ` w ) .x. ( M gsum w ) ) ) ) ) <-> E. g e. F X = ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) ) |
| 82 |
74 81
|
bitrd |
|- ( ( ph /\ X e. B ) -> ( X e. ( N ` A ) <-> E. g e. F X = ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) ) |
| 83 |
14 59 82
|
bibiad |
|- ( ph -> ( X e. ( N ` A ) <-> E. g e. F X = ( R gsum ( w e. Word A |-> ( ( g ` w ) .x. ( M gsum w ) ) ) ) ) ) |