Step |
Hyp |
Ref |
Expression |
1 |
|
gsumwspan.b |
|- B = ( Base ` M ) |
2 |
|
gsumwspan.k |
|- K = ( mrCls ` ( SubMnd ` M ) ) |
3 |
1
|
submacs |
|- ( M e. Mnd -> ( SubMnd ` M ) e. ( ACS ` B ) ) |
4 |
3
|
acsmred |
|- ( M e. Mnd -> ( SubMnd ` M ) e. ( Moore ` B ) ) |
5 |
4
|
adantr |
|- ( ( M e. Mnd /\ G C_ B ) -> ( SubMnd ` M ) e. ( Moore ` B ) ) |
6 |
|
simpr |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ x e. G ) -> x e. G ) |
7 |
6
|
s1cld |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ x e. G ) -> <" x "> e. Word G ) |
8 |
|
ssel2 |
|- ( ( G C_ B /\ x e. G ) -> x e. B ) |
9 |
8
|
adantll |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ x e. G ) -> x e. B ) |
10 |
1
|
gsumws1 |
|- ( x e. B -> ( M gsum <" x "> ) = x ) |
11 |
9 10
|
syl |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ x e. G ) -> ( M gsum <" x "> ) = x ) |
12 |
11
|
eqcomd |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ x e. G ) -> x = ( M gsum <" x "> ) ) |
13 |
|
oveq2 |
|- ( w = <" x "> -> ( M gsum w ) = ( M gsum <" x "> ) ) |
14 |
13
|
rspceeqv |
|- ( ( <" x "> e. Word G /\ x = ( M gsum <" x "> ) ) -> E. w e. Word G x = ( M gsum w ) ) |
15 |
7 12 14
|
syl2anc |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ x e. G ) -> E. w e. Word G x = ( M gsum w ) ) |
16 |
|
eqid |
|- ( w e. Word G |-> ( M gsum w ) ) = ( w e. Word G |-> ( M gsum w ) ) |
17 |
16
|
elrnmpt |
|- ( x e. _V -> ( x e. ran ( w e. Word G |-> ( M gsum w ) ) <-> E. w e. Word G x = ( M gsum w ) ) ) |
18 |
17
|
elv |
|- ( x e. ran ( w e. Word G |-> ( M gsum w ) ) <-> E. w e. Word G x = ( M gsum w ) ) |
19 |
15 18
|
sylibr |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ x e. G ) -> x e. ran ( w e. Word G |-> ( M gsum w ) ) ) |
20 |
19
|
ex |
|- ( ( M e. Mnd /\ G C_ B ) -> ( x e. G -> x e. ran ( w e. Word G |-> ( M gsum w ) ) ) ) |
21 |
20
|
ssrdv |
|- ( ( M e. Mnd /\ G C_ B ) -> G C_ ran ( w e. Word G |-> ( M gsum w ) ) ) |
22 |
2
|
mrccl |
|- ( ( ( SubMnd ` M ) e. ( Moore ` B ) /\ G C_ B ) -> ( K ` G ) e. ( SubMnd ` M ) ) |
23 |
4 22
|
sylan |
|- ( ( M e. Mnd /\ G C_ B ) -> ( K ` G ) e. ( SubMnd ` M ) ) |
24 |
2
|
mrcssid |
|- ( ( ( SubMnd ` M ) e. ( Moore ` B ) /\ G C_ B ) -> G C_ ( K ` G ) ) |
25 |
4 24
|
sylan |
|- ( ( M e. Mnd /\ G C_ B ) -> G C_ ( K ` G ) ) |
26 |
|
sswrd |
|- ( G C_ ( K ` G ) -> Word G C_ Word ( K ` G ) ) |
27 |
25 26
|
syl |
|- ( ( M e. Mnd /\ G C_ B ) -> Word G C_ Word ( K ` G ) ) |
28 |
27
|
sselda |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ w e. Word G ) -> w e. Word ( K ` G ) ) |
29 |
|
gsumwsubmcl |
|- ( ( ( K ` G ) e. ( SubMnd ` M ) /\ w e. Word ( K ` G ) ) -> ( M gsum w ) e. ( K ` G ) ) |
30 |
23 28 29
|
syl2an2r |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ w e. Word G ) -> ( M gsum w ) e. ( K ` G ) ) |
31 |
30
|
fmpttd |
|- ( ( M e. Mnd /\ G C_ B ) -> ( w e. Word G |-> ( M gsum w ) ) : Word G --> ( K ` G ) ) |
32 |
31
|
frnd |
|- ( ( M e. Mnd /\ G C_ B ) -> ran ( w e. Word G |-> ( M gsum w ) ) C_ ( K ` G ) ) |
33 |
4 2
|
mrcssvd |
|- ( M e. Mnd -> ( K ` G ) C_ B ) |
34 |
33
|
adantr |
|- ( ( M e. Mnd /\ G C_ B ) -> ( K ` G ) C_ B ) |
35 |
32 34
|
sstrd |
|- ( ( M e. Mnd /\ G C_ B ) -> ran ( w e. Word G |-> ( M gsum w ) ) C_ B ) |
36 |
|
wrd0 |
|- (/) e. Word G |
37 |
|
eqid |
|- ( 0g ` M ) = ( 0g ` M ) |
38 |
37
|
gsum0 |
|- ( M gsum (/) ) = ( 0g ` M ) |
39 |
38
|
eqcomi |
|- ( 0g ` M ) = ( M gsum (/) ) |
40 |
39
|
a1i |
|- ( ( M e. Mnd /\ G C_ B ) -> ( 0g ` M ) = ( M gsum (/) ) ) |
41 |
|
oveq2 |
|- ( w = (/) -> ( M gsum w ) = ( M gsum (/) ) ) |
42 |
41
|
rspceeqv |
|- ( ( (/) e. Word G /\ ( 0g ` M ) = ( M gsum (/) ) ) -> E. w e. Word G ( 0g ` M ) = ( M gsum w ) ) |
43 |
36 40 42
|
sylancr |
|- ( ( M e. Mnd /\ G C_ B ) -> E. w e. Word G ( 0g ` M ) = ( M gsum w ) ) |
44 |
|
fvex |
|- ( 0g ` M ) e. _V |
45 |
16
|
elrnmpt |
|- ( ( 0g ` M ) e. _V -> ( ( 0g ` M ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> E. w e. Word G ( 0g ` M ) = ( M gsum w ) ) ) |
46 |
44 45
|
ax-mp |
|- ( ( 0g ` M ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> E. w e. Word G ( 0g ` M ) = ( M gsum w ) ) |
47 |
43 46
|
sylibr |
|- ( ( M e. Mnd /\ G C_ B ) -> ( 0g ` M ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) |
48 |
|
ccatcl |
|- ( ( z e. Word G /\ v e. Word G ) -> ( z ++ v ) e. Word G ) |
49 |
|
simpll |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> M e. Mnd ) |
50 |
|
sswrd |
|- ( G C_ B -> Word G C_ Word B ) |
51 |
50
|
ad2antlr |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> Word G C_ Word B ) |
52 |
|
simprl |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> z e. Word G ) |
53 |
51 52
|
sseldd |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> z e. Word B ) |
54 |
|
simprr |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> v e. Word G ) |
55 |
51 54
|
sseldd |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> v e. Word B ) |
56 |
|
eqid |
|- ( +g ` M ) = ( +g ` M ) |
57 |
1 56
|
gsumccat |
|- ( ( M e. Mnd /\ z e. Word B /\ v e. Word B ) -> ( M gsum ( z ++ v ) ) = ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) ) |
58 |
49 53 55 57
|
syl3anc |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> ( M gsum ( z ++ v ) ) = ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) ) |
59 |
58
|
eqcomd |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) = ( M gsum ( z ++ v ) ) ) |
60 |
|
oveq2 |
|- ( w = ( z ++ v ) -> ( M gsum w ) = ( M gsum ( z ++ v ) ) ) |
61 |
60
|
rspceeqv |
|- ( ( ( z ++ v ) e. Word G /\ ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) = ( M gsum ( z ++ v ) ) ) -> E. w e. Word G ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) = ( M gsum w ) ) |
62 |
48 59 61
|
syl2an2 |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> E. w e. Word G ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) = ( M gsum w ) ) |
63 |
|
ovex |
|- ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) e. _V |
64 |
16
|
elrnmpt |
|- ( ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) e. _V -> ( ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> E. w e. Word G ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) = ( M gsum w ) ) ) |
65 |
63 64
|
ax-mp |
|- ( ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> E. w e. Word G ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) = ( M gsum w ) ) |
66 |
62 65
|
sylibr |
|- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) |
67 |
66
|
ralrimivva |
|- ( ( M e. Mnd /\ G C_ B ) -> A. z e. Word G A. v e. Word G ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) |
68 |
|
oveq2 |
|- ( w = z -> ( M gsum w ) = ( M gsum z ) ) |
69 |
68
|
cbvmptv |
|- ( w e. Word G |-> ( M gsum w ) ) = ( z e. Word G |-> ( M gsum z ) ) |
70 |
69
|
rneqi |
|- ran ( w e. Word G |-> ( M gsum w ) ) = ran ( z e. Word G |-> ( M gsum z ) ) |
71 |
70
|
raleqi |
|- ( A. x e. ran ( w e. Word G |-> ( M gsum w ) ) A. y e. ran ( w e. Word G |-> ( M gsum w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> A. x e. ran ( z e. Word G |-> ( M gsum z ) ) A. y e. ran ( w e. Word G |-> ( M gsum w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) |
72 |
|
oveq2 |
|- ( w = v -> ( M gsum w ) = ( M gsum v ) ) |
73 |
72
|
cbvmptv |
|- ( w e. Word G |-> ( M gsum w ) ) = ( v e. Word G |-> ( M gsum v ) ) |
74 |
73
|
rneqi |
|- ran ( w e. Word G |-> ( M gsum w ) ) = ran ( v e. Word G |-> ( M gsum v ) ) |
75 |
74
|
raleqi |
|- ( A. y e. ran ( w e. Word G |-> ( M gsum w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> A. y e. ran ( v e. Word G |-> ( M gsum v ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) |
76 |
|
eqid |
|- ( v e. Word G |-> ( M gsum v ) ) = ( v e. Word G |-> ( M gsum v ) ) |
77 |
|
oveq2 |
|- ( y = ( M gsum v ) -> ( x ( +g ` M ) y ) = ( x ( +g ` M ) ( M gsum v ) ) ) |
78 |
77
|
eleq1d |
|- ( y = ( M gsum v ) -> ( ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> ( x ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) ) |
79 |
76 78
|
ralrnmptw |
|- ( A. v e. Word G ( M gsum v ) e. _V -> ( A. y e. ran ( v e. Word G |-> ( M gsum v ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> A. v e. Word G ( x ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) ) |
80 |
|
ovexd |
|- ( v e. Word G -> ( M gsum v ) e. _V ) |
81 |
79 80
|
mprg |
|- ( A. y e. ran ( v e. Word G |-> ( M gsum v ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> A. v e. Word G ( x ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) |
82 |
75 81
|
bitri |
|- ( A. y e. ran ( w e. Word G |-> ( M gsum w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> A. v e. Word G ( x ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) |
83 |
82
|
ralbii |
|- ( A. x e. ran ( z e. Word G |-> ( M gsum z ) ) A. y e. ran ( w e. Word G |-> ( M gsum w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> A. x e. ran ( z e. Word G |-> ( M gsum z ) ) A. v e. Word G ( x ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) |
84 |
|
eqid |
|- ( z e. Word G |-> ( M gsum z ) ) = ( z e. Word G |-> ( M gsum z ) ) |
85 |
|
oveq1 |
|- ( x = ( M gsum z ) -> ( x ( +g ` M ) ( M gsum v ) ) = ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) ) |
86 |
85
|
eleq1d |
|- ( x = ( M gsum z ) -> ( ( x ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) ) |
87 |
86
|
ralbidv |
|- ( x = ( M gsum z ) -> ( A. v e. Word G ( x ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> A. v e. Word G ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) ) |
88 |
84 87
|
ralrnmptw |
|- ( A. z e. Word G ( M gsum z ) e. _V -> ( A. x e. ran ( z e. Word G |-> ( M gsum z ) ) A. v e. Word G ( x ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> A. z e. Word G A. v e. Word G ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) ) |
89 |
|
ovexd |
|- ( z e. Word G -> ( M gsum z ) e. _V ) |
90 |
88 89
|
mprg |
|- ( A. x e. ran ( z e. Word G |-> ( M gsum z ) ) A. v e. Word G ( x ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> A. z e. Word G A. v e. Word G ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) |
91 |
71 83 90
|
3bitri |
|- ( A. x e. ran ( w e. Word G |-> ( M gsum w ) ) A. y e. ran ( w e. Word G |-> ( M gsum w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsum w ) ) <-> A. z e. Word G A. v e. Word G ( ( M gsum z ) ( +g ` M ) ( M gsum v ) ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) |
92 |
67 91
|
sylibr |
|- ( ( M e. Mnd /\ G C_ B ) -> A. x e. ran ( w e. Word G |-> ( M gsum w ) ) A. y e. ran ( w e. Word G |-> ( M gsum w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) |
93 |
1 37 56
|
issubm |
|- ( M e. Mnd -> ( ran ( w e. Word G |-> ( M gsum w ) ) e. ( SubMnd ` M ) <-> ( ran ( w e. Word G |-> ( M gsum w ) ) C_ B /\ ( 0g ` M ) e. ran ( w e. Word G |-> ( M gsum w ) ) /\ A. x e. ran ( w e. Word G |-> ( M gsum w ) ) A. y e. ran ( w e. Word G |-> ( M gsum w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) ) ) |
94 |
93
|
adantr |
|- ( ( M e. Mnd /\ G C_ B ) -> ( ran ( w e. Word G |-> ( M gsum w ) ) e. ( SubMnd ` M ) <-> ( ran ( w e. Word G |-> ( M gsum w ) ) C_ B /\ ( 0g ` M ) e. ran ( w e. Word G |-> ( M gsum w ) ) /\ A. x e. ran ( w e. Word G |-> ( M gsum w ) ) A. y e. ran ( w e. Word G |-> ( M gsum w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsum w ) ) ) ) ) |
95 |
35 47 92 94
|
mpbir3and |
|- ( ( M e. Mnd /\ G C_ B ) -> ran ( w e. Word G |-> ( M gsum w ) ) e. ( SubMnd ` M ) ) |
96 |
2
|
mrcsscl |
|- ( ( ( SubMnd ` M ) e. ( Moore ` B ) /\ G C_ ran ( w e. Word G |-> ( M gsum w ) ) /\ ran ( w e. Word G |-> ( M gsum w ) ) e. ( SubMnd ` M ) ) -> ( K ` G ) C_ ran ( w e. Word G |-> ( M gsum w ) ) ) |
97 |
5 21 95 96
|
syl3anc |
|- ( ( M e. Mnd /\ G C_ B ) -> ( K ` G ) C_ ran ( w e. Word G |-> ( M gsum w ) ) ) |
98 |
97 32
|
eqssd |
|- ( ( M e. Mnd /\ G C_ B ) -> ( K ` G ) = ran ( w e. Word G |-> ( M gsum w ) ) ) |