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Theorem gsumwspan 14719
Description: The submonoid generated by a set of elements is precisely the set of elements which can be expressed as finite products of the generator. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Hypotheses
Ref Expression
gsumwspan.b  |-  B  =  ( Base `  M
)
gsumwspan.k  |-  K  =  (mrCls `  (SubMnd `  M
) )
Assertion
Ref Expression
gsumwspan  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  =  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) ) )
Distinct variable groups:    w, G    w, B    w, M    w, K

Proof of Theorem gsumwspan
Dummy variables  v  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumwspan.b . . . . . 6  |-  B  =  ( Base `  M
)
21submacs 14693 . . . . 5  |-  ( M  e.  Mnd  ->  (SubMnd `  M )  e.  (ACS
`  B ) )
32acsmred 13809 . . . 4  |-  ( M  e.  Mnd  ->  (SubMnd `  M )  e.  (Moore `  B ) )
43adantr 452 . . 3  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
(SubMnd `  M )  e.  (Moore `  B )
)
5 simpr 448 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  e.  G )
65s1cld 11684 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  <" x ">  e. Word  G )
7 ssel2 3287 . . . . . . . . . 10  |-  ( ( G  C_  B  /\  x  e.  G )  ->  x  e.  B )
87adantll 695 . . . . . . . . 9  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  e.  B )
91gsumws1 14713 . . . . . . . . 9  |-  ( x  e.  B  ->  ( M  gsumg 
<" x "> )  =  x )
108, 9syl 16 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  ( M  gsumg  <" x "> )  =  x )
1110eqcomd 2393 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  =  ( M  gsumg 
<" x "> ) )
12 oveq2 6029 . . . . . . . . 9  |-  ( w  =  <" x ">  ->  ( M  gsumg  w )  =  ( M 
gsumg  <" x "> ) )
1312eqeq2d 2399 . . . . . . . 8  |-  ( w  =  <" x ">  ->  ( x  =  ( M  gsumg  w )  <-> 
x  =  ( M 
gsumg  <" x "> ) ) )
1413rspcev 2996 . . . . . . 7  |-  ( (
<" x ">  e. Word  G  /\  x  =  ( M  gsumg 
<" x "> ) )  ->  E. w  e. Word  G x  =  ( M  gsumg  w ) )
156, 11, 14syl2anc 643 . . . . . 6  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  E. w  e. Word  G x  =  ( M  gsumg  w ) )
16 vex 2903 . . . . . . 7  |-  x  e. 
_V
17 eqid 2388 . . . . . . . 8  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( w  e. Word  G  |->  ( M  gsumg  w ) )
1817elrnmpt 5058 . . . . . . 7  |-  ( x  e.  _V  ->  (
x  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  <->  E. w  e. Word  G x  =  ( M  gsumg  w ) ) )
1916, 18ax-mp 8 . . . . . 6  |-  ( x  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  <->  E. w  e. Word  G x  =  ( M  gsumg  w ) )
2015, 19sylibr 204 . . . . 5  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  e.  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) )
2120ex 424 . . . 4  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( x  e.  G  ->  x  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) ) ) )
2221ssrdv 3298 . . 3  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  G  C_  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) )
23 gsumwspan.k . . . . . . . . . . 11  |-  K  =  (mrCls `  (SubMnd `  M
) )
2423mrccl 13764 . . . . . . . . . 10  |-  ( ( (SubMnd `  M )  e.  (Moore `  B )  /\  G  C_  B )  ->  ( K `  G )  e.  (SubMnd `  M ) )
253, 24sylan 458 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  e.  (SubMnd `  M ) )
2625adantr 452 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  w  e. Word  G )  ->  ( K `  G )  e.  (SubMnd `  M ) )
2723mrcssid 13770 . . . . . . . . . . 11  |-  ( ( (SubMnd `  M )  e.  (Moore `  B )  /\  G  C_  B )  ->  G  C_  ( K `  G )
)
283, 27sylan 458 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  G  C_  ( K `  G ) )
29 sswrd 11665 . . . . . . . . . 10  |-  ( G 
C_  ( K `  G )  -> Word  G  C_ Word  ( K `  G ) )
3028, 29syl 16 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> Word  G 
C_ Word  ( K `  G
) )
3130sselda 3292 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  w  e. Word  G )  ->  w  e. Word  ( K `  G )
)
32 gsumwsubmcl 14712 . . . . . . . 8  |-  ( ( ( K `  G
)  e.  (SubMnd `  M )  /\  w  e. Word  ( K `  G
) )  ->  ( M  gsumg  w )  e.  ( K `  G ) )
3326, 31, 32syl2anc 643 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  w  e. Word  G )  ->  ( M  gsumg  w )  e.  ( K `  G ) )
3433, 17fmptd 5833 . . . . . 6  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( w  e. Word  G  |->  ( M  gsumg  w ) ) :Word 
G --> ( K `  G ) )
35 frn 5538 . . . . . 6  |-  ( ( w  e. Word  G  |->  ( M  gsumg  w ) ) :Word 
G --> ( K `  G )  ->  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  C_  ( K `  G ) )
3634, 35syl 16 . . . . 5  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  C_  ( K `  G ) )
373, 23mrcssvd 13776 . . . . . 6  |-  ( M  e.  Mnd  ->  ( K `  G )  C_  B )
3837adantr 452 . . . . 5  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  C_  B )
3936, 38sstrd 3302 . . . 4  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  C_  B )
40 wrd0 11660 . . . . . 6  |-  (/)  e. Word  G
41 eqid 2388 . . . . . . . . 9  |-  ( 0g
`  M )  =  ( 0g `  M
)
4241gsum0 14708 . . . . . . . 8  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
4342eqcomi 2392 . . . . . . 7  |-  ( 0g
`  M )  =  ( M  gsumg  (/) )
4443a1i 11 . . . . . 6  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( 0g `  M
)  =  ( M 
gsumg  (/) ) )
45 oveq2 6029 . . . . . . . 8  |-  ( w  =  (/)  ->  ( M 
gsumg  w )  =  ( M  gsumg  (/) ) )
4645eqeq2d 2399 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( 0g `  M )  =  ( M  gsumg  w )  <-> 
( 0g `  M
)  =  ( M 
gsumg  (/) ) ) )
4746rspcev 2996 . . . . . 6  |-  ( (
(/)  e. Word  G  /\  ( 0g `  M )  =  ( M  gsumg  (/) ) )  ->  E. w  e. Word  G ( 0g `  M )  =  ( M  gsumg  w ) )
4840, 44, 47sylancr 645 . . . . 5  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  E. w  e. Word  G ( 0g `  M )  =  ( M  gsumg  w ) )
49 fvex 5683 . . . . . 6  |-  ( 0g
`  M )  e. 
_V
5017elrnmpt 5058 . . . . . 6  |-  ( ( 0g `  M )  e.  _V  ->  (
( 0g `  M
)  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  <->  E. w  e. Word  G ( 0g `  M )  =  ( M  gsumg  w ) ) )
5149, 50ax-mp 8 . . . . 5  |-  ( ( 0g `  M )  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  <->  E. w  e. Word  G ( 0g `  M )  =  ( M  gsumg  w ) )
5248, 51sylibr 204 . . . 4  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( 0g `  M
)  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) ) )
53 ccatcl 11671 . . . . . . . . 9  |-  ( ( z  e. Word  G  /\  v  e. Word  G )  ->  ( z concat  v )  e. Word  G )
5453adantl 453 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  ( z concat  v )  e. Word  G )
55 simpll 731 . . . . . . . . . 10  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  M  e.  Mnd )
56 sswrd 11665 . . . . . . . . . . . 12  |-  ( G 
C_  B  -> Word  G  C_ Word  B )
5756ad2antlr 708 . . . . . . . . . . 11  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  -> Word  G  C_ Word  B )
58 simprl 733 . . . . . . . . . . 11  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  z  e. Word  G )
5957, 58sseldd 3293 . . . . . . . . . 10  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  z  e. Word  B )
60 simprr 734 . . . . . . . . . . 11  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  v  e. Word  G )
6157, 60sseldd 3293 . . . . . . . . . 10  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  v  e. Word  B )
62 eqid 2388 . . . . . . . . . . 11  |-  ( +g  `  M )  =  ( +g  `  M )
631, 62gsumccat 14715 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  z  e. Word  B  /\  v  e. Word  B )  ->  ( M  gsumg  ( z concat  v ) )  =  ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) ) )
6455, 59, 61, 63syl3anc 1184 . . . . . . . . 9  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  ( M  gsumg  ( z concat  v ) )  =  ( ( M 
gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) ) )
6564eqcomd 2393 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  =  ( M  gsumg  ( z concat  v ) ) )
66 oveq2 6029 . . . . . . . . . 10  |-  ( w  =  ( z concat  v
)  ->  ( M  gsumg  w )  =  ( M 
gsumg  ( z concat  v )
) )
6766eqeq2d 2399 . . . . . . . . 9  |-  ( w  =  ( z concat  v
)  ->  ( (
( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  =  ( M  gsumg  w )  <->  ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  =  ( M  gsumg  ( z concat  v ) ) ) )
6867rspcev 2996 . . . . . . . 8  |-  ( ( ( z concat  v )  e. Word  G  /\  (
( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  =  ( M  gsumg  ( z concat  v ) ) )  ->  E. w  e. Word  G ( ( M 
gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  =  ( M  gsumg  w ) )
6954, 65, 68syl2anc 643 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  E. w  e. Word  G ( ( M 
gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  =  ( M  gsumg  w ) )
70 ovex 6046 . . . . . . . 8  |-  ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  e. 
_V
7117elrnmpt 5058 . . . . . . . 8  |-  ( ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  e. 
_V  ->  ( ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  <->  E. w  e. Word  G
( ( M  gsumg  z ) ( +g  `  M
) ( M  gsumg  v ) )  =  ( M 
gsumg  w ) ) )
7270, 71ax-mp 8 . . . . . . 7  |-  ( ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  <->  E. w  e. Word  G
( ( M  gsumg  z ) ( +g  `  M
) ( M  gsumg  v ) )  =  ( M 
gsumg  w ) )
7369, 72sylibr 204 . . . . . 6  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) )
7473ralrimivva 2742 . . . . 5  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  A. z  e. Word  G A. v  e. Word  G (
( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) )
75 oveq2 6029 . . . . . . . . 9  |-  ( w  =  z  ->  ( M  gsumg  w )  =  ( M  gsumg  z ) )
7675cbvmptv 4242 . . . . . . . 8  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( z  e. Word  G  |->  ( M  gsumg  z ) )
7776rneqi 5037 . . . . . . 7  |-  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  =  ran  ( z  e. Word  G  |->  ( M  gsumg  z ) )
7877raleqi 2852 . . . . . 6  |-  ( A. x  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) A. y  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) ( x ( +g  `  M
) y )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  <->  A. x  e.  ran  ( z  e. Word  G  |->  ( M  gsumg  z ) ) A. y  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) ( x ( +g  `  M
) y )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) )
79 oveq2 6029 . . . . . . . . . . 11  |-  ( w  =  v  ->  ( M  gsumg  w )  =  ( M  gsumg  v ) )
8079cbvmptv 4242 . . . . . . . . . 10  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( v  e. Word  G  |->  ( M  gsumg  v ) )
8180rneqi 5037 . . . . . . . . 9  |-  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  =  ran  ( v  e. Word  G  |->  ( M  gsumg  v ) )
8281raleqi 2852 . . . . . . . 8  |-  ( A. y  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) ( x ( +g  `  M
) y )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  <->  A. y  e.  ran  ( v  e. Word  G  |->  ( M  gsumg  v ) ) ( x ( +g  `  M
) y )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) )
83 eqid 2388 . . . . . . . . . 10  |-  ( v  e. Word  G  |->  ( M 
gsumg  v ) )  =  ( v  e. Word  G  |->  ( M  gsumg  v ) )
84 oveq2 6029 . . . . . . . . . . 11  |-  ( y  =  ( M  gsumg  v )  ->  ( x ( +g  `  M ) y )  =  ( x ( +g  `  M
) ( M  gsumg  v ) ) )
8584eleq1d 2454 . . . . . . . . . 10  |-  ( y  =  ( M  gsumg  v )  ->  ( ( x ( +g  `  M
) y )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  <->  ( x ( +g  `  M ) ( M  gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) ) )
8683, 85ralrnmpt 5818 . . . . . . . . 9  |-  ( A. v  e. Word  G ( M  gsumg  v )  e.  _V  ->  ( A. y  e. 
ran  ( v  e. Word  G  |->  ( M  gsumg  v ) ) ( x ( +g  `  M ) y )  e.  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  <->  A. v  e. Word  G ( x ( +g  `  M ) ( M  gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) ) )
87 ovex 6046 . . . . . . . . . 10  |-  ( M 
gsumg  v )  e.  _V
8887a1i 11 . . . . . . . . 9  |-  ( v  e. Word  G  ->  ( M  gsumg  v )  e.  _V )
8986, 88mprg 2719 . . . . . . . 8  |-  ( A. y  e.  ran  ( v  e. Word  G  |->  ( M 
gsumg  v ) ) ( x ( +g  `  M
) y )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  <->  A. v  e. Word  G
( x ( +g  `  M ) ( M 
gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) )
9082, 89bitri 241 . . . . . . 7  |-  ( A. y  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) ( x ( +g  `  M
) y )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  <->  A. v  e. Word  G
( x ( +g  `  M ) ( M 
gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) )
9190ralbii 2674 . . . . . 6  |-  ( A. x  e.  ran  ( z  e. Word  G  |->  ( M 
gsumg  z ) ) A. y  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) ( x ( +g  `  M
) y )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  <->  A. x  e.  ran  ( z  e. Word  G  |->  ( M  gsumg  z ) ) A. v  e. Word  G (
x ( +g  `  M
) ( M  gsumg  v ) )  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) ) )
92 eqid 2388 . . . . . . . 8  |-  ( z  e. Word  G  |->  ( M 
gsumg  z ) )  =  ( z  e. Word  G  |->  ( M  gsumg  z ) )
93 oveq1 6028 . . . . . . . . . 10  |-  ( x  =  ( M  gsumg  z )  ->  ( x ( +g  `  M ) ( M  gsumg  v ) )  =  ( ( M  gsumg  z ) ( +g  `  M
) ( M  gsumg  v ) ) )
9493eleq1d 2454 . . . . . . . . 9  |-  ( x  =  ( M  gsumg  z )  ->  ( ( x ( +g  `  M
) ( M  gsumg  v ) )  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  <->  ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) ) )
9594ralbidv 2670 . . . . . . . 8  |-  ( x  =  ( M  gsumg  z )  ->  ( A. v  e. Word  G ( x ( +g  `  M ) ( M  gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  <->  A. v  e. Word  G
( ( M  gsumg  z ) ( +g  `  M
) ( M  gsumg  v ) )  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) ) ) )
9692, 95ralrnmpt 5818 . . . . . . 7  |-  ( A. z  e. Word  G ( M  gsumg  z )  e.  _V  ->  ( A. x  e. 
ran  ( z  e. Word  G  |->  ( M  gsumg  z ) ) A. v  e. Word  G ( x ( +g  `  M ) ( M  gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  <->  A. z  e. Word  G A. v  e. Word  G ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) ) )
97 ovex 6046 . . . . . . . 8  |-  ( M 
gsumg  z )  e.  _V
9897a1i 11 . . . . . . 7  |-  ( z  e. Word  G  ->  ( M  gsumg  z )  e.  _V )
9996, 98mprg 2719 . . . . . 6  |-  ( A. x  e.  ran  ( z  e. Word  G  |->  ( M 
gsumg  z ) ) A. v  e. Word  G (
x ( +g  `  M
) ( M  gsumg  v ) )  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  <->  A. z  e. Word  G A. v  e. Word  G ( ( M 
gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) )
10078, 91, 993bitri 263 . . . . 5  |-  ( A. x  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) A. y  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) ( x ( +g  `  M
) y )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  <->  A. z  e. Word  G A. v  e. Word  G ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) )
10174, 100sylibr 204 . . . 4  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  A. x  e.  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) A. y  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) ( x ( +g  `  M
) y )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) )
1021, 41, 62issubm 14676 . . . . 5  |-  ( M  e.  Mnd  ->  ( ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  e.  (SubMnd `  M )  <->  ( ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  C_  B  /\  ( 0g `  M )  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  /\  A. x  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) ) A. y  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) ( x ( +g  `  M
) y )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) ) ) )
103102adantr 452 . . . 4  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  e.  (SubMnd `  M )  <->  ( ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  C_  B  /\  ( 0g `  M )  e.  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  /\  A. x  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) ) A. y  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) ( x ( +g  `  M
) y )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) ) ) )
10439, 52, 101, 103mpbir3and 1137 . . 3  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  e.  (SubMnd `  M )
)
10523mrcsscl 13773 . . 3  |-  ( ( (SubMnd `  M )  e.  (Moore `  B )  /\  G  C_  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  /\  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  e.  (SubMnd `  M )
)  ->  ( K `  G )  C_  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) )
1064, 22, 104, 105syl3anc 1184 . 2  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  C_  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) )
107106, 36eqssd 3309 1  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  =  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650   E.wrex 2651   _Vcvv 2900    C_ wss 3264   (/)c0 3572    e. cmpt 4208   ran crn 4820   -->wf 5391   ` cfv 5395  (class class class)co 6021  Word cword 11645   concat cconcat 11646   <"cs1 11647   Basecbs 13397   +g cplusg 13457   0gc0g 13651    gsumg cgsu 13652  Moorecmre 13735  mrClscmrc 13736   Mndcmnd 14612  SubMndcsubmnd 14665
This theorem is referenced by:  psgneldm2  27097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-fzo 11067  df-seq 11252  df-hash 11547  df-word 11651  df-concat 11652  df-s1 11653  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-0g 13655  df-gsum 13656  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667
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