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Theorem gsumwspan 15885
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 15867 . . . . 5  |-  ( M  e.  Mnd  ->  (SubMnd `  M )  e.  (ACS
`  B ) )
32acsmred 14927 . . . 4  |-  ( M  e.  Mnd  ->  (SubMnd `  M )  e.  (Moore `  B ) )
43adantr 465 . . 3  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
(SubMnd `  M )  e.  (Moore `  B )
)
5 simpr 461 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  e.  G )
65s1cld 12594 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  <" x ">  e. Word  G )
7 ssel2 3504 . . . . . . . . . 10  |-  ( ( G  C_  B  /\  x  e.  G )  ->  x  e.  B )
87adantll 713 . . . . . . . . 9  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  e.  B )
91gsumws1 15878 . . . . . . . . 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 2475 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  =  ( M  gsumg 
<" x "> ) )
12 oveq2 6303 . . . . . . . . 9  |-  ( w  =  <" x ">  ->  ( M  gsumg  w )  =  ( M 
gsumg  <" x "> ) )
1312eqeq2d 2481 . . . . . . . 8  |-  ( w  =  <" x ">  ->  ( x  =  ( M  gsumg  w )  <-> 
x  =  ( M 
gsumg  <" x "> ) ) )
1413rspcev 3219 . . . . . . 7  |-  ( (
<" x ">  e. Word  G  /\  x  =  ( M  gsumg 
<" x "> ) )  ->  E. w  e. Word  G x  =  ( M  gsumg  w ) )
156, 11, 14syl2anc 661 . . . . . 6  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  E. w  e. Word  G x  =  ( M  gsumg  w ) )
16 vex 3121 . . . . . . 7  |-  x  e. 
_V
17 eqid 2467 . . . . . . . 8  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( w  e. Word  G  |->  ( M  gsumg  w ) )
1817elrnmpt 5255 . . . . . . 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 5 . . . . . 6  |-  ( x  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  <->  E. w  e. Word  G x  =  ( M  gsumg  w ) )
2015, 19sylibr 212 . . . . 5  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  e.  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) )
2120ex 434 . . . 4  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( x  e.  G  ->  x  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) ) ) )
2221ssrdv 3515 . . 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 14882 . . . . . . . . . 10  |-  ( ( (SubMnd `  M )  e.  (Moore `  B )  /\  G  C_  B )  ->  ( K `  G )  e.  (SubMnd `  M ) )
253, 24sylan 471 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  e.  (SubMnd `  M ) )
2625adantr 465 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  w  e. Word  G )  ->  ( K `  G )  e.  (SubMnd `  M ) )
2723mrcssid 14888 . . . . . . . . . . 11  |-  ( ( (SubMnd `  M )  e.  (Moore `  B )  /\  G  C_  B )  ->  G  C_  ( K `  G )
)
283, 27sylan 471 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  G  C_  ( K `  G ) )
29 sswrd 12535 . . . . . . . . . 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 3509 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  w  e. Word  G )  ->  w  e. Word  ( K `  G )
)
32 gsumwsubmcl 15877 . . . . . . . 8  |-  ( ( ( K `  G
)  e.  (SubMnd `  M )  /\  w  e. Word  ( K `  G
) )  ->  ( M  gsumg  w )  e.  ( K `  G ) )
3326, 31, 32syl2anc 661 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  w  e. Word  G )  ->  ( M  gsumg  w )  e.  ( K `  G ) )
3433, 17fmptd 6056 . . . . . 6  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( w  e. Word  G  |->  ( M  gsumg  w ) ) :Word 
G --> ( K `  G ) )
35 frn 5743 . . . . . 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 14894 . . . . . 6  |-  ( M  e.  Mnd  ->  ( K `  G )  C_  B )
3837adantr 465 . . . . 5  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  C_  B )
3936, 38sstrd 3519 . . . 4  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  C_  B )
40 wrd0 12545 . . . . . 6  |-  (/)  e. Word  G
41 eqid 2467 . . . . . . . . 9  |-  ( 0g
`  M )  =  ( 0g `  M
)
4241gsum0 15778 . . . . . . . 8  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
4342eqcomi 2480 . . . . . . 7  |-  ( 0g
`  M )  =  ( M  gsumg  (/) )
4443a1i 11 . . . . . 6  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( 0g `  M
)  =  ( M 
gsumg  (/) ) )
45 oveq2 6303 . . . . . . . 8  |-  ( w  =  (/)  ->  ( M 
gsumg  w )  =  ( M  gsumg  (/) ) )
4645eqeq2d 2481 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( 0g `  M )  =  ( M  gsumg  w )  <-> 
( 0g `  M
)  =  ( M 
gsumg  (/) ) ) )
4746rspcev 3219 . . . . . 6  |-  ( (
(/)  e. Word  G  /\  ( 0g `  M )  =  ( M  gsumg  (/) ) )  ->  E. w  e. Word  G ( 0g `  M )  =  ( M  gsumg  w ) )
4840, 44, 47sylancr 663 . . . . 5  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  E. w  e. Word  G ( 0g `  M )  =  ( M  gsumg  w ) )
49 fvex 5882 . . . . . 6  |-  ( 0g
`  M )  e. 
_V
5017elrnmpt 5255 . . . . . 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 5 . . . . 5  |-  ( ( 0g `  M )  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  <->  E. w  e. Word  G ( 0g `  M )  =  ( M  gsumg  w ) )
5248, 51sylibr 212 . . . 4  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( 0g `  M
)  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) ) )
53 ccatcl 12572 . . . . . . . . 9  |-  ( ( z  e. Word  G  /\  v  e. Word  G )  ->  ( z concat  v )  e. Word  G )
5453adantl 466 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  ( z concat  v )  e. Word  G )
55 simpll 753 . . . . . . . . . 10  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  M  e.  Mnd )
56 sswrd 12535 . . . . . . . . . . . 12  |-  ( G 
C_  B  -> Word  G  C_ Word  B )
5756ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  -> Word  G  C_ Word  B )
58 simprl 755 . . . . . . . . . . 11  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  z  e. Word  G )
5957, 58sseldd 3510 . . . . . . . . . 10  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  z  e. Word  B )
60 simprr 756 . . . . . . . . . . 11  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  v  e. Word  G )
6157, 60sseldd 3510 . . . . . . . . . 10  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  v  e. Word  B )
62 eqid 2467 . . . . . . . . . . 11  |-  ( +g  `  M )  =  ( +g  `  M )
631, 62gsumccat 15880 . . . . . . . . . 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 1228 . . . . . . . . 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 2475 . . . . . . . 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 6303 . . . . . . . . . 10  |-  ( w  =  ( z concat  v
)  ->  ( M  gsumg  w )  =  ( M 
gsumg  ( z concat  v )
) )
6766eqeq2d 2481 . . . . . . . . 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 3219 . . . . . . . 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 661 . . . . . . 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 6320 . . . . . . . 8  |-  ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  e. 
_V
7117elrnmpt 5255 . . . . . . . 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 5 . . . . . . 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 212 . . . . . 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 2888 . . . . 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 6303 . . . . . . . . 9  |-  ( w  =  z  ->  ( M  gsumg  w )  =  ( M  gsumg  z ) )
7675cbvmptv 4544 . . . . . . . 8  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( z  e. Word  G  |->  ( M  gsumg  z ) )
7776rneqi 5235 . . . . . . 7  |-  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  =  ran  ( z  e. Word  G  |->  ( M  gsumg  z ) )
7877raleqi 3067 . . . . . 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 6303 . . . . . . . . . . 11  |-  ( w  =  v  ->  ( M  gsumg  w )  =  ( M  gsumg  v ) )
8079cbvmptv 4544 . . . . . . . . . 10  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( v  e. Word  G  |->  ( M  gsumg  v ) )
8180rneqi 5235 . . . . . . . . 9  |-  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  =  ran  ( v  e. Word  G  |->  ( M  gsumg  v ) )
8281raleqi 3067 . . . . . . . 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 2467 . . . . . . . . . 10  |-  ( v  e. Word  G  |->  ( M 
gsumg  v ) )  =  ( v  e. Word  G  |->  ( M  gsumg  v ) )
84 oveq2 6303 . . . . . . . . . . 11  |-  ( y  =  ( M  gsumg  v )  ->  ( x ( +g  `  M ) y )  =  ( x ( +g  `  M
) ( M  gsumg  v ) ) )
8584eleq1d 2536 . . . . . . . . . 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 6041 . . . . . . . . 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 6320 . . . . . . . . . 10  |-  ( M 
gsumg  v )  e.  _V
8887a1i 11 . . . . . . . . 9  |-  ( v  e. Word  G  ->  ( M  gsumg  v )  e.  _V )
8986, 88mprg 2830 . . . . . . . 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 249 . . . . . . 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 2898 . . . . . 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 2467 . . . . . . . 8  |-  ( z  e. Word  G  |->  ( M 
gsumg  z ) )  =  ( z  e. Word  G  |->  ( M  gsumg  z ) )
93 oveq1 6302 . . . . . . . . . 10  |-  ( x  =  ( M  gsumg  z )  ->  ( x ( +g  `  M ) ( M  gsumg  v ) )  =  ( ( M  gsumg  z ) ( +g  `  M
) ( M  gsumg  v ) ) )
9493eleq1d 2536 . . . . . . . . 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 2906 . . . . . . . 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 6041 . . . . . . 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 6320 . . . . . . . 8  |-  ( M 
gsumg  z )  e.  _V
9897a1i 11 . . . . . . 7  |-  ( z  e. Word  G  ->  ( M  gsumg  z )  e.  _V )
9996, 98mprg 2830 . . . . . 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 271 . . . . 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 212 . . . 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 15849 . . . . 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 465 . . . 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 1179 . . 3  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  e.  (SubMnd `  M )
)
10523mrcsscl 14891 . . 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 1228 . 2  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  C_  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) )
107106, 36eqssd 3526 1  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  =  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   _Vcvv 3118    C_ wss 3481   (/)c0 3790    |-> cmpt 4511   ran crn 5006   -->wf 5590   ` cfv 5594  (class class class)co 6295  Word cword 12514   concat cconcat 12516   <"cs1 12517   Basecbs 14506   +g cplusg 14571   0gc0g 14711    gsumg cgsu 14712  Moorecmre 14853  mrClscmrc 14854   Mndcmnd 15792  SubMndcsubmnd 15837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-word 12522  df-concat 12524  df-s1 12525  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-0g 14713  df-gsum 14714  df-mre 14857  df-mrc 14858  df-acs 14860  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-submnd 15839
This theorem is referenced by:  psgneldm2  16400  psgnfitr  16413
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