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Theorem gsumwspan 15517
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 15488 . . . . 5  |-  ( M  e.  Mnd  ->  (SubMnd `  M )  e.  (ACS
`  B ) )
32acsmred 14590 . . . 4  |-  ( M  e.  Mnd  ->  (SubMnd `  M )  e.  (Moore `  B ) )
43adantr 462 . . 3  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
(SubMnd `  M )  e.  (Moore `  B )
)
5 simpr 458 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  e.  G )
65s1cld 12290 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  <" x ">  e. Word  G )
7 ssel2 3348 . . . . . . . . . 10  |-  ( ( G  C_  B  /\  x  e.  G )  ->  x  e.  B )
87adantll 708 . . . . . . . . 9  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  e.  B )
91gsumws1 15510 . . . . . . . . 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 2446 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  =  ( M  gsumg 
<" x "> ) )
12 oveq2 6098 . . . . . . . . 9  |-  ( w  =  <" x ">  ->  ( M  gsumg  w )  =  ( M 
gsumg  <" x "> ) )
1312eqeq2d 2452 . . . . . . . 8  |-  ( w  =  <" x ">  ->  ( x  =  ( M  gsumg  w )  <-> 
x  =  ( M 
gsumg  <" x "> ) ) )
1413rspcev 3070 . . . . . . 7  |-  ( (
<" x ">  e. Word  G  /\  x  =  ( M  gsumg 
<" x "> ) )  ->  E. w  e. Word  G x  =  ( M  gsumg  w ) )
156, 11, 14syl2anc 656 . . . . . 6  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  E. w  e. Word  G x  =  ( M  gsumg  w ) )
16 vex 2973 . . . . . . 7  |-  x  e. 
_V
17 eqid 2441 . . . . . . . 8  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( w  e. Word  G  |->  ( M  gsumg  w ) )
1817elrnmpt 5082 . . . . . . 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 3359 . . 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 14545 . . . . . . . . . 10  |-  ( ( (SubMnd `  M )  e.  (Moore `  B )  /\  G  C_  B )  ->  ( K `  G )  e.  (SubMnd `  M ) )
253, 24sylan 468 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  e.  (SubMnd `  M ) )
2625adantr 462 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  w  e. Word  G )  ->  ( K `  G )  e.  (SubMnd `  M ) )
2723mrcssid 14551 . . . . . . . . . . 11  |-  ( ( (SubMnd `  M )  e.  (Moore `  B )  /\  G  C_  B )  ->  G  C_  ( K `  G )
)
283, 27sylan 468 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  G  C_  ( K `  G ) )
29 sswrd 12238 . . . . . . . . . 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 3353 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  w  e. Word  G )  ->  w  e. Word  ( K `  G )
)
32 gsumwsubmcl 15509 . . . . . . . 8  |-  ( ( ( K `  G
)  e.  (SubMnd `  M )  /\  w  e. Word  ( K `  G
) )  ->  ( M  gsumg  w )  e.  ( K `  G ) )
3326, 31, 32syl2anc 656 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  w  e. Word  G )  ->  ( M  gsumg  w )  e.  ( K `  G ) )
3433, 17fmptd 5864 . . . . . 6  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( w  e. Word  G  |->  ( M  gsumg  w ) ) :Word 
G --> ( K `  G ) )
35 frn 5562 . . . . . 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 14557 . . . . . 6  |-  ( M  e.  Mnd  ->  ( K `  G )  C_  B )
3837adantr 462 . . . . 5  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  C_  B )
3936, 38sstrd 3363 . . . 4  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  C_  B )
40 wrd0 12248 . . . . . 6  |-  (/)  e. Word  G
41 eqid 2441 . . . . . . . . 9  |-  ( 0g
`  M )  =  ( 0g `  M
)
4241gsum0 15503 . . . . . . . 8  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
4342eqcomi 2445 . . . . . . 7  |-  ( 0g
`  M )  =  ( M  gsumg  (/) )
4443a1i 11 . . . . . 6  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( 0g `  M
)  =  ( M 
gsumg  (/) ) )
45 oveq2 6098 . . . . . . . 8  |-  ( w  =  (/)  ->  ( M 
gsumg  w )  =  ( M  gsumg  (/) ) )
4645eqeq2d 2452 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( 0g `  M )  =  ( M  gsumg  w )  <-> 
( 0g `  M
)  =  ( M 
gsumg  (/) ) ) )
4746rspcev 3070 . . . . . 6  |-  ( (
(/)  e. Word  G  /\  ( 0g `  M )  =  ( M  gsumg  (/) ) )  ->  E. w  e. Word  G ( 0g `  M )  =  ( M  gsumg  w ) )
4840, 44, 47sylancr 658 . . . . 5  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  E. w  e. Word  G ( 0g `  M )  =  ( M  gsumg  w ) )
49 fvex 5698 . . . . . 6  |-  ( 0g
`  M )  e. 
_V
5017elrnmpt 5082 . . . . . 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 12270 . . . . . . . . 9  |-  ( ( z  e. Word  G  /\  v  e. Word  G )  ->  ( z concat  v )  e. Word  G )
5453adantl 463 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  ( z concat  v )  e. Word  G )
55 simpll 748 . . . . . . . . . 10  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  M  e.  Mnd )
56 sswrd 12238 . . . . . . . . . . . 12  |-  ( G 
C_  B  -> Word  G  C_ Word  B )
5756ad2antlr 721 . . . . . . . . . . 11  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  -> Word  G  C_ Word  B )
58 simprl 750 . . . . . . . . . . 11  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  z  e. Word  G )
5957, 58sseldd 3354 . . . . . . . . . 10  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  z  e. Word  B )
60 simprr 751 . . . . . . . . . . 11  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  v  e. Word  G )
6157, 60sseldd 3354 . . . . . . . . . 10  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  v  e. Word  B )
62 eqid 2441 . . . . . . . . . . 11  |-  ( +g  `  M )  =  ( +g  `  M )
631, 62gsumccat 15512 . . . . . . . . . 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 1213 . . . . . . . . 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 2446 . . . . . . . 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 6098 . . . . . . . . . 10  |-  ( w  =  ( z concat  v
)  ->  ( M  gsumg  w )  =  ( M 
gsumg  ( z concat  v )
) )
6766eqeq2d 2452 . . . . . . . . 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 3070 . . . . . . . 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 656 . . . . . . 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 6115 . . . . . . . 8  |-  ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  e. 
_V
7117elrnmpt 5082 . . . . . . . 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 2806 . . . . 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 6098 . . . . . . . . 9  |-  ( w  =  z  ->  ( M  gsumg  w )  =  ( M  gsumg  z ) )
7675cbvmptv 4380 . . . . . . . 8  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( z  e. Word  G  |->  ( M  gsumg  z ) )
7776rneqi 5062 . . . . . . 7  |-  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  =  ran  ( z  e. Word  G  |->  ( M  gsumg  z ) )
7877raleqi 2919 . . . . . 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 6098 . . . . . . . . . . 11  |-  ( w  =  v  ->  ( M  gsumg  w )  =  ( M  gsumg  v ) )
8079cbvmptv 4380 . . . . . . . . . 10  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( v  e. Word  G  |->  ( M  gsumg  v ) )
8180rneqi 5062 . . . . . . . . 9  |-  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  =  ran  ( v  e. Word  G  |->  ( M  gsumg  v ) )
8281raleqi 2919 . . . . . . . 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 2441 . . . . . . . . . 10  |-  ( v  e. Word  G  |->  ( M 
gsumg  v ) )  =  ( v  e. Word  G  |->  ( M  gsumg  v ) )
84 oveq2 6098 . . . . . . . . . . 11  |-  ( y  =  ( M  gsumg  v )  ->  ( x ( +g  `  M ) y )  =  ( x ( +g  `  M
) ( M  gsumg  v ) ) )
8584eleq1d 2507 . . . . . . . . . 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 5849 . . . . . . . . 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 6115 . . . . . . . . . 10  |-  ( M 
gsumg  v )  e.  _V
8887a1i 11 . . . . . . . . 9  |-  ( v  e. Word  G  ->  ( M  gsumg  v )  e.  _V )
8986, 88mprg 2783 . . . . . . . 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 2737 . . . . . 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 2441 . . . . . . . 8  |-  ( z  e. Word  G  |->  ( M 
gsumg  z ) )  =  ( z  e. Word  G  |->  ( M  gsumg  z ) )
93 oveq1 6097 . . . . . . . . . 10  |-  ( x  =  ( M  gsumg  z )  ->  ( x ( +g  `  M ) ( M  gsumg  v ) )  =  ( ( M  gsumg  z ) ( +g  `  M
) ( M  gsumg  v ) ) )
9493eleq1d 2507 . . . . . . . . 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 2733 . . . . . . . 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 5849 . . . . . . 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 6115 . . . . . . . 8  |-  ( M 
gsumg  z )  e.  _V
9897a1i 11 . . . . . . 7  |-  ( z  e. Word  G  ->  ( M  gsumg  z )  e.  _V )
9996, 98mprg 2783 . . . . . 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 15470 . . . . 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 462 . . . 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 1166 . . 3  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  e.  (SubMnd `  M )
)
10523mrcsscl 14554 . . 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 1213 . 2  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  C_  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) )
107106, 36eqssd 3370 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 960    = wceq 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   _Vcvv 2970    C_ wss 3325   (/)c0 3634    e. cmpt 4347   ran crn 4837   -->wf 5411   ` cfv 5415  (class class class)co 6090  Word cword 12217   concat cconcat 12219   <"cs1 12220   Basecbs 14170   +g cplusg 14234   0gc0g 14374    gsumg cgsu 14375  Moorecmre 14516  mrClscmrc 14517   Mndcmnd 15405  SubMndcsubmnd 15459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-word 12225  df-concat 12227  df-s1 12228  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-gsum 14377  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461
This theorem is referenced by:  psgneldm2  16003  psgnfitr  16016
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