MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumwspan Unicode version

Theorem gsumwspan 14779
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 14753 . . . . 5  |-  ( M  e.  Mnd  ->  (SubMnd `  M )  e.  (ACS
`  B ) )
32acsmred 13869 . . . 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 11744 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  <" x ">  e. Word  G )
7 ssel2 3335 . . . . . . . . . 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 14773 . . . . . . . . 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 2440 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  =  ( M  gsumg 
<" x "> ) )
12 oveq2 6080 . . . . . . . . 9  |-  ( w  =  <" x ">  ->  ( M  gsumg  w )  =  ( M 
gsumg  <" x "> ) )
1312eqeq2d 2446 . . . . . . . 8  |-  ( w  =  <" x ">  ->  ( x  =  ( M  gsumg  w )  <-> 
x  =  ( M 
gsumg  <" x "> ) ) )
1413rspcev 3044 . . . . . . 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 2951 . . . . . . 7  |-  x  e. 
_V
17 eqid 2435 . . . . . . . 8  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( w  e. Word  G  |->  ( M  gsumg  w ) )
1817elrnmpt 5108 . . . . . . 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 3346 . . 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 13824 . . . . . . . . . 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 13830 . . . . . . . . . . 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 11725 . . . . . . . . . 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 3340 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  w  e. Word  G )  ->  w  e. Word  ( K `  G )
)
32 gsumwsubmcl 14772 . . . . . . . 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 5884 . . . . . 6  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( w  e. Word  G  |->  ( M  gsumg  w ) ) :Word 
G --> ( K `  G ) )
35 frn 5588 . . . . . 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 13836 . . . . . 6  |-  ( M  e.  Mnd  ->  ( K `  G )  C_  B )
3837adantr 452 . . . . 5  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  C_  B )
3936, 38sstrd 3350 . . . 4  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  C_  B )
40 wrd0 11720 . . . . . 6  |-  (/)  e. Word  G
41 eqid 2435 . . . . . . . . 9  |-  ( 0g
`  M )  =  ( 0g `  M
)
4241gsum0 14768 . . . . . . . 8  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
4342eqcomi 2439 . . . . . . 7  |-  ( 0g
`  M )  =  ( M  gsumg  (/) )
4443a1i 11 . . . . . 6  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( 0g `  M
)  =  ( M 
gsumg  (/) ) )
45 oveq2 6080 . . . . . . . 8  |-  ( w  =  (/)  ->  ( M 
gsumg  w )  =  ( M  gsumg  (/) ) )
4645eqeq2d 2446 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( 0g `  M )  =  ( M  gsumg  w )  <-> 
( 0g `  M
)  =  ( M 
gsumg  (/) ) ) )
4746rspcev 3044 . . . . . 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 5733 . . . . . 6  |-  ( 0g
`  M )  e. 
_V
5017elrnmpt 5108 . . . . . 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 11731 . . . . . . . . 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 11725 . . . . . . . . . . . 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 3341 . . . . . . . . . 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 3341 . . . . . . . . . 10  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  v  e. Word  B )
62 eqid 2435 . . . . . . . . . . 11  |-  ( +g  `  M )  =  ( +g  `  M )
631, 62gsumccat 14775 . . . . . . . . . 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 2440 . . . . . . . 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 6080 . . . . . . . . . 10  |-  ( w  =  ( z concat  v
)  ->  ( M  gsumg  w )  =  ( M 
gsumg  ( z concat  v )
) )
6766eqeq2d 2446 . . . . . . . . 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 3044 . . . . . . . 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 6097 . . . . . . . 8  |-  ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  e. 
_V
7117elrnmpt 5108 . . . . . . . 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 2790 . . . . 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 6080 . . . . . . . . 9  |-  ( w  =  z  ->  ( M  gsumg  w )  =  ( M  gsumg  z ) )
7675cbvmptv 4292 . . . . . . . 8  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( z  e. Word  G  |->  ( M  gsumg  z ) )
7776rneqi 5087 . . . . . . 7  |-  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  =  ran  ( z  e. Word  G  |->  ( M  gsumg  z ) )
7877raleqi 2900 . . . . . 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 6080 . . . . . . . . . . 11  |-  ( w  =  v  ->  ( M  gsumg  w )  =  ( M  gsumg  v ) )
8079cbvmptv 4292 . . . . . . . . . 10  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( v  e. Word  G  |->  ( M  gsumg  v ) )
8180rneqi 5087 . . . . . . . . 9  |-  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  =  ran  ( v  e. Word  G  |->  ( M  gsumg  v ) )
8281raleqi 2900 . . . . . . . 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 2435 . . . . . . . . . 10  |-  ( v  e. Word  G  |->  ( M 
gsumg  v ) )  =  ( v  e. Word  G  |->  ( M  gsumg  v ) )
84 oveq2 6080 . . . . . . . . . . 11  |-  ( y  =  ( M  gsumg  v )  ->  ( x ( +g  `  M ) y )  =  ( x ( +g  `  M
) ( M  gsumg  v ) ) )
8584eleq1d 2501 . . . . . . . . . 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 5869 . . . . . . . . 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 6097 . . . . . . . . . 10  |-  ( M 
gsumg  v )  e.  _V
8887a1i 11 . . . . . . . . 9  |-  ( v  e. Word  G  ->  ( M  gsumg  v )  e.  _V )
8986, 88mprg 2767 . . . . . . . 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 2721 . . . . . 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 2435 . . . . . . . 8  |-  ( z  e. Word  G  |->  ( M 
gsumg  z ) )  =  ( z  e. Word  G  |->  ( M  gsumg  z ) )
93 oveq1 6079 . . . . . . . . . 10  |-  ( x  =  ( M  gsumg  z )  ->  ( x ( +g  `  M ) ( M  gsumg  v ) )  =  ( ( M  gsumg  z ) ( +g  `  M
) ( M  gsumg  v ) ) )
9493eleq1d 2501 . . . . . . . . 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 2717 . . . . . . . 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 5869 . . . . . . 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 6097 . . . . . . . 8  |-  ( M 
gsumg  z )  e.  _V
9897a1i 11 . . . . . . 7  |-  ( z  e. Word  G  ->  ( M  gsumg  z )  e.  _V )
9996, 98mprg 2767 . . . . . 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 14736 . . . . 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 13833 . . 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 3357 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948    C_ wss 3312   (/)c0 3620    e. cmpt 4258   ran crn 4870   -->wf 5441   ` cfv 5445  (class class class)co 6072  Word cword 11705   concat cconcat 11706   <"cs1 11707   Basecbs 13457   +g cplusg 13517   0gc0g 13711    gsumg cgsu 13712  Moorecmre 13795  mrClscmrc 13796   Mndcmnd 14672  SubMndcsubmnd 14725
This theorem is referenced by:  psgneldm2  27342
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-card 7815  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-n0 10211  df-z 10272  df-uz 10478  df-fz 11033  df-fzo 11124  df-seq 11312  df-hash 11607  df-word 11711  df-concat 11712  df-s1 11713  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-0g 13715  df-gsum 13716  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-submnd 14727
  Copyright terms: Public domain W3C validator