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Theorem gsumwspan 14468
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 14442 . . . . 5  |-  ( M  e.  Mnd  ->  (SubMnd `  M )  e.  (ACS
`  B ) )
3 acsmre 13554 . . . . 5  |-  ( (SubMnd `  M )  e.  (ACS
`  B )  -> 
(SubMnd `  M )  e.  (Moore `  B )
)
42, 3syl 15 . . . 4  |-  ( M  e.  Mnd  ->  (SubMnd `  M )  e.  (Moore `  B ) )
54adantr 451 . . 3  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
(SubMnd `  M )  e.  (Moore `  B )
)
6 simpr 447 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  e.  G )
76s1cld 11442 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  <" x ">  e. Word  G )
8 ssel2 3175 . . . . . . . . . 10  |-  ( ( G  C_  B  /\  x  e.  G )  ->  x  e.  B )
98adantll 694 . . . . . . . . 9  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  e.  B )
101gsumws1 14462 . . . . . . . . 9  |-  ( x  e.  B  ->  ( M  gsumg 
<" x "> )  =  x )
119, 10syl 15 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  ( M  gsumg  <" x "> )  =  x )
1211eqcomd 2288 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  =  ( M  gsumg 
<" x "> ) )
13 oveq2 5866 . . . . . . . . 9  |-  ( w  =  <" x ">  ->  ( M  gsumg  w )  =  ( M 
gsumg  <" x "> ) )
1413eqeq2d 2294 . . . . . . . 8  |-  ( w  =  <" x ">  ->  ( x  =  ( M  gsumg  w )  <-> 
x  =  ( M 
gsumg  <" x "> ) ) )
1514rspcev 2884 . . . . . . 7  |-  ( (
<" x ">  e. Word  G  /\  x  =  ( M  gsumg 
<" x "> ) )  ->  E. w  e. Word  G x  =  ( M  gsumg  w ) )
167, 12, 15syl2anc 642 . . . . . 6  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  E. w  e. Word  G x  =  ( M  gsumg  w ) )
17 vex 2791 . . . . . . 7  |-  x  e. 
_V
18 eqid 2283 . . . . . . . 8  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( w  e. Word  G  |->  ( M  gsumg  w ) )
1918elrnmpt 4926 . . . . . . 7  |-  ( x  e.  _V  ->  (
x  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  <->  E. w  e. Word  G x  =  ( M  gsumg  w ) ) )
2017, 19ax-mp 8 . . . . . 6  |-  ( x  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  <->  E. w  e. Word  G x  =  ( M  gsumg  w ) )
2116, 20sylibr 203 . . . . 5  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  x  e.  G
)  ->  x  e.  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) )
2221ex 423 . . . 4  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( x  e.  G  ->  x  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) ) ) )
2322ssrdv 3185 . . 3  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  G  C_  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) )
24 gsumwspan.k . . . . . . . . . . 11  |-  K  =  (mrCls `  (SubMnd `  M
) )
2524mrccl 13513 . . . . . . . . . 10  |-  ( ( (SubMnd `  M )  e.  (Moore `  B )  /\  G  C_  B )  ->  ( K `  G )  e.  (SubMnd `  M ) )
264, 25sylan 457 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  e.  (SubMnd `  M ) )
2726adantr 451 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  w  e. Word  G )  ->  ( K `  G )  e.  (SubMnd `  M ) )
2824mrcssid 13519 . . . . . . . . . . 11  |-  ( ( (SubMnd `  M )  e.  (Moore `  B )  /\  G  C_  B )  ->  G  C_  ( K `  G )
)
294, 28sylan 457 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  G  C_  ( K `  G ) )
30 sswrd 11423 . . . . . . . . . 10  |-  ( G 
C_  ( K `  G )  -> Word  G  C_ Word  ( K `  G ) )
3129, 30syl 15 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> Word  G 
C_ Word  ( K `  G
) )
3231sselda 3180 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  w  e. Word  G )  ->  w  e. Word  ( K `  G )
)
33 gsumwsubmcl 14461 . . . . . . . 8  |-  ( ( ( K `  G
)  e.  (SubMnd `  M )  /\  w  e. Word  ( K `  G
) )  ->  ( M  gsumg  w )  e.  ( K `  G ) )
3427, 32, 33syl2anc 642 . . . . . . 7  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  w  e. Word  G )  ->  ( M  gsumg  w )  e.  ( K `  G ) )
3534, 18fmptd 5684 . . . . . 6  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( w  e. Word  G  |->  ( M  gsumg  w ) ) :Word 
G --> ( K `  G ) )
36 frn 5395 . . . . . 6  |-  ( ( w  e. Word  G  |->  ( M  gsumg  w ) ) :Word 
G --> ( K `  G )  ->  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  C_  ( K `  G ) )
3735, 36syl 15 . . . . 5  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  C_  ( K `  G ) )
3824mrcssv 13516 . . . . . . 7  |-  ( (SubMnd `  M )  e.  (Moore `  B )  ->  ( K `  G )  C_  B )
394, 38syl 15 . . . . . 6  |-  ( M  e.  Mnd  ->  ( K `  G )  C_  B )
4039adantr 451 . . . . 5  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  C_  B )
4137, 40sstrd 3189 . . . 4  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  C_  B )
42 wrd0 11418 . . . . . 6  |-  (/)  e. Word  G
43 eqid 2283 . . . . . . . . 9  |-  ( 0g
`  M )  =  ( 0g `  M
)
4443gsum0 14457 . . . . . . . 8  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
4544eqcomi 2287 . . . . . . 7  |-  ( 0g
`  M )  =  ( M  gsumg  (/) )
4645a1i 10 . . . . . 6  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( 0g `  M
)  =  ( M 
gsumg  (/) ) )
47 oveq2 5866 . . . . . . . 8  |-  ( w  =  (/)  ->  ( M 
gsumg  w )  =  ( M  gsumg  (/) ) )
4847eqeq2d 2294 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( 0g `  M )  =  ( M  gsumg  w )  <-> 
( 0g `  M
)  =  ( M 
gsumg  (/) ) ) )
4948rspcev 2884 . . . . . 6  |-  ( (
(/)  e. Word  G  /\  ( 0g `  M )  =  ( M  gsumg  (/) ) )  ->  E. w  e. Word  G ( 0g `  M )  =  ( M  gsumg  w ) )
5042, 46, 49sylancr 644 . . . . 5  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  E. w  e. Word  G ( 0g `  M )  =  ( M  gsumg  w ) )
51 fvex 5539 . . . . . 6  |-  ( 0g
`  M )  e. 
_V
5218elrnmpt 4926 . . . . . 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 ) ) )
5351, 52ax-mp 8 . . . . 5  |-  ( ( 0g `  M )  e.  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  <->  E. w  e. Word  G ( 0g `  M )  =  ( M  gsumg  w ) )
5450, 53sylibr 203 . . . 4  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( 0g `  M
)  e.  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) ) )
55 ccatcl 11429 . . . . . . . . 9  |-  ( ( z  e. Word  G  /\  v  e. Word  G )  ->  ( z concat  v )  e. Word  G )
5655adantl 452 . . . . . . . 8  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  ( z concat  v )  e. Word  G )
57 simpll 730 . . . . . . . . . 10  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  M  e.  Mnd )
58 sswrd 11423 . . . . . . . . . . . 12  |-  ( G 
C_  B  -> Word  G  C_ Word  B )
5958ad2antlr 707 . . . . . . . . . . 11  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  -> Word  G  C_ Word  B )
60 simprl 732 . . . . . . . . . . 11  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  z  e. Word  G )
6159, 60sseldd 3181 . . . . . . . . . 10  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  z  e. Word  B )
62 simprr 733 . . . . . . . . . . 11  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  v  e. Word  G )
6359, 62sseldd 3181 . . . . . . . . . 10  |-  ( ( ( M  e.  Mnd  /\  G  C_  B )  /\  ( z  e. Word  G  /\  v  e. Word  G ) )  ->  v  e. Word  B )
64 eqid 2283 . . . . . . . . . . 11  |-  ( +g  `  M )  =  ( +g  `  M )
651, 64gsumccat 14464 . . . . . . . . . 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 ) ) )
6657, 61, 63, 65syl3anc 1182 . . . . . . . . 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 ) ) )
6766eqcomd 2288 . . . . . . . 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 ) ) )
68 oveq2 5866 . . . . . . . . . 10  |-  ( w  =  ( z concat  v
)  ->  ( M  gsumg  w )  =  ( M 
gsumg  ( z concat  v )
) )
6968eqeq2d 2294 . . . . . . . . 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 ) ) ) )
7069rspcev 2884 . . . . . . . 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 ) )
7156, 67, 70syl2anc 642 . . . . . . 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 ) )
72 ovex 5883 . . . . . . . 8  |-  ( ( M  gsumg  z ) ( +g  `  M ) ( M 
gsumg  v ) )  e. 
_V
7318elrnmpt 4926 . . . . . . . 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 ) ) )
7472, 73ax-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 ) )
7571, 74sylibr 203 . . . . . 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 ) ) )
7675ralrimivva 2635 . . . . 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 ) ) )
77 oveq2 5866 . . . . . . . . 9  |-  ( w  =  z  ->  ( M  gsumg  w )  =  ( M  gsumg  z ) )
7877cbvmptv 4111 . . . . . . . 8  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( z  e. Word  G  |->  ( M  gsumg  z ) )
7978rneqi 4905 . . . . . . 7  |-  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  =  ran  ( z  e. Word  G  |->  ( M  gsumg  z ) )
8079raleqi 2740 . . . . . 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 ) ) )
81 oveq2 5866 . . . . . . . . . . . 12  |-  ( w  =  v  ->  ( M  gsumg  w )  =  ( M  gsumg  v ) )
8281cbvmptv 4111 . . . . . . . . . . 11  |-  ( w  e. Word  G  |->  ( M 
gsumg  w ) )  =  ( v  e. Word  G  |->  ( M  gsumg  v ) )
8382rneqi 4905 . . . . . . . . . 10  |-  ran  (
w  e. Word  G  |->  ( M  gsumg  w ) )  =  ran  ( v  e. Word  G  |->  ( M  gsumg  v ) )
8483raleqi 2740 . . . . . . . . 9  |-  ( 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 ) ) )
85 eqid 2283 . . . . . . . . . . 11  |-  ( v  e. Word  G  |->  ( M 
gsumg  v ) )  =  ( v  e. Word  G  |->  ( M  gsumg  v ) )
86 oveq2 5866 . . . . . . . . . . . 12  |-  ( y  =  ( M  gsumg  v )  ->  ( x ( +g  `  M ) y )  =  ( x ( +g  `  M
) ( M  gsumg  v ) ) )
8786eleq1d 2349 . . . . . . . . . . 11  |-  ( 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 ) ) ) )
8885, 87ralrnmpt 5669 . . . . . . . . . 10  |-  ( 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 ) ) ) )
89 ovex 5883 . . . . . . . . . . 11  |-  ( M 
gsumg  v )  e.  _V
9089a1i 10 . . . . . . . . . 10  |-  ( v  e. Word  G  ->  ( M  gsumg  v )  e.  _V )
9188, 90mprg 2612 . . . . . . . . 9  |-  ( 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 ) ) )
9284, 91bitri 240 . . . . . . . 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. v  e. Word  G
( x ( +g  `  M ) ( M 
gsumg  v ) )  e. 
ran  ( w  e. Word  G  |->  ( M  gsumg  w ) ) )
9392ralbii 2567 . . . . . . 7  |-  ( 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 ) ) )
94 eqid 2283 . . . . . . . . 9  |-  ( z  e. Word  G  |->  ( M 
gsumg  z ) )  =  ( z  e. Word  G  |->  ( M  gsumg  z ) )
95 oveq1 5865 . . . . . . . . . . 11  |-  ( x  =  ( M  gsumg  z )  ->  ( x ( +g  `  M ) ( M  gsumg  v ) )  =  ( ( M  gsumg  z ) ( +g  `  M
) ( M  gsumg  v ) ) )
9695eleq1d 2349 . . . . . . . . . 10  |-  ( 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 ) ) ) )
9796ralbidv 2563 . . . . . . . . 9  |-  ( 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 ) ) ) )
9894, 97ralrnmpt 5669 . . . . . . . 8  |-  ( 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 ) ) ) )
99 ovex 5883 . . . . . . . . 9  |-  ( M 
gsumg  z )  e.  _V
10099a1i 10 . . . . . . . 8  |-  ( z  e. Word  G  ->  ( M  gsumg  z )  e.  _V )
10198, 100mprg 2612 . . . . . . 7  |-  ( 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 ) ) )
10293, 101bitri 240 . . . . . 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. 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 ) ) )
10380, 102bitri 240 . . . . 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 ) ) )
10476, 103sylibr 203 . . . 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 ) ) )
1051, 43, 64issubm 14425 . . . . 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 ) ) ) ) )
106105adantr 451 . . . 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 ) ) ) ) )
10741, 54, 104, 106mpbir3and 1135 . . 3  |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  ran  ( w  e. Word  G  |->  ( M  gsumg  w ) )  e.  (SubMnd `  M )
)
10824mrcsscl 13522 . . 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 ) ) )
1095, 23, 107, 108syl3anc 1182 . 2  |-  ( ( M  e.  Mnd  /\  G  C_  B )  -> 
( K `  G
)  C_  ran  ( w  e. Word  G  |->  ( M 
gsumg  w ) ) )
110109, 37eqssd 3196 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   (/)c0 3455    e. cmpt 4077   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858  Word cword 11403   concat cconcat 11404   <"cs1 11405   Basecbs 13148   +g cplusg 13208   0gc0g 13400    gsumg cgsu 13401  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487   Mndcmnd 14361  SubMndcsubmnd 14414
This theorem is referenced by:  psgneldm2  27427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416
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