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Theorem gsmsymgrfix 16245
Description: The composition of permutations fixing one element also fixes this element. (Contributed by AV, 20-Jan-2019.)
Hypotheses
Ref Expression
gsmsymgrfix.s  |-  S  =  ( SymGrp `  N )
gsmsymgrfix.b  |-  B  =  ( Base `  S
)
Assertion
Ref Expression
gsmsymgrfix  |-  ( ( N  e.  Fin  /\  K  e.  N  /\  W  e. Word  B )  ->  ( A. i  e.  ( 0..^ ( # `  W ) ) ( ( W `  i
) `  K )  =  K  ->  ( ( S  gsumg  W ) `  K
)  =  K ) )
Distinct variable groups:    B, i    i, K    i, N    i, W
Allowed substitution hint:    S( i)

Proof of Theorem gsmsymgrfix
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3116 . . . . . . . . . . 11  |-  w  e. 
_V
2 hasheq0 12395 . . . . . . . . . . 11  |-  ( w  e.  _V  ->  (
( # `  w )  =  0  <->  w  =  (/) ) )
31, 2ax-mp 5 . . . . . . . . . 10  |-  ( (
# `  w )  =  0  <->  w  =  (/) )
43biimpri 206 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( # `  w )  =  0 )
54oveq2d 6298 . . . . . . . 8  |-  ( w  =  (/)  ->  ( 0..^ ( # `  w
) )  =  ( 0..^ 0 ) )
6 fzo0 11813 . . . . . . . 8  |-  ( 0..^ 0 )  =  (/)
75, 6syl6eq 2524 . . . . . . 7  |-  ( w  =  (/)  ->  ( 0..^ ( # `  w
) )  =  (/) )
8 fveq1 5863 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w `
 i )  =  ( (/) `  i ) )
98fveq1d 5866 . . . . . . . 8  |-  ( w  =  (/)  ->  ( ( w `  i ) `
 K )  =  ( ( (/) `  i
) `  K )
)
109eqeq1d 2469 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( ( w `  i
) `  K )  =  K  <->  ( ( (/) `  i ) `  K
)  =  K ) )
117, 10raleqbidv 3072 . . . . . 6  |-  ( w  =  (/)  ->  ( A. i  e.  ( 0..^ ( # `  w
) ) ( ( w `  i ) `
 K )  =  K  <->  A. i  e.  (/)  ( ( (/) `  i
) `  K )  =  K ) )
12 oveq2 6290 . . . . . . . 8  |-  ( w  =  (/)  ->  ( S 
gsumg  w )  =  ( S  gsumg  (/) ) )
1312fveq1d 5866 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( S  gsumg  w ) `  K
)  =  ( ( S  gsumg  (/) ) `  K ) )
1413eqeq1d 2469 . . . . . 6  |-  ( w  =  (/)  ->  ( ( ( S  gsumg  w ) `  K
)  =  K  <->  ( ( S  gsumg  (/) ) `  K )  =  K ) )
1511, 14imbi12d 320 . . . . 5  |-  ( w  =  (/)  ->  ( ( A. i  e.  ( 0..^ ( # `  w
) ) ( ( w `  i ) `
 K )  =  K  ->  ( ( S  gsumg  w ) `  K
)  =  K )  <-> 
( A. i  e.  (/)  ( ( (/) `  i
) `  K )  =  K  ->  ( ( S  gsumg  (/) ) `  K )  =  K ) ) )
1615imbi2d 316 . . . 4  |-  ( w  =  (/)  ->  ( ( ( N  e.  Fin  /\  K  e.  N )  ->  ( A. i  e.  ( 0..^ ( # `  w ) ) ( ( w `  i
) `  K )  =  K  ->  ( ( S  gsumg  w ) `  K
)  =  K ) )  <->  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( A. i  e.  (/)  ( (
(/) `  i ) `  K )  =  K  ->  ( ( S 
gsumg  (/) ) `  K )  =  K ) ) ) )
17 fveq2 5864 . . . . . . . 8  |-  ( w  =  y  ->  ( # `
 w )  =  ( # `  y
) )
1817oveq2d 6298 . . . . . . 7  |-  ( w  =  y  ->  (
0..^ ( # `  w
) )  =  ( 0..^ ( # `  y
) ) )
19 fveq1 5863 . . . . . . . . 9  |-  ( w  =  y  ->  (
w `  i )  =  ( y `  i ) )
2019fveq1d 5866 . . . . . . . 8  |-  ( w  =  y  ->  (
( w `  i
) `  K )  =  ( ( y `
 i ) `  K ) )
2120eqeq1d 2469 . . . . . . 7  |-  ( w  =  y  ->  (
( ( w `  i ) `  K
)  =  K  <->  ( (
y `  i ) `  K )  =  K ) )
2218, 21raleqbidv 3072 . . . . . 6  |-  ( w  =  y  ->  ( A. i  e.  (
0..^ ( # `  w
) ) ( ( w `  i ) `
 K )  =  K  <->  A. i  e.  ( 0..^ ( # `  y
) ) ( ( y `  i ) `
 K )  =  K ) )
23 oveq2 6290 . . . . . . . 8  |-  ( w  =  y  ->  ( S  gsumg  w )  =  ( S  gsumg  y ) )
2423fveq1d 5866 . . . . . . 7  |-  ( w  =  y  ->  (
( S  gsumg  w ) `  K
)  =  ( ( S  gsumg  y ) `  K
) )
2524eqeq1d 2469 . . . . . 6  |-  ( w  =  y  ->  (
( ( S  gsumg  w ) `
 K )  =  K  <->  ( ( S 
gsumg  y ) `  K
)  =  K ) )
2622, 25imbi12d 320 . . . . 5  |-  ( w  =  y  ->  (
( A. i  e.  ( 0..^ ( # `  w ) ) ( ( w `  i
) `  K )  =  K  ->  ( ( S  gsumg  w ) `  K
)  =  K )  <-> 
( A. i  e.  ( 0..^ ( # `  y ) ) ( ( y `  i
) `  K )  =  K  ->  ( ( S  gsumg  y ) `  K
)  =  K ) ) )
2726imbi2d 316 . . . 4  |-  ( w  =  y  ->  (
( ( N  e. 
Fin  /\  K  e.  N )  ->  ( A. i  e.  (
0..^ ( # `  w
) ) ( ( w `  i ) `
 K )  =  K  ->  ( ( S  gsumg  w ) `  K
)  =  K ) )  <->  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( A. i  e.  (
0..^ ( # `  y
) ) ( ( y `  i ) `
 K )  =  K  ->  ( ( S  gsumg  y ) `  K
)  =  K ) ) ) )
28 fveq2 5864 . . . . . . . 8  |-  ( w  =  ( y concat  <" z "> )  ->  ( # `  w
)  =  ( # `  ( y concat  <" z "> ) ) )
2928oveq2d 6298 . . . . . . 7  |-  ( w  =  ( y concat  <" z "> )  ->  ( 0..^ ( # `  w ) )  =  ( 0..^ ( # `  ( y concat  <" z "> ) ) ) )
30 fveq1 5863 . . . . . . . . 9  |-  ( w  =  ( y concat  <" z "> )  ->  ( w `  i
)  =  ( ( y concat  <" z "> ) `  i
) )
3130fveq1d 5866 . . . . . . . 8  |-  ( w  =  ( y concat  <" z "> )  ->  ( ( w `  i ) `  K
)  =  ( ( ( y concat  <" z "> ) `  i
) `  K )
)
3231eqeq1d 2469 . . . . . . 7  |-  ( w  =  ( y concat  <" z "> )  ->  ( ( ( w `
 i ) `  K )  =  K  <-> 
( ( ( y concat  <" z "> ) `  i ) `  K )  =  K ) )
3329, 32raleqbidv 3072 . . . . . 6  |-  ( w  =  ( y concat  <" z "> )  ->  ( A. i  e.  ( 0..^ ( # `  w ) ) ( ( w `  i
) `  K )  =  K  <->  A. i  e.  ( 0..^ ( # `  (
y concat  <" z "> ) ) ) ( ( ( y concat  <" z "> ) `  i ) `  K )  =  K ) )
34 oveq2 6290 . . . . . . . 8  |-  ( w  =  ( y concat  <" z "> )  ->  ( S  gsumg  w )  =  ( S  gsumg  ( y concat  <" z "> ) ) )
3534fveq1d 5866 . . . . . . 7  |-  ( w  =  ( y concat  <" z "> )  ->  ( ( S  gsumg  w ) `
 K )  =  ( ( S  gsumg  ( y concat  <" z "> ) ) `  K
) )
3635eqeq1d 2469 . . . . . 6  |-  ( w  =  ( y concat  <" z "> )  ->  ( ( ( S 
gsumg  w ) `  K
)  =  K  <->  ( ( S  gsumg  ( y concat  <" z "> ) ) `  K )  =  K ) )
3733, 36imbi12d 320 . . . . 5  |-  ( w  =  ( y concat  <" z "> )  ->  ( ( A. i  e.  ( 0..^ ( # `  w ) ) ( ( w `  i
) `  K )  =  K  ->  ( ( S  gsumg  w ) `  K
)  =  K )  <-> 
( A. i  e.  ( 0..^ ( # `  ( y concat  <" z "> ) ) ) ( ( ( y concat  <" z "> ) `  i ) `  K )  =  K  ->  ( ( S 
gsumg  ( y concat  <" z "> ) ) `  K )  =  K ) ) )
3837imbi2d 316 . . . 4  |-  ( w  =  ( y concat  <" z "> )  ->  ( ( ( N  e.  Fin  /\  K  e.  N )  ->  ( A. i  e.  (
0..^ ( # `  w
) ) ( ( w `  i ) `
 K )  =  K  ->  ( ( S  gsumg  w ) `  K
)  =  K ) )  <->  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( A. i  e.  (
0..^ ( # `  (
y concat  <" z "> ) ) ) ( ( ( y concat  <" z "> ) `  i ) `  K )  =  K  ->  ( ( S 
gsumg  ( y concat  <" z "> ) ) `  K )  =  K ) ) ) )
39 fveq2 5864 . . . . . . . 8  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
4039oveq2d 6298 . . . . . . 7  |-  ( w  =  W  ->  (
0..^ ( # `  w
) )  =  ( 0..^ ( # `  W
) ) )
41 fveq1 5863 . . . . . . . . 9  |-  ( w  =  W  ->  (
w `  i )  =  ( W `  i ) )
4241fveq1d 5866 . . . . . . . 8  |-  ( w  =  W  ->  (
( w `  i
) `  K )  =  ( ( W `
 i ) `  K ) )
4342eqeq1d 2469 . . . . . . 7  |-  ( w  =  W  ->  (
( ( w `  i ) `  K
)  =  K  <->  ( ( W `  i ) `  K )  =  K ) )
4440, 43raleqbidv 3072 . . . . . 6  |-  ( w  =  W  ->  ( A. i  e.  (
0..^ ( # `  w
) ) ( ( w `  i ) `
 K )  =  K  <->  A. i  e.  ( 0..^ ( # `  W
) ) ( ( W `  i ) `
 K )  =  K ) )
45 oveq2 6290 . . . . . . . 8  |-  ( w  =  W  ->  ( S  gsumg  w )  =  ( S  gsumg  W ) )
4645fveq1d 5866 . . . . . . 7  |-  ( w  =  W  ->  (
( S  gsumg  w ) `  K
)  =  ( ( S  gsumg  W ) `  K
) )
4746eqeq1d 2469 . . . . . 6  |-  ( w  =  W  ->  (
( ( S  gsumg  w ) `
 K )  =  K  <->  ( ( S 
gsumg  W ) `  K
)  =  K ) )
4844, 47imbi12d 320 . . . . 5  |-  ( w  =  W  ->  (
( A. i  e.  ( 0..^ ( # `  w ) ) ( ( w `  i
) `  K )  =  K  ->  ( ( S  gsumg  w ) `  K
)  =  K )  <-> 
( A. i  e.  ( 0..^ ( # `  W ) ) ( ( W `  i
) `  K )  =  K  ->  ( ( S  gsumg  W ) `  K
)  =  K ) ) )
4948imbi2d 316 . . . 4  |-  ( w  =  W  ->  (
( ( N  e. 
Fin  /\  K  e.  N )  ->  ( A. i  e.  (
0..^ ( # `  w
) ) ( ( w `  i ) `
 K )  =  K  ->  ( ( S  gsumg  w ) `  K
)  =  K ) )  <->  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( A. i  e.  (
0..^ ( # `  W
) ) ( ( W `  i ) `
 K )  =  K  ->  ( ( S  gsumg  W ) `  K
)  =  K ) ) ) )
50 gsmsymgrfix.s . . . . . . . . . 10  |-  S  =  ( SymGrp `  N )
5150symgid 16218 . . . . . . . . 9  |-  ( N  e.  Fin  ->  (  _I  |`  N )  =  ( 0g `  S
) )
5251adantr 465 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  (  _I  |`  N )  =  ( 0g `  S ) )
53 eqid 2467 . . . . . . . . 9  |-  ( 0g
`  S )  =  ( 0g `  S
)
5453gsum0 15820 . . . . . . . 8  |-  ( S 
gsumg  (/) )  =  ( 0g
`  S )
5552, 54syl6reqr 2527 . . . . . . 7  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( S  gsumg  (/) )  =  (  _I  |`  N ) )
5655fveq1d 5866 . . . . . 6  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( ( S  gsumg  (/) ) `  K )  =  ( (  _I  |`  N ) `
 K ) )
57 fvresi 6085 . . . . . . 7  |-  ( K  e.  N  ->  (
(  _I  |`  N ) `
 K )  =  K )
5857adantl 466 . . . . . 6  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( (  _I  |`  N ) `
 K )  =  K )
5956, 58eqtrd 2508 . . . . 5  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( ( S  gsumg  (/) ) `  K )  =  K )
6059a1d 25 . . . 4  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( A. i  e.  (/)  ( ( (/) `  i
) `  K )  =  K  ->  ( ( S  gsumg  (/) ) `  K )  =  K ) )
61 ccatws1len 12583 . . . . . . . . . 10  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( # `  (
y concat  <" z "> ) )  =  ( ( # `  y
)  +  1 ) )
6261oveq2d 6298 . . . . . . . . 9  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( 0..^ ( # `  ( y concat  <" z "> ) ) )  =  ( 0..^ ( ( # `  y
)  +  1 ) ) )
6362raleqdv 3064 . . . . . . . 8  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( A. i  e.  ( 0..^ ( # `  ( y concat  <" z "> ) ) ) ( ( ( y concat  <" z "> ) `  i ) `  K )  =  K  <->  A. i  e.  (
0..^ ( ( # `  y )  +  1 ) ) ( ( ( y concat  <" z "> ) `  i
) `  K )  =  K ) )
6463adantr 465 . . . . . . 7  |-  ( ( ( y  e. Word  B  /\  z  e.  B
)  /\  ( ( N  e.  Fin  /\  K  e.  N )  /\  ( A. i  e.  (
0..^ ( # `  y
) ) ( ( y `  i ) `
 K )  =  K  ->  ( ( S  gsumg  y ) `  K
)  =  K ) ) )  ->  ( A. i  e.  (
0..^ ( # `  (
y concat  <" z "> ) ) ) ( ( ( y concat  <" z "> ) `  i ) `  K )  =  K  <->  A. i  e.  (
0..^ ( ( # `  y )  +  1 ) ) ( ( ( y concat  <" z "> ) `  i
) `  K )  =  K ) )
65 gsmsymgrfix.b . . . . . . . . 9  |-  B  =  ( Base `  S
)
6650, 65gsmsymgrfixlem1 16244 . . . . . . . 8  |-  ( ( ( y  e. Word  B  /\  z  e.  B
)  /\  ( N  e.  Fin  /\  K  e.  N )  /\  ( A. i  e.  (
0..^ ( # `  y
) ) ( ( y `  i ) `
 K )  =  K  ->  ( ( S  gsumg  y ) `  K
)  =  K ) )  ->  ( A. i  e.  ( 0..^ ( ( # `  y
)  +  1 ) ) ( ( ( y concat  <" z "> ) `  i
) `  K )  =  K  ->  ( ( S  gsumg  ( y concat  <" z "> ) ) `  K )  =  K ) )
67663expb 1197 . . . . . . 7  |-  ( ( ( y  e. Word  B  /\  z  e.  B
)  /\  ( ( N  e.  Fin  /\  K  e.  N )  /\  ( A. i  e.  (
0..^ ( # `  y
) ) ( ( y `  i ) `
 K )  =  K  ->  ( ( S  gsumg  y ) `  K
)  =  K ) ) )  ->  ( A. i  e.  (
0..^ ( ( # `  y )  +  1 ) ) ( ( ( y concat  <" z "> ) `  i
) `  K )  =  K  ->  ( ( S  gsumg  ( y concat  <" z "> ) ) `  K )  =  K ) )
6864, 67sylbid 215 . . . . . 6  |-  ( ( ( y  e. Word  B  /\  z  e.  B
)  /\  ( ( N  e.  Fin  /\  K  e.  N )  /\  ( A. i  e.  (
0..^ ( # `  y
) ) ( ( y `  i ) `
 K )  =  K  ->  ( ( S  gsumg  y ) `  K
)  =  K ) ) )  ->  ( A. i  e.  (
0..^ ( # `  (
y concat  <" z "> ) ) ) ( ( ( y concat  <" z "> ) `  i ) `  K )  =  K  ->  ( ( S 
gsumg  ( y concat  <" z "> ) ) `  K )  =  K ) )
6968exp32 605 . . . . 5  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( ( N  e. 
Fin  /\  K  e.  N )  ->  (
( A. i  e.  ( 0..^ ( # `  y ) ) ( ( y `  i
) `  K )  =  K  ->  ( ( S  gsumg  y ) `  K
)  =  K )  ->  ( A. i  e.  ( 0..^ ( # `  ( y concat  <" z "> ) ) ) ( ( ( y concat  <" z "> ) `  i ) `  K )  =  K  ->  ( ( S 
gsumg  ( y concat  <" z "> ) ) `  K )  =  K ) ) ) )
7069a2d 26 . . . 4  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( ( ( N  e.  Fin  /\  K  e.  N )  ->  ( A. i  e.  (
0..^ ( # `  y
) ) ( ( y `  i ) `
 K )  =  K  ->  ( ( S  gsumg  y ) `  K
)  =  K ) )  ->  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( A. i  e.  (
0..^ ( # `  (
y concat  <" z "> ) ) ) ( ( ( y concat  <" z "> ) `  i ) `  K )  =  K  ->  ( ( S 
gsumg  ( y concat  <" z "> ) ) `  K )  =  K ) ) ) )
7116, 27, 38, 49, 60, 70wrdind 12659 . . 3  |-  ( W  e. Word  B  ->  (
( N  e.  Fin  /\  K  e.  N )  ->  ( A. i  e.  ( 0..^ ( # `  W ) ) ( ( W `  i
) `  K )  =  K  ->  ( ( S  gsumg  W ) `  K
)  =  K ) ) )
7271com12 31 . 2  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( W  e. Word  B  ->  ( A. i  e.  ( 0..^ ( # `  W ) ) ( ( W `  i
) `  K )  =  K  ->  ( ( S  gsumg  W ) `  K
)  =  K ) ) )
73723impia 1193 1  |-  ( ( N  e.  Fin  /\  K  e.  N  /\  W  e. Word  B )  ->  ( A. i  e.  ( 0..^ ( # `  W ) ) ( ( W `  i
) `  K )  =  K  ->  ( ( S  gsumg  W ) `  K
)  =  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   (/)c0 3785    _I cid 4790    |` cres 5001   ` cfv 5586  (class class class)co 6282   Fincfn 7513   0cc0 9488   1c1 9489    + caddc 9491  ..^cfzo 11788   #chash 12367  Word cword 12494   concat cconcat 12496   <"cs1 12497   Basecbs 14483   0gc0g 14688    gsumg cgsu 14689   SymGrpcsymg 16194
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12071  df-hash 12368  df-word 12502  df-concat 12504  df-s1 12505  df-substr 12506  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-tset 14567  df-0g 14690  df-gsum 14691  df-mnd 15725  df-submnd 15775  df-grp 15855  df-symg 16195
This theorem is referenced by:  psgndiflemB  18400
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