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Theorem gsmsymgrfix 15938
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 2980 . . . . . . . . . . 11  |-  w  e. 
_V
2 hasheq0 12136 . . . . . . . . . . 11  |-  ( w  e.  _V  ->  (
( # `  w )  =  0  <->  w  =  (/) ) )
31, 2ax-mp 5 . . . . . . . . . 10  |-  ( (
# `  w )  =  0  <->  w  =  (/) )
43biimpri 206 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( # `  w )  =  0 )
54oveq2d 6112 . . . . . . . 8  |-  ( w  =  (/)  ->  ( 0..^ ( # `  w
) )  =  ( 0..^ 0 ) )
6 fzo0 11578 . . . . . . . 8  |-  ( 0..^ 0 )  =  (/)
75, 6syl6eq 2491 . . . . . . 7  |-  ( w  =  (/)  ->  ( 0..^ ( # `  w
) )  =  (/) )
8 fveq1 5695 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w `
 i )  =  ( (/) `  i ) )
98fveq1d 5698 . . . . . . . 8  |-  ( w  =  (/)  ->  ( ( w `  i ) `
 K )  =  ( ( (/) `  i
) `  K )
)
109eqeq1d 2451 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( ( w `  i
) `  K )  =  K  <->  ( ( (/) `  i ) `  K
)  =  K ) )
117, 10raleqbidv 2936 . . . . . 6  |-  ( w  =  (/)  ->  ( A. i  e.  ( 0..^ ( # `  w
) ) ( ( w `  i ) `
 K )  =  K  <->  A. i  e.  (/)  ( ( (/) `  i
) `  K )  =  K ) )
12 oveq2 6104 . . . . . . . 8  |-  ( w  =  (/)  ->  ( S 
gsumg  w )  =  ( S  gsumg  (/) ) )
1312fveq1d 5698 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( S  gsumg  w ) `  K
)  =  ( ( S  gsumg  (/) ) `  K ) )
1413eqeq1d 2451 . . . . . 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 5696 . . . . . . . 8  |-  ( w  =  y  ->  ( # `
 w )  =  ( # `  y
) )
1817oveq2d 6112 . . . . . . 7  |-  ( w  =  y  ->  (
0..^ ( # `  w
) )  =  ( 0..^ ( # `  y
) ) )
19 fveq1 5695 . . . . . . . . 9  |-  ( w  =  y  ->  (
w `  i )  =  ( y `  i ) )
2019fveq1d 5698 . . . . . . . 8  |-  ( w  =  y  ->  (
( w `  i
) `  K )  =  ( ( y `
 i ) `  K ) )
2120eqeq1d 2451 . . . . . . 7  |-  ( w  =  y  ->  (
( ( w `  i ) `  K
)  =  K  <->  ( (
y `  i ) `  K )  =  K ) )
2218, 21raleqbidv 2936 . . . . . 6  |-  ( w  =  y  ->  ( A. i  e.  (
0..^ ( # `  w
) ) ( ( w `  i ) `
 K )  =  K  <->  A. i  e.  ( 0..^ ( # `  y
) ) ( ( y `  i ) `
 K )  =  K ) )
23 oveq2 6104 . . . . . . . 8  |-  ( w  =  y  ->  ( S  gsumg  w )  =  ( S  gsumg  y ) )
2423fveq1d 5698 . . . . . . 7  |-  ( w  =  y  ->  (
( S  gsumg  w ) `  K
)  =  ( ( S  gsumg  y ) `  K
) )
2524eqeq1d 2451 . . . . . 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 5696 . . . . . . . 8  |-  ( w  =  ( y concat  <" z "> )  ->  ( # `  w
)  =  ( # `  ( y concat  <" z "> ) ) )
2928oveq2d 6112 . . . . . . 7  |-  ( w  =  ( y concat  <" z "> )  ->  ( 0..^ ( # `  w ) )  =  ( 0..^ ( # `  ( y concat  <" z "> ) ) ) )
30 fveq1 5695 . . . . . . . . 9  |-  ( w  =  ( y concat  <" z "> )  ->  ( w `  i
)  =  ( ( y concat  <" z "> ) `  i
) )
3130fveq1d 5698 . . . . . . . 8  |-  ( w  =  ( y concat  <" z "> )  ->  ( ( w `  i ) `  K
)  =  ( ( ( y concat  <" z "> ) `  i
) `  K )
)
3231eqeq1d 2451 . . . . . . 7  |-  ( w  =  ( y concat  <" z "> )  ->  ( ( ( w `
 i ) `  K )  =  K  <-> 
( ( ( y concat  <" z "> ) `  i ) `  K )  =  K ) )
3329, 32raleqbidv 2936 . . . . . 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 6104 . . . . . . . 8  |-  ( w  =  ( y concat  <" z "> )  ->  ( S  gsumg  w )  =  ( S  gsumg  ( y concat  <" z "> ) ) )
3534fveq1d 5698 . . . . . . 7  |-  ( w  =  ( y concat  <" z "> )  ->  ( ( S  gsumg  w ) `
 K )  =  ( ( S  gsumg  ( y concat  <" z "> ) ) `  K
) )
3635eqeq1d 2451 . . . . . 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 5696 . . . . . . . 8  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
4039oveq2d 6112 . . . . . . 7  |-  ( w  =  W  ->  (
0..^ ( # `  w
) )  =  ( 0..^ ( # `  W
) ) )
41 fveq1 5695 . . . . . . . . 9  |-  ( w  =  W  ->  (
w `  i )  =  ( W `  i ) )
4241fveq1d 5698 . . . . . . . 8  |-  ( w  =  W  ->  (
( w `  i
) `  K )  =  ( ( W `
 i ) `  K ) )
4342eqeq1d 2451 . . . . . . 7  |-  ( w  =  W  ->  (
( ( w `  i ) `  K
)  =  K  <->  ( ( W `  i ) `  K )  =  K ) )
4440, 43raleqbidv 2936 . . . . . 6  |-  ( w  =  W  ->  ( A. i  e.  (
0..^ ( # `  w
) ) ( ( w `  i ) `
 K )  =  K  <->  A. i  e.  ( 0..^ ( # `  W
) ) ( ( W `  i ) `
 K )  =  K ) )
45 oveq2 6104 . . . . . . . 8  |-  ( w  =  W  ->  ( S  gsumg  w )  =  ( S  gsumg  W ) )
4645fveq1d 5698 . . . . . . 7  |-  ( w  =  W  ->  (
( S  gsumg  w ) `  K
)  =  ( ( S  gsumg  W ) `  K
) )
4746eqeq1d 2451 . . . . . 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 15911 . . . . . . . . 9  |-  ( N  e.  Fin  ->  (  _I  |`  N )  =  ( 0g `  S
) )
5251adantr 465 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  (  _I  |`  N )  =  ( 0g `  S ) )
53 eqid 2443 . . . . . . . . 9  |-  ( 0g
`  S )  =  ( 0g `  S
)
5453gsum0 15515 . . . . . . . 8  |-  ( S 
gsumg  (/) )  =  ( 0g
`  S )
5552, 54syl6reqr 2494 . . . . . . 7  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( S  gsumg  (/) )  =  (  _I  |`  N ) )
5655fveq1d 5698 . . . . . 6  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( ( S  gsumg  (/) ) `  K )  =  ( (  _I  |`  N ) `
 K ) )
57 fvresi 5909 . . . . . . 7  |-  ( K  e.  N  ->  (
(  _I  |`  N ) `
 K )  =  K )
5857adantl 466 . . . . . 6  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( (  _I  |`  N ) `
 K )  =  K )
5956, 58eqtrd 2475 . . . . 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 12309 . . . . . . . . . 10  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( # `  (
y concat  <" z "> ) )  =  ( ( # `  y
)  +  1 ) )
6261oveq2d 6112 . . . . . . . . 9  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( 0..^ ( # `  ( y concat  <" z "> ) ) )  =  ( 0..^ ( ( # `  y
)  +  1 ) ) )
6362raleqdv 2928 . . . . . . . 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 15937 . . . . . . . 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 1188 . . . . . . 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 12376 . . 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 1184 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 965    = wceq 1369    e. wcel 1756   A.wral 2720   _Vcvv 2977   (/)c0 3642    _I cid 4636    |` cres 4847   ` cfv 5423  (class class class)co 6096   Fincfn 7315   0cc0 9287   1c1 9288    + caddc 9290  ..^cfzo 11553   #chash 12108  Word cword 12226   concat cconcat 12228   <"cs1 12229   Basecbs 14179   0gc0g 14383    gsumg cgsu 14384   SymGrpcsymg 15887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-word 12234  df-concat 12236  df-s1 12237  df-substr 12238  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-tset 14262  df-0g 14385  df-gsum 14386  df-mnd 15420  df-submnd 15470  df-grp 15550  df-symg 15888
This theorem is referenced by:  psgndiflemB  18035
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