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Theorem gsmsymgrfix 17013
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 3081 . . . . . . . . . . 11  |-  w  e. 
_V
2 hasheq0 12530 . . . . . . . . . . 11  |-  ( w  e.  _V  ->  (
( # `  w )  =  0  <->  w  =  (/) ) )
31, 2ax-mp 5 . . . . . . . . . 10  |-  ( (
# `  w )  =  0  <->  w  =  (/) )
43biimpri 209 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( # `  w )  =  0 )
54oveq2d 6312 . . . . . . . 8  |-  ( w  =  (/)  ->  ( 0..^ ( # `  w
) )  =  ( 0..^ 0 ) )
6 fzo0 11929 . . . . . . . 8  |-  ( 0..^ 0 )  =  (/)
75, 6syl6eq 2477 . . . . . . 7  |-  ( w  =  (/)  ->  ( 0..^ ( # `  w
) )  =  (/) )
8 fveq1 5871 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w `
 i )  =  ( (/) `  i ) )
98fveq1d 5874 . . . . . . . 8  |-  ( w  =  (/)  ->  ( ( w `  i ) `
 K )  =  ( ( (/) `  i
) `  K )
)
109eqeq1d 2422 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( ( w `  i
) `  K )  =  K  <->  ( ( (/) `  i ) `  K
)  =  K ) )
117, 10raleqbidv 3037 . . . . . 6  |-  ( w  =  (/)  ->  ( A. i  e.  ( 0..^ ( # `  w
) ) ( ( w `  i ) `
 K )  =  K  <->  A. i  e.  (/)  ( ( (/) `  i
) `  K )  =  K ) )
12 oveq2 6304 . . . . . . . 8  |-  ( w  =  (/)  ->  ( S 
gsumg  w )  =  ( S  gsumg  (/) ) )
1312fveq1d 5874 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( S  gsumg  w ) `  K
)  =  ( ( S  gsumg  (/) ) `  K ) )
1413eqeq1d 2422 . . . . . 6  |-  ( w  =  (/)  ->  ( ( ( S  gsumg  w ) `  K
)  =  K  <->  ( ( S  gsumg  (/) ) `  K )  =  K ) )
1511, 14imbi12d 321 . . . . 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 317 . . . 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 5872 . . . . . . . 8  |-  ( w  =  y  ->  ( # `
 w )  =  ( # `  y
) )
1817oveq2d 6312 . . . . . . 7  |-  ( w  =  y  ->  (
0..^ ( # `  w
) )  =  ( 0..^ ( # `  y
) ) )
19 fveq1 5871 . . . . . . . . 9  |-  ( w  =  y  ->  (
w `  i )  =  ( y `  i ) )
2019fveq1d 5874 . . . . . . . 8  |-  ( w  =  y  ->  (
( w `  i
) `  K )  =  ( ( y `
 i ) `  K ) )
2120eqeq1d 2422 . . . . . . 7  |-  ( w  =  y  ->  (
( ( w `  i ) `  K
)  =  K  <->  ( (
y `  i ) `  K )  =  K ) )
2218, 21raleqbidv 3037 . . . . . 6  |-  ( w  =  y  ->  ( A. i  e.  (
0..^ ( # `  w
) ) ( ( w `  i ) `
 K )  =  K  <->  A. i  e.  ( 0..^ ( # `  y
) ) ( ( y `  i ) `
 K )  =  K ) )
23 oveq2 6304 . . . . . . . 8  |-  ( w  =  y  ->  ( S  gsumg  w )  =  ( S  gsumg  y ) )
2423fveq1d 5874 . . . . . . 7  |-  ( w  =  y  ->  (
( S  gsumg  w ) `  K
)  =  ( ( S  gsumg  y ) `  K
) )
2524eqeq1d 2422 . . . . . 6  |-  ( w  =  y  ->  (
( ( S  gsumg  w ) `
 K )  =  K  <->  ( ( S 
gsumg  y ) `  K
)  =  K ) )
2622, 25imbi12d 321 . . . . 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 317 . . . 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 5872 . . . . . . . 8  |-  ( w  =  ( y ++  <" z "> )  ->  ( # `  w
)  =  ( # `  ( y ++  <" z "> ) ) )
2928oveq2d 6312 . . . . . . 7  |-  ( w  =  ( y ++  <" z "> )  ->  ( 0..^ ( # `  w ) )  =  ( 0..^ ( # `  ( y ++  <" z "> ) ) ) )
30 fveq1 5871 . . . . . . . . 9  |-  ( w  =  ( y ++  <" z "> )  ->  ( w `  i
)  =  ( ( y ++  <" z "> ) `  i
) )
3130fveq1d 5874 . . . . . . . 8  |-  ( w  =  ( y ++  <" z "> )  ->  ( ( w `  i ) `  K
)  =  ( ( ( y ++  <" z "> ) `  i
) `  K )
)
3231eqeq1d 2422 . . . . . . 7  |-  ( w  =  ( y ++  <" z "> )  ->  ( ( ( w `
 i ) `  K )  =  K  <-> 
( ( ( y ++ 
<" z "> ) `  i ) `  K )  =  K ) )
3329, 32raleqbidv 3037 . . . . . 6  |-  ( w  =  ( y ++  <" z "> )  ->  ( A. i  e.  ( 0..^ ( # `  w ) ) ( ( w `  i
) `  K )  =  K  <->  A. i  e.  ( 0..^ ( # `  (
y ++  <" z "> ) ) ) ( ( ( y ++ 
<" z "> ) `  i ) `  K )  =  K ) )
34 oveq2 6304 . . . . . . . 8  |-  ( w  =  ( y ++  <" z "> )  ->  ( S  gsumg  w )  =  ( S  gsumg  ( y ++  <" z "> ) ) )
3534fveq1d 5874 . . . . . . 7  |-  ( w  =  ( y ++  <" z "> )  ->  ( ( S  gsumg  w ) `
 K )  =  ( ( S  gsumg  ( y ++ 
<" z "> ) ) `  K
) )
3635eqeq1d 2422 . . . . . 6  |-  ( w  =  ( y ++  <" z "> )  ->  ( ( ( S 
gsumg  w ) `  K
)  =  K  <->  ( ( S  gsumg  ( y ++  <" z "> ) ) `  K )  =  K ) )
3733, 36imbi12d 321 . . . . 5  |-  ( w  =  ( y ++  <" z "> )  ->  ( ( A. i  e.  ( 0..^ ( # `  w ) ) ( ( w `  i
) `  K )  =  K  ->  ( ( S  gsumg  w ) `  K
)  =  K )  <-> 
( A. i  e.  ( 0..^ ( # `  ( y ++  <" z "> ) ) ) ( ( ( y ++ 
<" z "> ) `  i ) `  K )  =  K  ->  ( ( S 
gsumg  ( y ++  <" z "> ) ) `  K )  =  K ) ) )
3837imbi2d 317 . . . 4  |-  ( w  =  ( y ++  <" 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 ++  <" z "> ) ) ) ( ( ( y ++ 
<" z "> ) `  i ) `  K )  =  K  ->  ( ( S 
gsumg  ( y ++  <" z "> ) ) `  K )  =  K ) ) ) )
39 fveq2 5872 . . . . . . . 8  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
4039oveq2d 6312 . . . . . . 7  |-  ( w  =  W  ->  (
0..^ ( # `  w
) )  =  ( 0..^ ( # `  W
) ) )
41 fveq1 5871 . . . . . . . . 9  |-  ( w  =  W  ->  (
w `  i )  =  ( W `  i ) )
4241fveq1d 5874 . . . . . . . 8  |-  ( w  =  W  ->  (
( w `  i
) `  K )  =  ( ( W `
 i ) `  K ) )
4342eqeq1d 2422 . . . . . . 7  |-  ( w  =  W  ->  (
( ( w `  i ) `  K
)  =  K  <->  ( ( W `  i ) `  K )  =  K ) )
4440, 43raleqbidv 3037 . . . . . 6  |-  ( w  =  W  ->  ( A. i  e.  (
0..^ ( # `  w
) ) ( ( w `  i ) `
 K )  =  K  <->  A. i  e.  ( 0..^ ( # `  W
) ) ( ( W `  i ) `
 K )  =  K ) )
45 oveq2 6304 . . . . . . . 8  |-  ( w  =  W  ->  ( S  gsumg  w )  =  ( S  gsumg  W ) )
4645fveq1d 5874 . . . . . . 7  |-  ( w  =  W  ->  (
( S  gsumg  w ) `  K
)  =  ( ( S  gsumg  W ) `  K
) )
4746eqeq1d 2422 . . . . . 6  |-  ( w  =  W  ->  (
( ( S  gsumg  w ) `
 K )  =  K  <->  ( ( S 
gsumg  W ) `  K
)  =  K ) )
4844, 47imbi12d 321 . . . . 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 317 . . . 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 16986 . . . . . . . . 9  |-  ( N  e.  Fin  ->  (  _I  |`  N )  =  ( 0g `  S
) )
5251adantr 466 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  (  _I  |`  N )  =  ( 0g `  S ) )
53 eqid 2420 . . . . . . . . 9  |-  ( 0g
`  S )  =  ( 0g `  S
)
5453gsum0 16465 . . . . . . . 8  |-  ( S 
gsumg  (/) )  =  ( 0g
`  S )
5552, 54syl6reqr 2480 . . . . . . 7  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( S  gsumg  (/) )  =  (  _I  |`  N ) )
5655fveq1d 5874 . . . . . 6  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( ( S  gsumg  (/) ) `  K )  =  ( (  _I  |`  N ) `
 K ) )
57 fvresi 6096 . . . . . . 7  |-  ( K  e.  N  ->  (
(  _I  |`  N ) `
 K )  =  K )
5857adantl 467 . . . . . 6  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( (  _I  |`  N ) `
 K )  =  K )
5956, 58eqtrd 2461 . . . . 5  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( ( S  gsumg  (/) ) `  K )  =  K )
6059a1d 26 . . . 4  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( A. i  e.  (/)  ( ( (/) `  i
) `  K )  =  K  ->  ( ( S  gsumg  (/) ) `  K )  =  K ) )
61 ccatws1len 12729 . . . . . . . . . 10  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( # `  (
y ++  <" z "> ) )  =  ( ( # `  y
)  +  1 ) )
6261oveq2d 6312 . . . . . . . . 9  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( 0..^ ( # `  ( y ++  <" z "> ) ) )  =  ( 0..^ ( ( # `  y
)  +  1 ) ) )
6362raleqdv 3029 . . . . . . . 8  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( A. i  e.  ( 0..^ ( # `  ( y ++  <" z "> ) ) ) ( ( ( y ++ 
<" z "> ) `  i ) `  K )  =  K  <->  A. i  e.  (
0..^ ( ( # `  y )  +  1 ) ) ( ( ( y ++  <" z "> ) `  i
) `  K )  =  K ) )
6463adantr 466 . . . . . . 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 ++  <" z "> ) ) ) ( ( ( y ++ 
<" z "> ) `  i ) `  K )  =  K  <->  A. i  e.  (
0..^ ( ( # `  y )  +  1 ) ) ( ( ( y ++  <" z "> ) `  i
) `  K )  =  K ) )
65 gsmsymgrfix.b . . . . . . . . 9  |-  B  =  ( Base `  S
)
6650, 65gsmsymgrfixlem1 17012 . . . . . . . 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 ++  <" z "> ) `  i
) `  K )  =  K  ->  ( ( S  gsumg  ( y ++  <" z "> ) ) `  K )  =  K ) )
67663expb 1206 . . . . . . 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 ++  <" z "> ) `  i
) `  K )  =  K  ->  ( ( S  gsumg  ( y ++  <" z "> ) ) `  K )  =  K ) )
6864, 67sylbid 218 . . . . . 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 ++  <" z "> ) ) ) ( ( ( y ++ 
<" z "> ) `  i ) `  K )  =  K  ->  ( ( S 
gsumg  ( y ++  <" z "> ) ) `  K )  =  K ) )
6968exp32 608 . . . . 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 ++  <" z "> ) ) ) ( ( ( y ++ 
<" z "> ) `  i ) `  K )  =  K  ->  ( ( S 
gsumg  ( y ++  <" z "> ) ) `  K )  =  K ) ) ) )
7069a2d 29 . . . 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 ++  <" z "> ) ) ) ( ( ( y ++ 
<" z "> ) `  i ) `  K )  =  K  ->  ( ( S 
gsumg  ( y ++  <" z "> ) ) `  K )  =  K ) ) ) )
7116, 27, 38, 49, 60, 70wrdind 12807 . . 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 32 . 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 1202 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773   _Vcvv 3078   (/)c0 3758    _I cid 4755    |` cres 4847   ` cfv 5592  (class class class)co 6296   Fincfn 7568   0cc0 9528   1c1 9529    + caddc 9531  ..^cfzo 11902   #chash 12501  Word cword 12632   ++ cconcat 12634   <"cs1 12635   Basecbs 15073   0gc0g 15290    gsumg cgsu 15291   SymGrpcsymg 16962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  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-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11772  df-fzo 11903  df-seq 12200  df-hash 12502  df-word 12640  df-lsw 12641  df-concat 12642  df-s1 12643  df-substr 12644  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-ress 15080  df-plusg 15155  df-tset 15161  df-0g 15292  df-gsum 15293  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-submnd 16527  df-grp 16617  df-symg 16963
This theorem is referenced by:  psgndiflemB  19092
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