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

Theorem gsmsymgrfix 16579
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 3112 . . . . . . . . . . 11  |-  w  e. 
_V
2 hasheq0 12435 . . . . . . . . . . 11  |-  ( w  e.  _V  ->  (
( # `  w )  =  0  <->  w  =  (/) ) )
31, 2ax-mp 5 . . . . . . . . . 10  |-  ( (
# `  w )  =  0  <->  w  =  (/) )
43biimpri 206 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( # `  w )  =  0 )
54oveq2d 6312 . . . . . . . 8  |-  ( w  =  (/)  ->  ( 0..^ ( # `  w
) )  =  ( 0..^ 0 ) )
6 fzo0 11847 . . . . . . . 8  |-  ( 0..^ 0 )  =  (/)
75, 6syl6eq 2514 . . . . . . 7  |-  ( w  =  (/)  ->  ( 0..^ ( # `  w
) )  =  (/) )
8 fveq1 5871 . . . . . . . . 9  |-  ( w  =  (/)  ->  ( w `
 i )  =  ( (/) `  i ) )
98fveq1d 5874 . . . . . . . 8  |-  ( w  =  (/)  ->  ( ( w `  i ) `
 K )  =  ( ( (/) `  i
) `  K )
)
109eqeq1d 2459 . . . . . . 7  |-  ( w  =  (/)  ->  ( ( ( w `  i
) `  K )  =  K  <->  ( ( (/) `  i ) `  K
)  =  K ) )
117, 10raleqbidv 3068 . . . . . 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 2459 . . . . . 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 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 2459 . . . . . . 7  |-  ( w  =  y  ->  (
( ( w `  i ) `  K
)  =  K  <->  ( (
y `  i ) `  K )  =  K ) )
2218, 21raleqbidv 3068 . . . . . 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 2459 . . . . . 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 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 2459 . . . . . . 7  |-  ( w  =  ( y ++  <" z "> )  ->  ( ( ( w `
 i ) `  K )  =  K  <-> 
( ( ( y ++ 
<" z "> ) `  i ) `  K )  =  K ) )
3329, 32raleqbidv 3068 . . . . . 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 2459 . . . . . 6  |-  ( w  =  ( y ++  <" z "> )  ->  ( ( ( S 
gsumg  w ) `  K
)  =  K  <->  ( ( S  gsumg  ( y ++  <" z "> ) ) `  K )  =  K ) )
3733, 36imbi12d 320 . . . . 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 316 . . . 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 2459 . . . . . . 7  |-  ( w  =  W  ->  (
( ( w `  i ) `  K
)  =  K  <->  ( ( W `  i ) `  K )  =  K ) )
4440, 43raleqbidv 3068 . . . . . 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 2459 . . . . . 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 16552 . . . . . . . . 9  |-  ( N  e.  Fin  ->  (  _I  |`  N )  =  ( 0g `  S
) )
5251adantr 465 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  (  _I  |`  N )  =  ( 0g `  S ) )
53 eqid 2457 . . . . . . . . 9  |-  ( 0g
`  S )  =  ( 0g `  S
)
5453gsum0 16031 . . . . . . . 8  |-  ( S 
gsumg  (/) )  =  ( 0g
`  S )
5552, 54syl6reqr 2517 . . . . . . 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 6098 . . . . . . 7  |-  ( K  e.  N  ->  (
(  _I  |`  N ) `
 K )  =  K )
5857adantl 466 . . . . . 6  |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( (  _I  |`  N ) `
 K )  =  K )
5956, 58eqtrd 2498 . . . . 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 12634 . . . . . . . . . 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 3060 . . . . . . . 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 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 ++  <" 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 16578 . . . . . . . 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 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 ++  <" z "> ) `  i
) `  K )  =  K  ->  ( ( S  gsumg  ( y ++  <" 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 ++  <" z "> ) ) ) ( ( ( y ++ 
<" z "> ) `  i ) `  K )  =  K  ->  ( ( S 
gsumg  ( y ++  <" 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 ++  <" z "> ) ) ) ( ( ( y ++ 
<" z "> ) `  i ) `  K )  =  K  ->  ( ( S 
gsumg  ( y ++  <" 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 ++  <" z "> ) ) ) ( ( ( y ++ 
<" z "> ) `  i ) `  K )  =  K  ->  ( ( S 
gsumg  ( y ++  <" z "> ) ) `  K )  =  K ) ) ) )
7116, 27, 38, 49, 60, 70wrdind 12713 . . 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 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109   (/)c0 3793    _I cid 4799    |` cres 5010   ` cfv 5594  (class class class)co 6296   Fincfn 7535   0cc0 9509   1c1 9510    + caddc 9512  ..^cfzo 11820   #chash 12407  Word cword 12537   ++ cconcat 12539   <"cs1 12540   Basecbs 14643   0gc0g 14856    gsumg cgsu 14857   SymGrpcsymg 16528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11821  df-seq 12110  df-hash 12408  df-word 12545  df-lsw 12546  df-concat 12547  df-s1 12548  df-substr 12549  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-tset 14730  df-0g 14858  df-gsum 14859  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-grp 16183  df-symg 16529
This theorem is referenced by:  psgndiflemB  18762
  Copyright terms: Public domain W3C validator