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

Theorem subgmulg 16087
Description: A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
Hypotheses
Ref Expression
subgmulgcl.t  |-  .x.  =  (.g
`  G )
subgmulg.h  |-  H  =  ( Gs  S )
subgmulg.t  |-  .xb  =  (.g
`  H )
Assertion
Ref Expression
subgmulg  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  .x.  X )  =  ( N  .xb  X
) )

Proof of Theorem subgmulg
StepHypRef Expression
1 subgmulg.h . . . . . 6  |-  H  =  ( Gs  S )
2 eqid 2467 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
31, 2subg0 16079 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
433ad2ant1 1017 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( 0g `  G )  =  ( 0g `  H
) )
54ifeq1d 3963 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  if ( N  =  0 ,  ( 0g `  G ) ,  if ( 0  <  N ,  (  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  N
) ,  ( ( invg `  G
) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) )  =  if ( N  =  0 ,  ( 0g `  H
) ,  if ( 0  <  N , 
(  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( invg `  G ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) ) )
6 eqid 2467 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  G )
71, 6ressplusg 14614 . . . . . . . . . 10  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
873ad2ant1 1017 . . . . . . . . 9  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( +g  `  G )  =  ( +g  `  H
) )
98seqeq2d 12094 . . . . . . . 8  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )  =  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) )
109adantr 465 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  -.  N  =  0 )  ->  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )  =  seq 1 ( ( +g  `  H
) ,  ( NN 
X.  { X }
) ) )
1110fveq1d 5874 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  -.  N  =  0 )  ->  (  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  N
)  =  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  N ) )
1211ifeq1d 3963 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  -.  N  =  0 )  ->  if ( 0  <  N ,  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( invg `  G ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )  =  if ( 0  <  N ,  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( invg `  G ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) )
13 simp2 997 . . . . . . . . . . . . 13  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  N  e.  ZZ )
1413zred 10978 . . . . . . . . . . . 12  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  N  e.  RR )
15 0re 9608 . . . . . . . . . . . 12  |-  0  e.  RR
16 axlttri 9668 . . . . . . . . . . . 12  |-  ( ( N  e.  RR  /\  0  e.  RR )  ->  ( N  <  0  <->  -.  ( N  =  0  \/  0  <  N
) ) )
1714, 15, 16sylancl 662 . . . . . . . . . . 11  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  <  0  <->  -.  ( N  =  0  \/  0  <  N ) ) )
18 ioran 490 . . . . . . . . . . 11  |-  ( -.  ( N  =  0  \/  0  <  N
)  <->  ( -.  N  =  0  /\  -.  0  <  N ) )
1917, 18syl6bb 261 . . . . . . . . . 10  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  <  0  <->  ( -.  N  =  0  /\  -.  0  <  N ) ) )
2019biimpar 485 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( -.  N  =  0  /\  -.  0  <  N
) )  ->  N  <  0 )
21 simpl1 999 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  S  e.  (SubGrp `  G )
)
2213adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  N  e.  ZZ )
2322znegcld 10980 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  -u N  e.  ZZ )
2414lt0neg1d 10134 . . . . . . . . . . . . . 14  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  <  0  <->  0  <  -u N ) )
2524biimpa 484 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  0  <  -u N )
26 elnnz 10886 . . . . . . . . . . . . 13  |-  ( -u N  e.  NN  <->  ( -u N  e.  ZZ  /\  0  <  -u N ) )
2723, 25, 26sylanbrc 664 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  -u N  e.  NN )
28 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( Base `  G )  =  (
Base `  G )
2928subgss 16074 . . . . . . . . . . . . . . 15  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
30293ad2ant1 1017 . . . . . . . . . . . . . 14  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  S  C_  ( Base `  G
) )
31 simp3 998 . . . . . . . . . . . . . 14  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  X  e.  S )
3230, 31sseldd 3510 . . . . . . . . . . . . 13  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  X  e.  ( Base `  G
) )
3332adantr 465 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  X  e.  ( Base `  G
) )
34 subgmulgcl.t . . . . . . . . . . . . 13  |-  .x.  =  (.g
`  G )
35 eqid 2467 . . . . . . . . . . . . 13  |-  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
3628, 6, 34, 35mulgnn 16020 . . . . . . . . . . . 12  |-  ( (
-u N  e.  NN  /\  X  e.  ( Base `  G ) )  -> 
( -u N  .x.  X
)  =  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) )
3727, 33, 36syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  ( -u N  .x.  X )  =  (  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  -u N
) )
3831adantr 465 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  X  e.  S )
3934subgmulgcl 16086 . . . . . . . . . . . 12  |-  ( ( S  e.  (SubGrp `  G )  /\  -u N  e.  ZZ  /\  X  e.  S )  ->  ( -u N  .x.  X )  e.  S )
4021, 23, 38, 39syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  ( -u N  .x.  X )  e.  S )
4137, 40eqeltrrd 2556 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N )  e.  S
)
42 eqid 2467 . . . . . . . . . . 11  |-  ( invg `  G )  =  ( invg `  G )
43 eqid 2467 . . . . . . . . . . 11  |-  ( invg `  H )  =  ( invg `  H )
441, 42, 43subginv 16080 . . . . . . . . . 10  |-  ( ( S  e.  (SubGrp `  G )  /\  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N )  e.  S
)  ->  ( ( invg `  G ) `
 (  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  -u N
) )  =  ( ( invg `  H ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )
4521, 41, 44syl2anc 661 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  (
( invg `  G ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) )  =  ( ( invg `  H ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )
4620, 45syldan 470 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( -.  N  =  0  /\  -.  0  <  N
) )  ->  (
( invg `  G ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) )  =  ( ( invg `  H ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )
479adantr 465 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( -.  N  =  0  /\  -.  0  <  N
) )  ->  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )  =  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) )
4847fveq1d 5874 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( -.  N  =  0  /\  -.  0  <  N
) )  ->  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N )  =  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `
 -u N ) )
4948fveq2d 5876 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( -.  N  =  0  /\  -.  0  <  N
) )  ->  (
( invg `  H ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) )  =  ( ( invg `  H ) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )
5046, 49eqtrd 2508 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( -.  N  =  0  /\  -.  0  <  N
) )  ->  (
( invg `  G ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) )  =  ( ( invg `  H ) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )
5150anassrs 648 . . . . . 6  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  -.  N  =  0 )  /\  -.  0  <  N )  -> 
( ( invg `  G ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) )  =  ( ( invg `  H ) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )
5251ifeq2da 3976 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  -.  N  =  0 )  ->  if ( 0  <  N ,  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( invg `  G ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )  =  if ( 0  <  N ,  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( invg `  H ) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) )
5312, 52eqtrd 2508 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  -.  N  =  0 )  ->  if ( 0  <  N ,  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( invg `  G ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )  =  if ( 0  <  N ,  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( invg `  H ) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) )
5453ifeq2da 3976 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  if ( N  =  0 ,  ( 0g `  H ) ,  if ( 0  <  N ,  (  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  N
) ,  ( ( invg `  G
) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) )  =  if ( N  =  0 ,  ( 0g `  H
) ,  if ( 0  <  N , 
(  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( invg `  H ) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) ) )
555, 54eqtrd 2508 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  if ( N  =  0 ,  ( 0g `  G ) ,  if ( 0  <  N ,  (  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  N
) ,  ( ( invg `  G
) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) )  =  if ( N  =  0 ,  ( 0g `  H
) ,  if ( 0  <  N , 
(  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( invg `  H ) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) ) )
5628, 6, 2, 42, 34, 35mulgval 16016 . . 3  |-  ( ( N  e.  ZZ  /\  X  e.  ( Base `  G ) )  -> 
( N  .x.  X
)  =  if ( N  =  0 ,  ( 0g `  G
) ,  if ( 0  <  N , 
(  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( invg `  G ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) ) )
5713, 32, 56syl2anc 661 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  .x.  X )  =  if ( N  =  0 ,  ( 0g
`  G ) ,  if ( 0  < 
N ,  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  N ) ,  ( ( invg `  G ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) ) )
581subgbas 16077 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
59583ad2ant1 1017 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  S  =  ( Base `  H
) )
6031, 59eleqtrd 2557 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  X  e.  ( Base `  H
) )
61 eqid 2467 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
62 eqid 2467 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
63 eqid 2467 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
64 subgmulg.t . . . 4  |-  .xb  =  (.g
`  H )
65 eqid 2467 . . . 4  |-  seq 1
( ( +g  `  H
) ,  ( NN 
X.  { X }
) )  =  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) )
6661, 62, 63, 43, 64, 65mulgval 16016 . . 3  |-  ( ( N  e.  ZZ  /\  X  e.  ( Base `  H ) )  -> 
( N  .xb  X
)  =  if ( N  =  0 ,  ( 0g `  H
) ,  if ( 0  <  N , 
(  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( invg `  H ) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) ) )
6713, 60, 66syl2anc 661 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  .xb  X )  =  if ( N  =  0 ,  ( 0g
`  H ) ,  if ( 0  < 
N ,  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  N ) ,  ( ( invg `  H ) `  (  seq 1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) ) )
6855, 57, 673eqtr4d 2518 1  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  .x.  X )  =  ( N  .xb  X
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3481   ifcif 3945   {csn 4033   class class class wbr 4453    X. cxp 5003   ` cfv 5594  (class class class)co 6295   RRcr 9503   0cc0 9504   1c1 9505    < clt 9640   -ucneg 9818   NNcn 10548   ZZcz 10876    seqcseq 12087   Basecbs 14507   ↾s cress 14508   +g cplusg 14572   0gc0g 14712   invgcminusg 15926  .gcmg 15928  SubGrpcsubg 16067
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-seq 12088  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-mulg 15932  df-subg 16070
This theorem is referenced by:  cycsubgcyg  16776  ablfac2  17012  zringmulg  18366  zrngmulg  18372  zringcyg  18382  zcyg  18387  mulgrhm2OLD  18405  remulg  18512  rezh  27777
  Copyright terms: Public domain W3C validator