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Theorem dprdss 17597
Description: Create a direct product by finding subgroups inside each factor of another direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdss.1  |-  ( ph  ->  G dom DProd  T )
dprdss.2  |-  ( ph  ->  dom  T  =  I )
dprdss.3  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
dprdss.4  |-  ( (
ph  /\  k  e.  I )  ->  ( S `  k )  C_  ( T `  k
) )
Assertion
Ref Expression
dprdss  |-  ( ph  ->  ( G dom DProd  S  /\  ( G DProd  S )  C_  ( G DProd  T ) ) )
Distinct variable groups:    k, G    ph, k    S, k    T, k   
k, I

Proof of Theorem dprdss
Dummy variables  f 
a  h  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2429 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2429 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2429 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 dprdss.1 . . . 4  |-  ( ph  ->  G dom DProd  T )
5 dprdgrp 17572 . . . 4  |-  ( G dom DProd  T  ->  G  e. 
Grp )
64, 5syl 17 . . 3  |-  ( ph  ->  G  e.  Grp )
7 dprdss.2 . . . 4  |-  ( ph  ->  dom  T  =  I )
84, 7dprddomcld 17568 . . 3  |-  ( ph  ->  I  e.  _V )
9 dprdss.3 . . 3  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
10 dprdss.4 . . . . . . 7  |-  ( (
ph  /\  k  e.  I )  ->  ( S `  k )  C_  ( T `  k
) )
1110ralrimiva 2846 . . . . . 6  |-  ( ph  ->  A. k  e.  I 
( S `  k
)  C_  ( T `  k ) )
12 fveq2 5881 . . . . . . . 8  |-  ( k  =  x  ->  ( S `  k )  =  ( S `  x ) )
13 fveq2 5881 . . . . . . . 8  |-  ( k  =  x  ->  ( T `  k )  =  ( T `  x ) )
1412, 13sseq12d 3499 . . . . . . 7  |-  ( k  =  x  ->  (
( S `  k
)  C_  ( T `  k )  <->  ( S `  x )  C_  ( T `  x )
) )
1514rspcv 3184 . . . . . 6  |-  ( x  e.  I  ->  ( A. k  e.  I 
( S `  k
)  C_  ( T `  k )  ->  ( S `  x )  C_  ( T `  x
) ) )
1611, 15mpan9 471 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  C_  ( T `  x
) )
17163ad2antr1 1170 . . . 4  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  x
)  C_  ( T `  x ) )
184adantr 466 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  G dom DProd  T )
197adantr 466 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  dom  T  =  I )
20 simpr1 1011 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  x  e.  I )
21 simpr2 1012 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
y  e.  I )
22 simpr3 1013 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  x  =/=  y )
2318, 19, 20, 21, 22, 1dprdcntz 17575 . . . . 5  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  x
)  C_  ( (Cntz `  G ) `  ( T `  y )
) )
244, 7dprdf2 17574 . . . . . . . . 9  |-  ( ph  ->  T : I --> (SubGrp `  G ) )
2524adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  T : I --> (SubGrp `  G ) )
2625, 21ffvelrnd 6038 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  y
)  e.  (SubGrp `  G ) )
27 eqid 2429 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
2827subgss 16769 . . . . . . 7  |-  ( ( T `  y )  e.  (SubGrp `  G
)  ->  ( T `  y )  C_  ( Base `  G ) )
2926, 28syl 17 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  y
)  C_  ( Base `  G ) )
3011adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  A. k  e.  I 
( S `  k
)  C_  ( T `  k ) )
31 fveq2 5881 . . . . . . . . 9  |-  ( k  =  y  ->  ( S `  k )  =  ( S `  y ) )
32 fveq2 5881 . . . . . . . . 9  |-  ( k  =  y  ->  ( T `  k )  =  ( T `  y ) )
3331, 32sseq12d 3499 . . . . . . . 8  |-  ( k  =  y  ->  (
( S `  k
)  C_  ( T `  k )  <->  ( S `  y )  C_  ( T `  y )
) )
3433rspcv 3184 . . . . . . 7  |-  ( y  e.  I  ->  ( A. k  e.  I 
( S `  k
)  C_  ( T `  k )  ->  ( S `  y )  C_  ( T `  y
) ) )
3521, 30, 34sylc 62 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  y
)  C_  ( T `  y ) )
3627, 1cntz2ss 16937 . . . . . 6  |-  ( ( ( T `  y
)  C_  ( Base `  G )  /\  ( S `  y )  C_  ( T `  y
) )  ->  (
(Cntz `  G ) `  ( T `  y
) )  C_  (
(Cntz `  G ) `  ( S `  y
) ) )
3729, 35, 36syl2anc 665 . . . . 5  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( (Cntz `  G
) `  ( T `  y ) )  C_  ( (Cntz `  G ) `  ( S `  y
) ) )
3823, 37sstrd 3480 . . . 4  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
) )
3917, 38sstrd 3480 . . 3  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
) )
406adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  Grp )
4127subgacs 16803 . . . . . . 7  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
42 acsmre 15509 . . . . . . 7  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
4340, 41, 423syl 18 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
44 difss 3598 . . . . . . . . 9  |-  ( I 
\  { x }
)  C_  I
4511adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  A. k  e.  I  ( S `  k )  C_  ( T `  k )
)
46 ssralv 3531 . . . . . . . . 9  |-  ( ( I  \  { x } )  C_  I  ->  ( A. k  e.  I  ( S `  k )  C_  ( T `  k )  ->  A. k  e.  ( I  \  { x } ) ( S `
 k )  C_  ( T `  k ) ) )
4744, 45, 46mpsyl 65 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  A. k  e.  ( I  \  {
x } ) ( S `  k ) 
C_  ( T `  k ) )
48 ss2iun 4318 . . . . . . . 8  |-  ( A. k  e.  ( I  \  { x } ) ( S `  k
)  C_  ( T `  k )  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k ) 
C_  U_ k  e.  ( I  \  { x } ) ( T `
 k ) )
4947, 48syl 17 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k ) 
C_  U_ k  e.  ( I  \  { x } ) ( T `
 k ) )
509adantr 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  S : I --> (SubGrp `  G ) )
51 ffun 5748 . . . . . . . 8  |-  ( S : I --> (SubGrp `  G )  ->  Fun  S )
52 funiunfv 6168 . . . . . . . 8  |-  ( Fun 
S  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k )  =  U. ( S
" ( I  \  { x } ) ) )
5350, 51, 523syl 18 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k )  =  U. ( S
" ( I  \  { x } ) ) )
5424adantr 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  T : I --> (SubGrp `  G ) )
55 ffun 5748 . . . . . . . 8  |-  ( T : I --> (SubGrp `  G )  ->  Fun  T )
56 funiunfv 6168 . . . . . . . 8  |-  ( Fun 
T  ->  U_ k  e.  ( I  \  {
x } ) ( T `  k )  =  U. ( T
" ( I  \  { x } ) ) )
5754, 55, 563syl 18 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  U_ k  e.  ( I  \  {
x } ) ( T `  k )  =  U. ( T
" ( I  \  { x } ) ) )
5849, 53, 573sstr3d 3512 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  U. ( T " ( I  \  { x } ) ) )
59 imassrn 5199 . . . . . . . 8  |-  ( T
" ( I  \  { x } ) )  C_  ran  T
60 frn 5752 . . . . . . . . . 10  |-  ( T : I --> (SubGrp `  G )  ->  ran  T 
C_  (SubGrp `  G )
)
6154, 60syl 17 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  ran  T 
C_  (SubGrp `  G )
)
62 mresspw 15449 . . . . . . . . . 10  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
6343, 62syl 17 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
6461, 63sstrd 3480 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  ran  T 
C_  ~P ( Base `  G
) )
6559, 64syl5ss 3481 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( T " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
66 sspwuni 4391 . . . . . . 7  |-  ( ( T " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( T " (
I  \  { x } ) )  C_  ( Base `  G )
)
6765, 66sylib 199 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  U. ( T " ( I  \  { x } ) )  C_  ( Base `  G ) )
6843, 3, 58, 67mrcssd 15481 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( T " ( I 
\  { x }
) ) ) )
69 ss2in 3695 . . . . 5  |-  ( ( ( S `  x
)  C_  ( T `  x )  /\  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( T " ( I 
\  { x }
) ) ) )  ->  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )  C_  ( ( T `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( T " (
I  \  { x } ) ) ) ) )
7016, 68, 69syl2anc 665 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  ( ( T `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( T " ( I 
\  { x }
) ) ) ) )
714adantr 466 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  G dom DProd  T )
727adantr 466 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  dom  T  =  I )
73 simpr 462 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
7471, 72, 73, 2, 3dprddisj 17576 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( T `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( T
" ( I  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
7570, 74sseqtrd 3506 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  { ( 0g `  G
) } )
761, 2, 3, 6, 8, 9, 39, 75dmdprdd 17566 . 2  |-  ( ph  ->  G dom DProd  S )
774a1d 26 . . . . 5  |-  ( ph  ->  ( G dom DProd  S  ->  G dom DProd  T ) )
78 ss2ixp 7543 . . . . . . 7  |-  ( A. k  e.  I  ( S `  k )  C_  ( T `  k
)  ->  X_ k  e.  I  ( S `  k )  C_  X_ k  e.  I  ( T `  k ) )
7911, 78syl 17 . . . . . 6  |-  ( ph  -> 
X_ k  e.  I 
( S `  k
)  C_  X_ k  e.  I  ( T `  k ) )
80 rabss2 3550 . . . . . 6  |-  ( X_ k  e.  I  ( S `  k )  C_  X_ k  e.  I 
( T `  k
)  ->  { h  e.  X_ k  e.  I 
( S `  k
)  |  h finSupp  ( 0g `  G ) } 
C_  { h  e.  X_ k  e.  I 
( T `  k
)  |  h finSupp  ( 0g `  G ) } )
81 ssrexv 3532 . . . . . 6  |-  ( { h  e.  X_ k  e.  I  ( S `  k )  |  h finSupp 
( 0g `  G
) }  C_  { h  e.  X_ k  e.  I 
( T `  k
)  |  h finSupp  ( 0g `  G ) }  ->  ( E. f  e.  { h  e.  X_ k  e.  I  ( S `  k )  |  h finSupp  ( 0g `  G ) } a  =  ( G  gsumg  f )  ->  E. f  e.  {
h  e.  X_ k  e.  I  ( T `  k )  |  h finSupp 
( 0g `  G
) } a  =  ( G  gsumg  f ) ) )
8279, 80, 813syl 18 . . . . 5  |-  ( ph  ->  ( E. f  e. 
{ h  e.  X_ k  e.  I  ( S `  k )  |  h finSupp  ( 0g `  G ) } a  =  ( G  gsumg  f )  ->  E. f  e.  {
h  e.  X_ k  e.  I  ( T `  k )  |  h finSupp 
( 0g `  G
) } a  =  ( G  gsumg  f ) ) )
8377, 82anim12d 565 . . . 4  |-  ( ph  ->  ( ( G dom DProd  S  /\  E. f  e. 
{ h  e.  X_ k  e.  I  ( S `  k )  |  h finSupp  ( 0g `  G ) } a  =  ( G  gsumg  f ) )  ->  ( G dom DProd  T  /\  E. f  e.  { h  e.  X_ k  e.  I  ( T `  k )  |  h finSupp  ( 0g `  G ) } a  =  ( G  gsumg  f ) ) ) )
84 fdm 5750 . . . . 5  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
85 eqid 2429 . . . . . 6  |-  { h  e.  X_ k  e.  I 
( S `  k
)  |  h finSupp  ( 0g `  G ) }  =  { h  e.  X_ k  e.  I 
( S `  k
)  |  h finSupp  ( 0g `  G ) }
862, 85eldprd 17571 . . . . 5  |-  ( dom 
S  =  I  -> 
( a  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  { h  e.  X_ k  e.  I  ( S `  k )  |  h finSupp  ( 0g `  G ) } a  =  ( G  gsumg  f ) ) ) )
879, 84, 863syl 18 . . . 4  |-  ( ph  ->  ( a  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  { h  e.  X_ k  e.  I  ( S `  k )  |  h finSupp  ( 0g `  G ) } a  =  ( G  gsumg  f ) ) ) )
88 eqid 2429 . . . . . 6  |-  { h  e.  X_ k  e.  I 
( T `  k
)  |  h finSupp  ( 0g `  G ) }  =  { h  e.  X_ k  e.  I 
( T `  k
)  |  h finSupp  ( 0g `  G ) }
892, 88eldprd 17571 . . . . 5  |-  ( dom 
T  =  I  -> 
( a  e.  ( G DProd  T )  <->  ( G dom DProd  T  /\  E. f  e.  { h  e.  X_ k  e.  I  ( T `  k )  |  h finSupp  ( 0g `  G ) } a  =  ( G  gsumg  f ) ) ) )
907, 89syl 17 . . . 4  |-  ( ph  ->  ( a  e.  ( G DProd  T )  <->  ( G dom DProd  T  /\  E. f  e.  { h  e.  X_ k  e.  I  ( T `  k )  |  h finSupp  ( 0g `  G ) } a  =  ( G  gsumg  f ) ) ) )
9183, 87, 903imtr4d 271 . . 3  |-  ( ph  ->  ( a  e.  ( G DProd  S )  -> 
a  e.  ( G DProd 
T ) ) )
9291ssrdv 3476 . 2  |-  ( ph  ->  ( G DProd  S ) 
C_  ( G DProd  T
) )
9376, 92jca 534 1  |-  ( ph  ->  ( G dom DProd  S  /\  ( G DProd  S )  C_  ( G DProd  T ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   E.wrex 2783   {crab 2786   _Vcvv 3087    \ cdif 3439    i^i cin 3441    C_ wss 3442   ~Pcpw 3985   {csn 4002   U.cuni 4222   U_ciun 4302   class class class wbr 4426   dom cdm 4854   ran crn 4855   "cima 4857   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6305   X_cixp 7530   finSupp cfsupp 7889   Basecbs 15084   0gc0g 15297    gsumg cgsu 15298  Moorecmre 15439  mrClscmrc 15440  ACScacs 15442   Grpcgrp 16620  SubGrpcsubg 16762  Cntzccntz 16920   DProd cdprd 17560
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-0g 15299  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-minusg 16625  df-subg 16765  df-cntz 16922  df-dprd 17562
This theorem is referenced by: (None)
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