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Theorem dprdss 16875
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 2467 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2467 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2467 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 dprdss.1 . . . 4  |-  ( ph  ->  G dom DProd  T )
5 dprdgrp 16838 . . . 4  |-  ( G dom DProd  T  ->  G  e. 
Grp )
64, 5syl 16 . . 3  |-  ( ph  ->  G  e.  Grp )
7 dprdss.2 . . . 4  |-  ( ph  ->  dom  T  =  I )
84, 7dprddomcld 16832 . . 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 2878 . . . . . 6  |-  ( ph  ->  A. k  e.  I 
( S `  k
)  C_  ( T `  k ) )
12 fveq2 5865 . . . . . . . 8  |-  ( k  =  x  ->  ( S `  k )  =  ( S `  x ) )
13 fveq2 5865 . . . . . . . 8  |-  ( k  =  x  ->  ( T `  k )  =  ( T `  x ) )
1412, 13sseq12d 3533 . . . . . . 7  |-  ( k  =  x  ->  (
( S `  k
)  C_  ( T `  k )  <->  ( S `  x )  C_  ( T `  x )
) )
1514rspcv 3210 . . . . . 6  |-  ( x  e.  I  ->  ( A. k  e.  I 
( S `  k
)  C_  ( T `  k )  ->  ( S `  x )  C_  ( T `  x
) ) )
1611, 15mpan9 469 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  C_  ( T `  x
) )
17163ad2antr1 1161 . . . 4  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  x
)  C_  ( T `  x ) )
184adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  G dom DProd  T )
197adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  dom  T  =  I )
20 simpr1 1002 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  x  e.  I )
21 simpr2 1003 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
y  e.  I )
22 simpr3 1004 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  x  =/=  y )
2318, 19, 20, 21, 22, 1dprdcntz 16841 . . . . 5  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  x
)  C_  ( (Cntz `  G ) `  ( T `  y )
) )
244, 7dprdf2 16840 . . . . . . . . 9  |-  ( ph  ->  T : I --> (SubGrp `  G ) )
2524adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  T : I --> (SubGrp `  G ) )
2625, 21ffvelrnd 6021 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  y
)  e.  (SubGrp `  G ) )
27 eqid 2467 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
2827subgss 16004 . . . . . . 7  |-  ( ( T `  y )  e.  (SubGrp `  G
)  ->  ( T `  y )  C_  ( Base `  G ) )
2926, 28syl 16 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  y
)  C_  ( Base `  G ) )
3011adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  A. k  e.  I 
( S `  k
)  C_  ( T `  k ) )
31 fveq2 5865 . . . . . . . . 9  |-  ( k  =  y  ->  ( S `  k )  =  ( S `  y ) )
32 fveq2 5865 . . . . . . . . 9  |-  ( k  =  y  ->  ( T `  k )  =  ( T `  y ) )
3331, 32sseq12d 3533 . . . . . . . 8  |-  ( k  =  y  ->  (
( S `  k
)  C_  ( T `  k )  <->  ( S `  y )  C_  ( T `  y )
) )
3433rspcv 3210 . . . . . . 7  |-  ( y  e.  I  ->  ( A. k  e.  I 
( S `  k
)  C_  ( T `  k )  ->  ( S `  y )  C_  ( T `  y
) ) )
3521, 30, 34sylc 60 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  y
)  C_  ( T `  y ) )
3627, 1cntz2ss 16172 . . . . . 6  |-  ( ( ( T `  y
)  C_  ( Base `  G )  /\  ( S `  y )  C_  ( T `  y
) )  ->  (
(Cntz `  G ) `  ( T `  y
) )  C_  (
(Cntz `  G ) `  ( S `  y
) ) )
3729, 35, 36syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( (Cntz `  G
) `  ( T `  y ) )  C_  ( (Cntz `  G ) `  ( S `  y
) ) )
3823, 37sstrd 3514 . . . 4  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
) )
3917, 38sstrd 3514 . . 3  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
) )
406adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  Grp )
4127subgacs 16038 . . . . . . 7  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
42 acsmre 14906 . . . . . . 7  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
4340, 41, 423syl 20 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
44 difss 3631 . . . . . . . . 9  |-  ( I 
\  { x }
)  C_  I
4511adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  A. k  e.  I  ( S `  k )  C_  ( T `  k )
)
46 ssralv 3564 . . . . . . . . 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 63 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  A. k  e.  ( I  \  {
x } ) ( S `  k ) 
C_  ( T `  k ) )
48 ss2iun 4341 . . . . . . . 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 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k ) 
C_  U_ k  e.  ( I  \  { x } ) ( T `
 k ) )
509adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  S : I --> (SubGrp `  G ) )
51 ffun 5732 . . . . . . . 8  |-  ( S : I --> (SubGrp `  G )  ->  Fun  S )
52 funiunfv 6147 . . . . . . . 8  |-  ( Fun 
S  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k )  =  U. ( S
" ( I  \  { x } ) ) )
5350, 51, 523syl 20 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k )  =  U. ( S
" ( I  \  { x } ) ) )
5424adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  T : I --> (SubGrp `  G ) )
55 ffun 5732 . . . . . . . 8  |-  ( T : I --> (SubGrp `  G )  ->  Fun  T )
56 funiunfv 6147 . . . . . . . 8  |-  ( Fun 
T  ->  U_ k  e.  ( I  \  {
x } ) ( T `  k )  =  U. ( T
" ( I  \  { x } ) ) )
5754, 55, 563syl 20 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  U_ k  e.  ( I  \  {
x } ) ( T `  k )  =  U. ( T
" ( I  \  { x } ) ) )
5849, 53, 573sstr3d 3546 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  U. ( T " ( I  \  { x } ) ) )
59 imassrn 5347 . . . . . . . 8  |-  ( T
" ( I  \  { x } ) )  C_  ran  T
60 frn 5736 . . . . . . . . . 10  |-  ( T : I --> (SubGrp `  G )  ->  ran  T 
C_  (SubGrp `  G )
)
6154, 60syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  ran  T 
C_  (SubGrp `  G )
)
62 mresspw 14846 . . . . . . . . . 10  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
6343, 62syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
6461, 63sstrd 3514 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  ran  T 
C_  ~P ( Base `  G
) )
6559, 64syl5ss 3515 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( T " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
66 sspwuni 4411 . . . . . . 7  |-  ( ( T " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( T " (
I  \  { x } ) )  C_  ( Base `  G )
)
6765, 66sylib 196 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  U. ( T " ( I  \  { x } ) )  C_  ( Base `  G ) )
6843, 3, 58, 67mrcssd 14878 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( T " ( I 
\  { x }
) ) ) )
69 ss2in 3725 . . . . 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 661 . . . 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 465 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  G dom DProd  T )
727adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  dom  T  =  I )
73 simpr 461 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
7471, 72, 73, 2, 3dprddisj 16842 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( T `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( T
" ( I  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
7570, 74sseqtrd 3540 . . 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 16830 . 2  |-  ( ph  ->  G dom DProd  S )
774a1d 25 . . . . 5  |-  ( ph  ->  ( G dom DProd  S  ->  G dom DProd  T ) )
78 ss2ixp 7482 . . . . . . 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 16 . . . . . 6  |-  ( ph  -> 
X_ k  e.  I 
( S `  k
)  C_  X_ k  e.  I  ( T `  k ) )
80 rabss2 3583 . . . . . 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 3565 . . . . . 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 20 . . . . 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 563 . . . 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 5734 . . . . 5  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
85 eqid 2467 . . . . . 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 16835 . . . . 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 20 . . . 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 2467 . . . . . 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 16835 . . . . 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 16 . . . 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 268 . . 3  |-  ( ph  ->  ( a  e.  ( G DProd  S )  -> 
a  e.  ( G DProd 
T ) ) )
9291ssrdv 3510 . 2  |-  ( ph  ->  ( G DProd  S ) 
C_  ( G DProd  T
) )
9376, 92jca 532 1  |-  ( ph  ->  ( G dom DProd  S  /\  ( G DProd  S )  C_  ( G DProd  T ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    \ cdif 3473    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   {csn 4027   U.cuni 4245   U_ciun 4325   class class class wbr 4447   dom cdm 4999   ran crn 5000   "cima 5002   Fun wfun 5581   -->wf 5583   ` cfv 5587  (class class class)co 6283   X_cixp 7469   finSupp cfsupp 7828   Basecbs 14489   0gc0g 14694    gsumg cgsu 14695  Moorecmre 14836  mrClscmrc 14837  ACScacs 14839   Grpcgrp 15726  SubGrpcsubg 15997  Cntzccntz 16155   DProd cdprd 16824
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-0g 14696  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-grp 15864  df-minusg 15865  df-subg 16000  df-cntz 16157  df-dprd 16826
This theorem is referenced by: (None)
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