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Theorem cntz2ss 16979
Description: Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.)
Hypotheses
Ref Expression
cntzrec.b  |-  B  =  ( Base `  M
)
cntzrec.z  |-  Z  =  (Cntz `  M )
Assertion
Ref Expression
cntz2ss  |-  ( ( S  C_  B  /\  T  C_  S )  -> 
( Z `  S
)  C_  ( Z `  T ) )

Proof of Theorem cntz2ss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2423 . . . . . 6  |-  ( +g  `  M )  =  ( +g  `  M )
2 cntzrec.z . . . . . 6  |-  Z  =  (Cntz `  M )
31, 2cntzi 16976 . . . . 5  |-  ( ( x  e.  ( Z `
 S )  /\  y  e.  S )  ->  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M ) x ) )
43ralrimiva 2840 . . . 4  |-  ( x  e.  ( Z `  S )  ->  A. y  e.  S  ( x
( +g  `  M ) y )  =  ( y ( +g  `  M
) x ) )
5 ssralv 3526 . . . . 5  |-  ( T 
C_  S  ->  ( A. y  e.  S  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M ) x )  ->  A. y  e.  T  ( x
( +g  `  M ) y )  =  ( y ( +g  `  M
) x ) ) )
65adantl 468 . . . 4  |-  ( ( S  C_  B  /\  T  C_  S )  -> 
( A. y  e.  S  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M
) x )  ->  A. y  e.  T  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M ) x ) ) )
74, 6syl5 34 . . 3  |-  ( ( S  C_  B  /\  T  C_  S )  -> 
( x  e.  ( Z `  S )  ->  A. y  e.  T  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M ) x ) ) )
87ralrimiv 2838 . 2  |-  ( ( S  C_  B  /\  T  C_  S )  ->  A. x  e.  ( Z `  S ) A. y  e.  T  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M ) x ) )
9 cntzrec.b . . . 4  |-  B  =  ( Base `  M
)
109, 2cntzssv 16975 . . 3  |-  ( Z `
 S )  C_  B
11 sstr 3473 . . . 4  |-  ( ( T  C_  S  /\  S  C_  B )  ->  T  C_  B )
1211ancoms 455 . . 3  |-  ( ( S  C_  B  /\  T  C_  S )  ->  T  C_  B )
139, 1, 2sscntz 16973 . . 3  |-  ( ( ( Z `  S
)  C_  B  /\  T  C_  B )  -> 
( ( Z `  S )  C_  ( Z `  T )  <->  A. x  e.  ( Z `
 S ) A. y  e.  T  (
x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x ) ) )
1410, 12, 13sylancr 668 . 2  |-  ( ( S  C_  B  /\  T  C_  S )  -> 
( ( Z `  S )  C_  ( Z `  T )  <->  A. x  e.  ( Z `
 S ) A. y  e.  T  (
x ( +g  `  M
) y )  =  ( y ( +g  `  M ) x ) ) )
158, 14mpbird 236 1  |-  ( ( S  C_  B  /\  T  C_  S )  -> 
( Z `  S
)  C_  ( Z `  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869   A.wral 2776    C_ wss 3437   ` cfv 5599  (class class class)co 6303   Basecbs 15114   +g cplusg 15183  Cntzccntz 16962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-cntz 16964
This theorem is referenced by:  cntzidss  16984  gsumzadd  17548  dprdfadd  17646  dprdss  17655  dprd2da  17668  dmdprdsplit2lem  17671  cntzsdrg  36032
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