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Theorem cntz2ss 14808
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 2283 . . . . . 6  |-  ( +g  `  M )  =  ( +g  `  M )
2 cntzrec.z . . . . . 6  |-  Z  =  (Cntz `  M )
31, 2cntzi 14805 . . . . 5  |-  ( ( x  e.  ( Z `
 S )  /\  y  e.  S )  ->  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M ) x ) )
43ralrimiva 2626 . . . 4  |-  ( x  e.  ( Z `  S )  ->  A. y  e.  S  ( x
( +g  `  M ) y )  =  ( y ( +g  `  M
) x ) )
5 ssralv 3237 . . . . 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 452 . . . 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 28 . . 3  |-  ( ( S  C_  B  /\  T  C_  S )  -> 
( x  e.  ( Z `  S )  ->  A. y  e.  T  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M ) x ) ) )
87ralrimiv 2625 . 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 14804 . . 3  |-  ( Z `
 S )  C_  B
11 sstr 3187 . . . 4  |-  ( ( T  C_  S  /\  S  C_  B )  ->  T  C_  B )
1211ancoms 439 . . 3  |-  ( ( S  C_  B  /\  T  C_  S )  ->  T  C_  B )
139, 1, 2sscntz 14802 . . 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 644 . 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 223 1  |-  ( ( S  C_  B  /\  T  C_  S )  -> 
( Z `  S
)  C_  ( Z `  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Cntzccntz 14791
This theorem is referenced by:  cntzidss  14813  gsumzadd  15204  dprdfadd  15255  dprdss  15264  dprd2da  15277  dmdprdsplit2lem  15280  cntzsdrg  27510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-cntz 14793
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