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Theorem cntz2ss 15123
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 2435 . . . . . 6  |-  ( +g  `  M )  =  ( +g  `  M )
2 cntzrec.z . . . . . 6  |-  Z  =  (Cntz `  M )
31, 2cntzi 15120 . . . . 5  |-  ( ( x  e.  ( Z `
 S )  /\  y  e.  S )  ->  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M ) x ) )
43ralrimiva 2781 . . . 4  |-  ( x  e.  ( Z `  S )  ->  A. y  e.  S  ( x
( +g  `  M ) y )  =  ( y ( +g  `  M
) x ) )
5 ssralv 3399 . . . . 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 453 . . . 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 30 . . 3  |-  ( ( S  C_  B  /\  T  C_  S )  -> 
( x  e.  ( Z `  S )  ->  A. y  e.  T  ( x ( +g  `  M ) y )  =  ( y ( +g  `  M ) x ) ) )
87ralrimiv 2780 . 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 15119 . . 3  |-  ( Z `
 S )  C_  B
11 sstr 3348 . . . 4  |-  ( ( T  C_  S  /\  S  C_  B )  ->  T  C_  B )
1211ancoms 440 . . 3  |-  ( ( S  C_  B  /\  T  C_  S )  ->  T  C_  B )
139, 1, 2sscntz 15117 . . 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 645 . 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 224 1  |-  ( ( S  C_  B  /\  T  C_  S )  -> 
( Z `  S
)  C_  ( Z `  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521  Cntzccntz 15106
This theorem is referenced by:  cntzidss  15128  gsumzadd  15519  dprdfadd  15570  dprdss  15579  dprd2da  15592  dmdprdsplit2lem  15595  cntzsdrg  27478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-cntz 15108
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