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Theorem cntzcmn 15459
Description: The centralizer of any subset in a commutative monoid is the whole monoid. (Contributed by Mario Carneiro, 3-Oct-2015.)
Hypotheses
Ref Expression
cntzcmn.b  |-  B  =  ( Base `  G
)
cntzcmn.z  |-  Z  =  (Cntz `  G )
Assertion
Ref Expression
cntzcmn  |-  ( ( G  e. CMnd  /\  S  C_  B )  ->  ( Z `  S )  =  B )

Proof of Theorem cntzcmn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cntzcmn.b . . . 4  |-  B  =  ( Base `  G
)
2 cntzcmn.z . . . 4  |-  Z  =  (Cntz `  G )
31, 2cntzssv 15127 . . 3  |-  ( Z `
 S )  C_  B
43a1i 11 . 2  |-  ( ( G  e. CMnd  /\  S  C_  B )  ->  ( Z `  S )  C_  B )
5 simpl1 960 . . . . . . 7  |-  ( ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  /\  y  e.  S )  ->  G  e. CMnd )
6 simpl3 962 . . . . . . 7  |-  ( ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  /\  y  e.  S )  ->  x  e.  B )
7 simp2 958 . . . . . . . 8  |-  ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  ->  S  C_  B )
87sselda 3348 . . . . . . 7  |-  ( ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  /\  y  e.  S )  ->  y  e.  B )
9 eqid 2436 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
101, 9cmncom 15428 . . . . . . 7  |-  ( ( G  e. CMnd  /\  x  e.  B  /\  y  e.  B )  ->  (
x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) )
115, 6, 8, 10syl3anc 1184 . . . . . 6  |-  ( ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  /\  y  e.  S )  ->  (
x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) )
1211ralrimiva 2789 . . . . 5  |-  ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  ->  A. y  e.  S  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) )
131, 9, 2cntzel 15122 . . . . . 6  |-  ( ( S  C_  B  /\  x  e.  B )  ->  ( x  e.  ( Z `  S )  <->  A. y  e.  S  ( x ( +g  `  G ) y )  =  ( y ( +g  `  G ) x ) ) )
14133adant1 975 . . . . 5  |-  ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  ->  (
x  e.  ( Z `
 S )  <->  A. y  e.  S  ( x
( +g  `  G ) y )  =  ( y ( +g  `  G
) x ) ) )
1512, 14mpbird 224 . . . 4  |-  ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  ->  x  e.  ( Z `  S
) )
16153expia 1155 . . 3  |-  ( ( G  e. CMnd  /\  S  C_  B )  ->  (
x  e.  B  ->  x  e.  ( Z `  S ) ) )
1716ssrdv 3354 . 2  |-  ( ( G  e. CMnd  /\  S  C_  B )  ->  B  C_  ( Z `  S
) )
184, 17eqssd 3365 1  |-  ( ( G  e. CMnd  /\  S  C_  B )  ->  ( Z `  S )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529  Cntzccntz 15114  CMndccmn 15412
This theorem is referenced by:  ablcntzd  15472  cntzcmnf  15515  gsumadd  15528
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-cntz 15116  df-cmn 15414
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