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Theorem cntzcmn 15136
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 14804 . . 3  |-  ( Z `
 S )  C_  B
43a1i 10 . 2  |-  ( ( G  e. CMnd  /\  S  C_  B )  ->  ( Z `  S )  C_  B )
5 simpl1 958 . . . . . . 7  |-  ( ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  /\  y  e.  S )  ->  G  e. CMnd )
6 simpl3 960 . . . . . . 7  |-  ( ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  /\  y  e.  S )  ->  x  e.  B )
7 simp2 956 . . . . . . . 8  |-  ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  ->  S  C_  B )
87sselda 3180 . . . . . . 7  |-  ( ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  /\  y  e.  S )  ->  y  e.  B )
9 eqid 2283 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
101, 9cmncom 15105 . . . . . . 7  |-  ( ( G  e. CMnd  /\  x  e.  B  /\  y  e.  B )  ->  (
x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) )
115, 6, 8, 10syl3anc 1182 . . . . . 6  |-  ( ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  /\  y  e.  S )  ->  (
x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) )
1211ralrimiva 2626 . . . . 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 14799 . . . . . 6  |-  ( ( S  C_  B  /\  x  e.  B )  ->  ( x  e.  ( Z `  S )  <->  A. y  e.  S  ( x ( +g  `  G ) y )  =  ( y ( +g  `  G ) x ) ) )
14133adant1 973 . . . . 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 223 . . . 4  |-  ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  ->  x  e.  ( Z `  S
) )
16153expia 1153 . . 3  |-  ( ( G  e. CMnd  /\  S  C_  B )  ->  (
x  e.  B  ->  x  e.  ( Z `  S ) ) )
1716ssrdv 3185 . 2  |-  ( ( G  e. CMnd  /\  S  C_  B )  ->  B  C_  ( Z `  S
) )
184, 17eqssd 3196 1  |-  ( ( G  e. CMnd  /\  S  C_  B )  ->  ( Z `  S )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = 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  CMndccmn 15089
This theorem is referenced by:  ablcntzd  15149  cntzcmnf  15192  gsumadd  15205
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  df-cmn 15091
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