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Theorem cntzcmn 15185
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 14853 . . 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 3214 . . . . . . 7  |-  ( ( ( G  e. CMnd  /\  S  C_  B  /\  x  e.  B )  /\  y  e.  S )  ->  y  e.  B )
9 eqid 2316 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
101, 9cmncom 15154 . . . . . . 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 2660 . . . . 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 14848 . . . . . 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 3219 . 2  |-  ( ( G  e. CMnd  /\  S  C_  B )  ->  B  C_  ( Z `  S
) )
184, 17eqssd 3230 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 1633    e. wcel 1701   A.wral 2577    C_ wss 3186   ` cfv 5292  (class class class)co 5900   Basecbs 13195   +g cplusg 13255  Cntzccntz 14840  CMndccmn 15138
This theorem is referenced by:  ablcntzd  15198  cntzcmnf  15241  gsumadd  15254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-cntz 14842  df-cmn 15140
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