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Theorem elcntz 15113
Description: Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
cntzfval.b  |-  B  =  ( Base `  M
)
cntzfval.p  |-  .+  =  ( +g  `  M )
cntzfval.z  |-  Z  =  (Cntz `  M )
Assertion
Ref Expression
elcntz  |-  ( S 
C_  B  ->  ( A  e.  ( Z `  S )  <->  ( A  e.  B  /\  A. y  e.  S  ( A  .+  y )  =  ( y  .+  A ) ) ) )
Distinct variable groups:    y,  .+    y, A    y, M    y, S
Allowed substitution hints:    B( y)    Z( y)

Proof of Theorem elcntz
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cntzfval.b . . . 4  |-  B  =  ( Base `  M
)
2 cntzfval.p . . . 4  |-  .+  =  ( +g  `  M )
3 cntzfval.z . . . 4  |-  Z  =  (Cntz `  M )
41, 2, 3cntzval 15112 . . 3  |-  ( S 
C_  B  ->  ( Z `  S )  =  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } )
54eleq2d 2502 . 2  |-  ( S 
C_  B  ->  ( A  e.  ( Z `  S )  <->  A  e.  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } ) )
6 oveq1 6080 . . . . 5  |-  ( x  =  A  ->  (
x  .+  y )  =  ( A  .+  y ) )
7 oveq2 6081 . . . . 5  |-  ( x  =  A  ->  (
y  .+  x )  =  ( y  .+  A ) )
86, 7eqeq12d 2449 . . . 4  |-  ( x  =  A  ->  (
( x  .+  y
)  =  ( y 
.+  x )  <->  ( A  .+  y )  =  ( y  .+  A ) ) )
98ralbidv 2717 . . 3  |-  ( x  =  A  ->  ( A. y  e.  S  ( x  .+  y )  =  ( y  .+  x )  <->  A. y  e.  S  ( A  .+  y )  =  ( y  .+  A ) ) )
109elrab 3084 . 2  |-  ( A  e.  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) }  <->  ( A  e.  B  /\  A. y  e.  S  ( A  .+  y )  =  ( y  .+  A ) ) )
115, 10syl6bb 253 1  |-  ( S 
C_  B  ->  ( A  e.  ( Z `  S )  <->  ( A  e.  B  /\  A. y  e.  S  ( A  .+  y )  =  ( y  .+  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701    C_ wss 3312   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521  Cntzccntz 15106
This theorem is referenced by:  cntzel  15114  cntzi  15120  resscntz  15122  cntzsubm  15126  cntzmhm  15129  oppgcntz  15152  dprdfcntz  15565
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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