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Theorem cntri 15058
Description: Defining property of the center of a group. (Contributed by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
cntri.b  |-  B  =  ( Base `  M
)
cntri.p  |-  .+  =  ( +g  `  M )
cntri.z  |-  Z  =  (Cntr `  M )
Assertion
Ref Expression
cntri  |-  ( ( X  e.  Z  /\  Y  e.  B )  ->  ( X  .+  Y
)  =  ( Y 
.+  X ) )

Proof of Theorem cntri
StepHypRef Expression
1 cntri.z . . . 4  |-  Z  =  (Cntr `  M )
2 cntri.b . . . . 5  |-  B  =  ( Base `  M
)
3 eqid 2389 . . . . 5  |-  (Cntz `  M )  =  (Cntz `  M )
42, 3cntrval 15047 . . . 4  |-  ( (Cntz `  M ) `  B
)  =  (Cntr `  M )
51, 4eqtr4i 2412 . . 3  |-  Z  =  ( (Cntz `  M
) `  B )
65eleq2i 2453 . 2  |-  ( X  e.  Z  <->  X  e.  ( (Cntz `  M ) `  B ) )
7 cntri.p . . 3  |-  .+  =  ( +g  `  M )
87, 3cntzi 15057 . 2  |-  ( ( X  e.  ( (Cntz `  M ) `  B
)  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X
) )
96, 8sylanb 459 1  |-  ( ( X  e.  Z  /\  Y  e.  B )  ->  ( X  .+  Y
)  =  ( Y 
.+  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   ` cfv 5396  (class class class)co 6022   Basecbs 13398   +g cplusg 13458  Cntzccntz 15043  Cntrccntr 15044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-cntz 15045  df-cntr 15046
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