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Theorem cntri 15121
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 2435 . . . . 5  |-  (Cntz `  M )  =  (Cntz `  M )
42, 3cntrval 15110 . . . 4  |-  ( (Cntz `  M ) `  B
)  =  (Cntr `  M )
51, 4eqtr4i 2458 . . 3  |-  Z  =  ( (Cntz `  M
) `  B )
65eleq2i 2499 . 2  |-  ( X  e.  Z  <->  X  e.  ( (Cntz `  M ) `  B ) )
7 cntri.p . . 3  |-  .+  =  ( +g  `  M )
87, 3cntzi 15120 . 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 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521  Cntzccntz 15106  Cntrccntr 15107
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  df-cntr 15109
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