Theorem cntri 17763

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

Step	Hyp	Ref	Expression
1		cntri.z	. . . 4 |- Z = (Cntr ` M )
2		cntri.b	. . . . 5 |- B = ( Base ` M )
3		eqid 2622	. . . . 5 |- (Cntz ` M ) = (Cntz ` M )
4	2, 3	cntrval 17752	. . . 4 |- ( (Cntz ` M ) ` B ) = (Cntr ` M )
5	1, 4	eqtr4i 2647	. . 3 |- Z = ( (Cntz ` M ) ` B )
6	5	eleq2i 2693	. 2 |- ( X e. Z <-> X e. ( (Cntz ` M ) ` B ) )
7		cntri.p	. . 3 |- .+ = ( +g ` M )
8	7, 3	cntzi 17762	. 2 |- ( ( X e. ( (Cntz ` M ) ` B ) /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )
9	6, 8	sylanb 489	1 |- ( ( X e. Z /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Cntzccntz 17748  Cntrccntr 17749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-cntz 17750  df-cntr 17751

This theorem is referenced by: (None)