Theorem cntziinsn 17767

Description: Express any centralizer as an intersection of singleton centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Hypotheses

Ref	Expression
cntzrec.b	|- B = ( Base ` M )
cntzrec.z	|- Z = (Cntz ` M )

Assertion

Ref	Expression
cntziinsn	|- ( S C_ B -> ( Z ` S ) = ( B i^i |^|_ x e. S ( Z ` { x } ) ) )

Distinct variable groups:   x, B   x, M   x, S   x, Z

Proof of Theorem cntziinsn

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntzrec.b	. . 3 |- B = ( Base ` M )
2		eqid 2622	. . 3 |- ( +g ` M ) = ( +g ` M )
3		cntzrec.z	. . 3 |- Z = (Cntz ` M )
4	1, 2, 3	cntzval 17754	. 2 |- ( S C_ B -> ( Z ` S ) = { y e. B | A. x e. S ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) } )
5		ssel2 3598	. . . . . 6 |- ( ( S C_ B /\ x e. S ) -> x e. B )
6	1, 2, 3	cntzsnval 17757	. . . . . 6 |- ( x e. B -> ( Z ` { x } ) = { y e. B | ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) } )
7	5, 6	syl 17	. . . . 5 |- ( ( S C_ B /\ x e. S ) -> ( Z ` { x } ) = { y e. B | ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) } )
8	7	iineq2dv 4543	. . . 4 |- ( S C_ B -> |^|_ x e. S ( Z ` { x } ) = |^|_ x e. S { y e. B | ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) } )
9	8	ineq2d 3814	. . 3 |- ( S C_ B -> ( B i^i |^|_ x e. S ( Z ` { x } ) ) = ( B i^i |^|_ x e. S { y e. B | ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) } ) )
10		riinrab 4596	. . 3 |- ( B i^i |^|_ x e. S { y e. B | ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) } ) = { y e. B | A. x e. S ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) }
11	9, 10	syl6eq 2672	. 2 |- ( S C_ B -> ( B i^i |^|_ x e. S ( Z ` { x } ) ) = { y e. B | A. x e. S ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) } )
12	4, 11	eqtr4d 2659	1 |- ( S C_ B -> ( Z ` S ) = ( B i^i |^|_ x e. S ( Z ` { x } ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   i^i cin 3573   C_ wss 3574   {csn 4177   |^|_ciin 4521   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Cntzccntz 17748

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-iin 4523  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

This theorem is referenced by: (None)