Theorem cntzrec 14169

Description: Reciprocity relationship for 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
cntzrec	|- ( ( S C_ B /\ T C_ B ) -> ( S C_ ( Z ` T ) <-> T C_ ( Z ` S ) ) )

Proof of Theorem cntzrec

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ralcom 2714	. . . 4 |- ( A. x e. S A. y e. T ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) <-> A. y e. T A. x e. S ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) )
2		eqcom 2240	. . . . 5 |- ( ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) <-> ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) )
3	2	2ralbii 2558	. . . 4 |- ( A. y e. T A. x e. S ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) <-> A. y e. T A. x e. S ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) )
4	1, 3	bitri 184	. . 3 |- ( A. x e. S A. y e. T ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) <-> A. y e. T A. x e. S ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) )
5	4	a1i 9	. 2 |- ( ( S C_ B /\ T C_ B ) -> ( A. x e. S A. y e. T ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) <-> A. y e. T A. x e. S ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) ) )
6		cntzrec.b	. . 3 |- B = ( Base ` M )
7		eqid 2238	. . 3 |- ( +g ` M ) = ( +g ` M )
8		cntzrec.z	. . 3 |- Z = (Cntz ` M )
9	6, 7, 8	sscntz 14158	. 2 |- ( ( S C_ B /\ T C_ B ) -> ( S C_ ( Z ` T ) <-> A. x e. S A. y e. T ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) ) )
10	6, 7, 8	sscntz 14158	. . 3 |- ( ( T C_ B /\ S C_ B ) -> ( T C_ ( Z ` S ) <-> A. y e. T A. x e. S ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) ) )
11	10	ancoms 268	. 2 |- ( ( S C_ B /\ T C_ B ) -> ( T C_ ( Z ` S ) <-> A. y e. T A. x e. S ( y ( +g ` M ) x ) = ( x ( +g ` M ) y ) ) )
12	5, 9, 11	3bitr4d 220	1 |- ( ( S C_ B /\ T C_ B ) -> ( S C_ ( Z ` T ) <-> T C_ ( Z ` S ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   A.wral 2528   C_ wss 3220   ` cfv 5380  (class class class)co 6088   Basecbs 13410   +g cplusg 13490  Cntzccntz 14146

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-inn 9314  df-ndx 13413  df-slot 13414  df-base 13416  df-cntz 14148

This theorem is used by: (None)