Theorem cntz2ss 14162

Description: Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.)

Hypotheses

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

Assertion

Ref	Expression
cntz2ss	|- ( ( S C_ B /\ T C_ S ) -> ( Z ` S ) C_ ( Z ` T ) )

Proof of Theorem cntz2ss

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

Step	Hyp	Ref	Expression
1		eqid 2238	. . . . . 6 |- ( +g ` M ) = ( +g ` M )
2		cntzrec.z	. . . . . 6 |- Z = (Cntz ` M )
3	1, 2	cntzi 14156	. . . . 5 |- ( ( x e. ( Z ` S ) /\ y e. S ) -> ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) )
4	3	ralrimiva 2623	. . . 4 |- ( x e. ( Z ` S ) -> A. y e. S ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) )
5		ssralv 3312	. . . . 5 |- ( T C_ S -> ( A. y e. S ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) -> A. y e. T ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) ) )
6	5	adantl 277	. . . 4 |- ( ( S C_ B /\ T C_ S ) -> ( A. y e. S ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) -> A. y e. T ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) ) )
7	4, 6	syl5 32	. . 3 |- ( ( S C_ B /\ T C_ S ) -> ( x e. ( Z ` S ) -> A. y e. T ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) ) )
8	7	ralrimiv 2622	. 2 |- ( ( S C_ B /\ T C_ S ) -> A. x e. ( Z ` S ) A. y e. T ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) )
9		cntzrec.b	. . . 4 |- B = ( Base ` M )
10	9, 2	cntzssv 14154	. . 3 |- ( Z ` S ) C_ B
11		sstr 3256	. . . 4 |- ( ( T C_ S /\ S C_ B ) -> T C_ B )
12	11	ancoms 268	. . 3 |- ( ( S C_ B /\ T C_ S ) -> T C_ B )
13	9, 1, 2	sscntz 14152	. . 3 |- ( ( ( Z ` S ) C_ B /\ T C_ B ) -> ( ( Z ` S ) C_ ( Z ` T ) <-> A. x e. ( Z ` S ) A. y e. T ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) ) )
14	10, 12, 13	sylancr 418	. 2 |- ( ( S C_ B /\ T C_ S ) -> ( ( Z ` S ) C_ ( Z ` T ) <-> A. x e. ( Z ` S ) A. y e. T ( x ( +g ` M ) y ) = ( y ( +g ` M ) x ) ) )
15	8, 14	mpbird 167	1 |- ( ( S C_ B /\ T C_ S ) -> ( Z ` S ) C_ ( Z ` T ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   ` cfv 5377  (class class class)co 6085   Basecbs 13404   +g cplusg 13484  Cntzccntz 14140

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 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8271  ax-resscn 8272  ax-1re 8274  ax-addrcl 8277

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 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-inn 9308  df-ndx 13407  df-slot 13408  df-base 13410  df-cntz 14142

This theorem is used by:  cntzidss  14166