Theorem cntzfval 14146

Description: First level substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Hypotheses

Ref	Expression
cntzfval.b	|- B = ( Base ` M )
cntzfval.p	|- .+ = ( +g ` M )
cntzfval.z	|- Z = (Cntz ` M )

Assertion

Ref	Expression
cntzfval	|- ( M e. V -> Z = ( s e. ~P B |-> { x e. B | A. y e. s ( x .+ y ) = ( y .+ x ) } ) )

Distinct variable groups:   x, s, y, .+   B, s, x   M, s, x, y

Allowed substitution hints:   B( y)   V( x, y, s)   Z( x, y, s)

Proof of Theorem cntzfval

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntzfval.z	. 2 |- Z = (Cntz ` M )
2		df-cntz 14142	. . 3 |- Cntz = ( m e. _V |-> ( s e. ~P ( Base ` m ) |-> { x e. ( Base ` m ) | A. y e. s ( x ( +g ` m ) y ) = ( y ( +g ` m ) x ) } ) )
3		fveq2 5695	. . . . . 6 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
4		cntzfval.b	. . . . . 6 |- B = ( Base ` M )
5	3, 4	eqtr4di 2289	. . . . 5 |- ( m = M -> ( Base ` m ) = B )
6	5	pweqd 3693	. . . 4 |- ( m = M -> ~P ( Base ` m ) = ~P B )
7		fveq2 5695	. . . . . . . . 9 |- ( m = M -> ( +g ` m ) = ( +g ` M ) )
8		cntzfval.p	. . . . . . . . 9 |- .+ = ( +g ` M )
9	7, 8	eqtr4di 2289	. . . . . . . 8 |- ( m = M -> ( +g ` m ) = .+ )
10	9	oveqd 6102	. . . . . . 7 |- ( m = M -> ( x ( +g ` m ) y ) = ( x .+ y ) )
11	9	oveqd 6102	. . . . . . 7 |- ( m = M -> ( y ( +g ` m ) x ) = ( y .+ x ) )
12	10, 11	eqeq12d 2253	. . . . . 6 |- ( m = M -> ( ( x ( +g ` m ) y ) = ( y ( +g ` m ) x ) <-> ( x .+ y ) = ( y .+ x ) ) )
13	12	ralbidv 2550	. . . . 5 |- ( m = M -> ( A. y e. s ( x ( +g ` m ) y ) = ( y ( +g ` m ) x ) <-> A. y e. s ( x .+ y ) = ( y .+ x ) ) )
14	5, 13	rabeqbidv 2816	. . . 4 |- ( m = M -> { x e. ( Base ` m ) | A. y e. s ( x ( +g ` m ) y ) = ( y ( +g ` m ) x ) } = { x e. B | A. y e. s ( x .+ y ) = ( y .+ x ) } )
15	6, 14	mpteq12dv 4213	. . 3 |- ( m = M -> ( s e. ~P ( Base ` m ) |-> { x e. ( Base ` m ) | A. y e. s ( x ( +g ` m ) y ) = ( y ( +g ` m ) x ) } ) = ( s e. ~P B |-> { x e. B | A. y e. s ( x .+ y ) = ( y .+ x ) } ) )
16		elex 2833	. . 3 |- ( M e. V -> M e. _V )
17		basfn 13463	. . . . . . 7 |- Base Fn _V
18		funfvex 5712	. . . . . . . 8 |- ( ( Fun Base /\ M e. dom Base ) -> ( Base ` M ) e. _V )
19	18	funfni 5483	. . . . . . 7 |- ( ( Base Fn _V /\ M e. _V ) -> ( Base ` M ) e. _V )
20	17, 16, 19	sylancr 418	. . . . . 6 |- ( M e. V -> ( Base ` M ) e. _V )
21	4, 20	eqeltrid 2325	. . . . 5 |- ( M e. V -> B e. _V )
22	21	pwexd 4318	. . . 4 |- ( M e. V -> ~P B e. _V )
23	22	mptexd 5944	. . 3 |- ( M e. V -> ( s e. ~P B |-> { x e. B | A. y e. s ( x .+ y ) = ( y .+ x ) } ) e. _V )
24	2, 15, 16, 23	fvmptd3 5799	. 2 |- ( M e. V -> (Cntz ` M ) = ( s e. ~P B |-> { x e. B | A. y e. s ( x .+ y ) = ( y .+ x ) } ) )
25	1, 24	eqtrid 2283	1 |- ( M e. V -> Z = ( s e. ~P B |-> { x e. B | A. y e. s ( x .+ y ) = ( y .+ x ) } ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   ~Pcpw 3688   |-> cmpt 4192   Fn wfn 5372   ` 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:  cntzval  14147  cntzrcl  14153