Theorem mendval 31336

Description: Value of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)

Hypotheses

Ref	Expression
mendval.b	|- B = ( M LMHom M )
mendval.p	|- .+ = ( x e. B , y e. B |-> ( x oF ( +g ` M ) y ) )
mendval.t	|- .X. = ( x e. B , y e. B |-> ( x o. y ) )
mendval.s	|- S = (Scalar ` M )
mendval.v	|- .x. = ( x e. ( Base ` S ) , y e. B |-> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) )

Assertion

Ref	Expression
mendval	|- ( M e. X -> (MEndo ` M ) = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } ) )

Distinct variable groups:   x, y, B   x, M, y

Allowed substitution hints:   .+ ( x, y)   S( x, y)   .x. ( x, y)   .X. ( x, y)   X( x, y)

Proof of Theorem mendval

Dummy variables m b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3118	. 2 |- ( M e. X -> M e. _V )
2		oveq12 6305	. . . . . . 7 |- ( ( m = M /\ m = M ) -> ( m LMHom m ) = ( M LMHom M ) )
3	2	anidms 645	. . . . . 6 |- ( m = M -> ( m LMHom m ) = ( M LMHom M ) )
4		mendval.b	. . . . . 6 |- B = ( M LMHom M )
5	3, 4	syl6eqr 2516	. . . . 5 |- ( m = M -> ( m LMHom m ) = B )
6	5	csbeq1d 3437	. . . 4 |- ( m = M -> [_ ( m LMHom m ) / b ]_ ( { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) >. , <. ( .r ` ndx ) , ( x e. b , y e. b |-> ( x o. y ) ) >. } u. { <. (Scalar ` ndx ) , (Scalar ` m ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` m ) ) , y e. b |-> ( ( ( Base ` m ) X. { x } ) oF ( .s ` m ) y ) ) >. } ) = [_ B / b ]_ ( { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) >. , <. ( .r ` ndx ) , ( x e. b , y e. b |-> ( x o. y ) ) >. } u. { <. (Scalar ` ndx ) , (Scalar ` m ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` m ) ) , y e. b |-> ( ( ( Base ` m ) X. { x } ) oF ( .s ` m ) y ) ) >. } ) )
7		ovex 6324	. . . . . 6 |- ( m LMHom m ) e. _V
8	5, 7	syl6eqelr 2554	. . . . 5 |- ( m = M -> B e. _V )
9		simpr 461	. . . . . . . 8 |- ( ( m = M /\ b = B ) -> b = B )
10	9	opeq2d 4226	. . . . . . 7 |- ( ( m = M /\ b = B ) -> <. ( Base ` ndx ) , b >. = <. ( Base ` ndx ) , B >. )
11		fveq2 5872	. . . . . . . . . . . 12 |- ( m = M -> ( +g ` m ) = ( +g ` M ) )
12		ofeq 6541	. . . . . . . . . . . 12 |- ( ( +g ` m ) = ( +g ` M ) -> oF ( +g ` m ) = oF ( +g ` M ) )
13	11, 12	syl 16	. . . . . . . . . . 11 |- ( m = M -> oF ( +g ` m ) = oF ( +g ` M ) )
14	13	oveqdr 6320	. . . . . . . . . 10 |- ( ( m = M /\ b = B ) -> ( x oF ( +g ` m ) y ) = ( x oF ( +g ` M ) y ) )
15	9, 9, 14	mpt2eq123dv 6358	. . . . . . . . 9 |- ( ( m = M /\ b = B ) -> ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) = ( x e. B , y e. B |-> ( x oF ( +g ` M ) y ) ) )
16		mendval.p	. . . . . . . . 9 |- .+ = ( x e. B , y e. B |-> ( x oF ( +g ` M ) y ) )
17	15, 16	syl6eqr 2516	. . . . . . . 8 |- ( ( m = M /\ b = B ) -> ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) = .+ )
18	17	opeq2d 4226	. . . . . . 7 |- ( ( m = M /\ b = B ) -> <. ( +g ` ndx ) , ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) >. = <. ( +g ` ndx ) , .+ >. )
19		eqidd 2458	. . . . . . . . . 10 |- ( ( m = M /\ b = B ) -> ( x o. y ) = ( x o. y ) )
20	9, 9, 19	mpt2eq123dv 6358	. . . . . . . . 9 |- ( ( m = M /\ b = B ) -> ( x e. b , y e. b |-> ( x o. y ) ) = ( x e. B , y e. B |-> ( x o. y ) ) )
21		mendval.t	. . . . . . . . 9 |- .X. = ( x e. B , y e. B |-> ( x o. y ) )
22	20, 21	syl6eqr 2516	. . . . . . . 8 |- ( ( m = M /\ b = B ) -> ( x e. b , y e. b |-> ( x o. y ) ) = .X. )
23	22	opeq2d 4226	. . . . . . 7 |- ( ( m = M /\ b = B ) -> <. ( .r ` ndx ) , ( x e. b , y e. b |-> ( x o. y ) ) >. = <. ( .r ` ndx ) , .X. >. )
24	10, 18, 23	tpeq123d 4126	. . . . . 6 |- ( ( m = M /\ b = B ) -> { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) >. , <. ( .r ` ndx ) , ( x e. b , y e. b |-> ( x o. y ) ) >. } = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } )
25		fveq2 5872	. . . . . . . . . 10 |- ( m = M -> (Scalar ` m ) = (Scalar ` M ) )
26	25	adantr 465	. . . . . . . . 9 |- ( ( m = M /\ b = B ) -> (Scalar ` m ) = (Scalar ` M ) )
27		mendval.s	. . . . . . . . 9 |- S = (Scalar ` M )
28	26, 27	syl6eqr 2516	. . . . . . . 8 |- ( ( m = M /\ b = B ) -> (Scalar ` m ) = S )
29	28	opeq2d 4226	. . . . . . 7 |- ( ( m = M /\ b = B ) -> <. (Scalar ` ndx ) , (Scalar ` m ) >. = <. (Scalar ` ndx ) , S >. )
30	28	fveq2d 5876	. . . . . . . . . 10 |- ( ( m = M /\ b = B ) -> ( Base ` (Scalar ` m ) ) = ( Base ` S ) )
31		fveq2 5872	. . . . . . . . . . . . 13 |- ( m = M -> ( .s ` m ) = ( .s ` M ) )
32	31	adantr 465	. . . . . . . . . . . 12 |- ( ( m = M /\ b = B ) -> ( .s ` m ) = ( .s ` M ) )
33		ofeq 6541	. . . . . . . . . . . 12 |- ( ( .s ` m ) = ( .s ` M ) -> oF ( .s ` m ) = oF ( .s ` M ) )
34	32, 33	syl 16	. . . . . . . . . . 11 |- ( ( m = M /\ b = B ) -> oF ( .s ` m ) = oF ( .s ` M ) )
35		fveq2 5872	. . . . . . . . . . . . 13 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
36	35	adantr 465	. . . . . . . . . . . 12 |- ( ( m = M /\ b = B ) -> ( Base ` m ) = ( Base ` M ) )
37	36	xpeq1d 5031	. . . . . . . . . . 11 |- ( ( m = M /\ b = B ) -> ( ( Base ` m ) X. { x } ) = ( ( Base ` M ) X. { x } ) )
38		eqidd 2458	. . . . . . . . . . 11 |- ( ( m = M /\ b = B ) -> y = y )
39	34, 37, 38	oveq123d 6317	. . . . . . . . . 10 |- ( ( m = M /\ b = B ) -> ( ( ( Base ` m ) X. { x } ) oF ( .s ` m ) y ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) )
40	30, 9, 39	mpt2eq123dv 6358	. . . . . . . . 9 |- ( ( m = M /\ b = B ) -> ( x e. ( Base ` (Scalar ` m ) ) , y e. b |-> ( ( ( Base ` m ) X. { x } ) oF ( .s ` m ) y ) ) = ( x e. ( Base ` S ) , y e. B |-> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) ) )
41		mendval.v	. . . . . . . . 9 |- .x. = ( x e. ( Base ` S ) , y e. B |-> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) )
42	40, 41	syl6eqr 2516	. . . . . . . 8 |- ( ( m = M /\ b = B ) -> ( x e. ( Base ` (Scalar ` m ) ) , y e. b |-> ( ( ( Base ` m ) X. { x } ) oF ( .s ` m ) y ) ) = .x. )
43	42	opeq2d 4226	. . . . . . 7 |- ( ( m = M /\ b = B ) -> <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` m ) ) , y e. b |-> ( ( ( Base ` m ) X. { x } ) oF ( .s ` m ) y ) ) >. = <. ( .s ` ndx ) , .x. >. )
44	29, 43	preq12d 4119	. . . . . 6 |- ( ( m = M /\ b = B ) -> { <. (Scalar ` ndx ) , (Scalar ` m ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` m ) ) , y e. b |-> ( ( ( Base ` m ) X. { x } ) oF ( .s ` m ) y ) ) >. } = { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } )
45	24, 44	uneq12d 3655	. . . . 5 |- ( ( m = M /\ b = B ) -> ( { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) >. , <. ( .r ` ndx ) , ( x e. b , y e. b |-> ( x o. y ) ) >. } u. { <. (Scalar ` ndx ) , (Scalar ` m ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` m ) ) , y e. b |-> ( ( ( Base ` m ) X. { x } ) oF ( .s ` m ) y ) ) >. } ) = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } ) )
46	8, 45	csbied 3457	. . . 4 |- ( m = M -> [_ B / b ]_ ( { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) >. , <. ( .r ` ndx ) , ( x e. b , y e. b |-> ( x o. y ) ) >. } u. { <. (Scalar ` ndx ) , (Scalar ` m ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` m ) ) , y e. b |-> ( ( ( Base ` m ) X. { x } ) oF ( .s ` m ) y ) ) >. } ) = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } ) )
47	6, 46	eqtrd 2498	. . 3 |- ( m = M -> [_ ( m LMHom m ) / b ]_ ( { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) >. , <. ( .r ` ndx ) , ( x e. b , y e. b |-> ( x o. y ) ) >. } u. { <. (Scalar ` ndx ) , (Scalar ` m ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` m ) ) , y e. b |-> ( ( ( Base ` m ) X. { x } ) oF ( .s ` m ) y ) ) >. } ) = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } ) )
48		df-mend 31329	. . 3 |- MEndo = ( m e. _V |-> [_ ( m LMHom m ) / b ]_ ( { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) >. , <. ( .r ` ndx ) , ( x e. b , y e. b |-> ( x o. y ) ) >. } u. { <. (Scalar ` ndx ) , (Scalar ` m ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` m ) ) , y e. b |-> ( ( ( Base ` m ) X. { x } ) oF ( .s ` m ) y ) ) >. } ) )
49		tpex 6598	. . . 4 |- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } e. _V
50		prex 4698	. . . 4 |- { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } e. _V
51	49, 50	unex 6597	. . 3 |- ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } ) e. _V
52	47, 48, 51	fvmpt 5956	. 2 |- ( M e. _V -> (MEndo ` M ) = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } ) )
53	1, 52	syl 16	1 |- ( M e. X -> (MEndo ` M ) = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1395   e. wcel 1819   _Vcvv 3109   [_csb 3430   u. cun 3469   {csn 4032   {cpr 4034   {ctp 4036   <.cop 4038   X. cxp 5006   o. ccom 5012   ` cfv 5594  (class class class)co 6296   |-> cmpt2 6298   oFcof 6537   ndxcnx 14641   Basecbs 14644   +g cplusg 14712   .rcmulr 14713  Scalarcsca 14715   .scvsca 14716   LMHom clmhm 17792  MEndocmend 31328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-mend 31329

This theorem is referenced by:  mendbas  31337  mendplusgfval  31338  mendmulrfval  31340  mendsca  31342  mendvscafval  31343