Theorem mendval 36094

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 3066	. 2 |- ( M e. X -> M e. _V )
2		oveq12 6324	. . . . . . 7 |- ( ( m = M /\ m = M ) -> ( m LMHom m ) = ( M LMHom M ) )
3	2	anidms 655	. . . . . 6 |- ( m = M -> ( m LMHom m ) = ( M LMHom M ) )
4		mendval.b	. . . . . 6 |- B = ( M LMHom M )
5	3, 4	syl6eqr 2514	. . . . 5 |- ( m = M -> ( m LMHom m ) = B )
6	5	csbeq1d 3382	. . . 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 6343	. . . . . 6 |- ( m LMHom m ) e. _V
8	5, 7	syl6eqelr 2549	. . . . 5 |- ( m = M -> B e. _V )
9		simpr 467	. . . . . . . 8 |- ( ( m = M /\ b = B ) -> b = B )
10	9	opeq2d 4187	. . . . . . 7 |- ( ( m = M /\ b = B ) -> <. ( Base ` ndx ) , b >. = <. ( Base ` ndx ) , B >. )
11		fveq2 5888	. . . . . . . . . . . 12 |- ( m = M -> ( +g ` m ) = ( +g ` M ) )
12		ofeq 6560	. . . . . . . . . . . 12 |- ( ( +g ` m ) = ( +g ` M ) -> oF ( +g ` m ) = oF ( +g ` M ) )
13	11, 12	syl 17	. . . . . . . . . . 11 |- ( m = M -> oF ( +g ` m ) = oF ( +g ` M ) )
14	13	oveqdr 6339	. . . . . . . . . 10 |- ( ( m = M /\ b = B ) -> ( x oF ( +g ` m ) y ) = ( x oF ( +g ` M ) y ) )
15	9, 9, 14	mpt2eq123dv 6380	. . . . . . . . 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 2514	. . . . . . . 8 |- ( ( m = M /\ b = B ) -> ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) = .+ )
18	17	opeq2d 4187	. . . . . . 7 |- ( ( m = M /\ b = B ) -> <. ( +g ` ndx ) , ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) >. = <. ( +g ` ndx ) , .+ >. )
19		eqidd 2463	. . . . . . . . . 10 |- ( ( m = M /\ b = B ) -> ( x o. y ) = ( x o. y ) )
20	9, 9, 19	mpt2eq123dv 6380	. . . . . . . . 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 2514	. . . . . . . 8 |- ( ( m = M /\ b = B ) -> ( x e. b , y e. b |-> ( x o. y ) ) = .X. )
23	22	opeq2d 4187	. . . . . . 7 |- ( ( m = M /\ b = B ) -> <. ( .r ` ndx ) , ( x e. b , y e. b |-> ( x o. y ) ) >. = <. ( .r ` ndx ) , .X. >. )
24	10, 18, 23	tpeq123d 4079	. . . . . 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 5888	. . . . . . . . . 10 |- ( m = M -> (Scalar ` m ) = (Scalar ` M ) )
26	25	adantr 471	. . . . . . . . 9 |- ( ( m = M /\ b = B ) -> (Scalar ` m ) = (Scalar ` M ) )
27		mendval.s	. . . . . . . . 9 |- S = (Scalar ` M )
28	26, 27	syl6eqr 2514	. . . . . . . 8 |- ( ( m = M /\ b = B ) -> (Scalar ` m ) = S )
29	28	opeq2d 4187	. . . . . . 7 |- ( ( m = M /\ b = B ) -> <. (Scalar ` ndx ) , (Scalar ` m ) >. = <. (Scalar ` ndx ) , S >. )
30	28	fveq2d 5892	. . . . . . . . . 10 |- ( ( m = M /\ b = B ) -> ( Base ` (Scalar ` m ) ) = ( Base ` S ) )
31		fveq2 5888	. . . . . . . . . . . . 13 |- ( m = M -> ( .s ` m ) = ( .s ` M ) )
32	31	adantr 471	. . . . . . . . . . . 12 |- ( ( m = M /\ b = B ) -> ( .s ` m ) = ( .s ` M ) )
33		ofeq 6560	. . . . . . . . . . . 12 |- ( ( .s ` m ) = ( .s ` M ) -> oF ( .s ` m ) = oF ( .s ` M ) )
34	32, 33	syl 17	. . . . . . . . . . 11 |- ( ( m = M /\ b = B ) -> oF ( .s ` m ) = oF ( .s ` M ) )
35		fveq2 5888	. . . . . . . . . . . . 13 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
36	35	adantr 471	. . . . . . . . . . . 12 |- ( ( m = M /\ b = B ) -> ( Base ` m ) = ( Base ` M ) )
37	36	xpeq1d 4876	. . . . . . . . . . 11 |- ( ( m = M /\ b = B ) -> ( ( Base ` m ) X. { x } ) = ( ( Base ` M ) X. { x } ) )
38		eqidd 2463	. . . . . . . . . . 11 |- ( ( m = M /\ b = B ) -> y = y )
39	34, 37, 38	oveq123d 6336	. . . . . . . . . 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 6380	. . . . . . . . 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 2514	. . . . . . . 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 4187	. . . . . . 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 4072	. . . . . 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 3601	. . . . 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 3402	. . . 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 2496	. . 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 36087	. . 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 6617	. . . 4 |- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } e. _V
50		prex 4656	. . . 4 |- { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } e. _V
51	49, 50	unex 6616	. . 3 |- ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } ) e. _V
52	47, 48, 51	fvmpt 5971	. 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 17	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 375   = wceq 1455   e. wcel 1898   _Vcvv 3057   [_csb 3375   u. cun 3414   {csn 3980   {cpr 3982   {ctp 3984   <.cop 3986   X. cxp 4851   o. ccom 4857   ` cfv 5601  (class class class)co 6315   |-> cmpt2 6317   oFcof 6556   ndxcnx 15167   Basecbs 15170   +g cplusg 15239   .rcmulr 15240  Scalarcsca 15242   .scvsca 15243   LMHom clmhm 18291  MEndocmend 36086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653  ax-un 6610

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-of 6558  df-mend 36087

This theorem is referenced by:  mendbas  36095  mendplusgfval  36096  mendmulrfval  36098  mendsca  36100  mendvscafval  36101