Theorem mendval 37753

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 3212	. 2 |- ( M e. X -> M e. _V )
2		oveq12 6659	. . . . . . 7 |- ( ( m = M /\ m = M ) -> ( m LMHom m ) = ( M LMHom M ) )
3	2	anidms 677	. . . . . 6 |- ( m = M -> ( m LMHom m ) = ( M LMHom M ) )
4		mendval.b	. . . . . 6 |- B = ( M LMHom M )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( m = M -> ( m LMHom m ) = B )
6	5	csbeq1d 3540	. . . 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 6678	. . . . . 6 |- ( m LMHom m ) e. _V
8	5, 7	syl6eqelr 2710	. . . . 5 |- ( m = M -> B e. _V )
9		simpr 477	. . . . . . . 8 |- ( ( m = M /\ b = B ) -> b = B )
10	9	opeq2d 4409	. . . . . . 7 |- ( ( m = M /\ b = B ) -> <. ( Base ` ndx ) , b >. = <. ( Base ` ndx ) , B >. )
11		fveq2 6191	. . . . . . . . . . . 12 |- ( m = M -> ( +g ` m ) = ( +g ` M ) )
12		ofeq 6899	. . . . . . . . . . . 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 6674	. . . . . . . . . 10 |- ( ( m = M /\ b = B ) -> ( x oF ( +g ` m ) y ) = ( x oF ( +g ` M ) y ) )
15	9, 9, 14	mpt2eq123dv 6717	. . . . . . . . 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 2674	. . . . . . . 8 |- ( ( m = M /\ b = B ) -> ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) = .+ )
18	17	opeq2d 4409	. . . . . . 7 |- ( ( m = M /\ b = B ) -> <. ( +g ` ndx ) , ( x e. b , y e. b |-> ( x oF ( +g ` m ) y ) ) >. = <. ( +g ` ndx ) , .+ >. )
19		eqidd 2623	. . . . . . . . . 10 |- ( ( m = M /\ b = B ) -> ( x o. y ) = ( x o. y ) )
20	9, 9, 19	mpt2eq123dv 6717	. . . . . . . . 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 2674	. . . . . . . 8 |- ( ( m = M /\ b = B ) -> ( x e. b , y e. b |-> ( x o. y ) ) = .X. )
23	22	opeq2d 4409	. . . . . . 7 |- ( ( m = M /\ b = B ) -> <. ( .r ` ndx ) , ( x e. b , y e. b |-> ( x o. y ) ) >. = <. ( .r ` ndx ) , .X. >. )
24	10, 18, 23	tpeq123d 4283	. . . . . 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 6191	. . . . . . . . . 10 |- ( m = M -> (Scalar ` m ) = (Scalar ` M ) )
26	25	adantr 481	. . . . . . . . 9 |- ( ( m = M /\ b = B ) -> (Scalar ` m ) = (Scalar ` M ) )
27		mendval.s	. . . . . . . . 9 |- S = (Scalar ` M )
28	26, 27	syl6eqr 2674	. . . . . . . 8 |- ( ( m = M /\ b = B ) -> (Scalar ` m ) = S )
29	28	opeq2d 4409	. . . . . . 7 |- ( ( m = M /\ b = B ) -> <. (Scalar ` ndx ) , (Scalar ` m ) >. = <. (Scalar ` ndx ) , S >. )
30	28	fveq2d 6195	. . . . . . . . . 10 |- ( ( m = M /\ b = B ) -> ( Base ` (Scalar ` m ) ) = ( Base ` S ) )
31		fveq2 6191	. . . . . . . . . . . . 13 |- ( m = M -> ( .s ` m ) = ( .s ` M ) )
32	31	adantr 481	. . . . . . . . . . . 12 |- ( ( m = M /\ b = B ) -> ( .s ` m ) = ( .s ` M ) )
33		ofeq 6899	. . . . . . . . . . . 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 6191	. . . . . . . . . . . . 13 |- ( m = M -> ( Base ` m ) = ( Base ` M ) )
36	35	adantr 481	. . . . . . . . . . . 12 |- ( ( m = M /\ b = B ) -> ( Base ` m ) = ( Base ` M ) )
37	36	xpeq1d 5138	. . . . . . . . . . 11 |- ( ( m = M /\ b = B ) -> ( ( Base ` m ) X. { x } ) = ( ( Base ` M ) X. { x } ) )
38		eqidd 2623	. . . . . . . . . . 11 |- ( ( m = M /\ b = B ) -> y = y )
39	34, 37, 38	oveq123d 6671	. . . . . . . . . 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 6717	. . . . . . . . 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 2674	. . . . . . . 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 4409	. . . . . . 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 4276	. . . . . 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 3768	. . . . 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 3560	. . . 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 2656	. . 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 37746	. . 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 6957	. . . 4 |- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } e. _V
50		prex 4909	. . . 4 |- { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } e. _V
51	49, 50	unex 6956	. . 3 |- ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } ) e. _V
52	47, 48, 51	fvmpt 6282	. 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 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   u. cun 3572   {csn 4177   {cpr 4179   {ctp 4181   <.cop 4183   X. cxp 5112   o. ccom 5118   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   LMHom clmhm 19019  MEndocmend 37745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-mend 37746

This theorem is referenced by:  mendbas  37754  mendplusgfval  37755  mendmulrfval  37757  mendsca  37759  mendvscafval  37760