Theorem mendvscafval 31343

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

Hypotheses

Ref	Expression
mendvscafval.a	|- A = (MEndo ` M )
mendvscafval.v	|- .x. = ( .s ` M )
mendvscafval.b	|- B = ( Base ` A )
mendvscafval.s	|- S = (Scalar ` M )
mendvscafval.k	|- K = ( Base ` S )
mendvscafval.e	|- E = ( Base ` M )

Assertion

Ref	Expression
mendvscafval	|- ( .s ` A ) = ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) )

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

Allowed substitution hints:   A( x, y)   S( x, y)   .x. ( x, y)   E( x, y)

Proof of Theorem mendvscafval

Step	Hyp	Ref	Expression
1		mendvscafval.a	. . 3 |- A = (MEndo ` M )
2	1	fveq2i 5875	. 2 |- ( .s ` A ) = ( .s ` (MEndo ` M ) )
3		mendvscafval.b	. . . . . . 7 |- B = ( Base ` A )
4	1	mendbas 31337	. . . . . . 7 |- ( M LMHom M ) = ( Base ` A )
5	3, 4	eqtr4i 2489	. . . . . 6 |- B = ( M LMHom M )
6		eqid 2457	. . . . . 6 |- ( x e. B , y e. B |-> ( x oF ( +g ` M ) y ) ) = ( x e. B , y e. B |-> ( x oF ( +g ` M ) y ) )
7		eqid 2457	. . . . . 6 |- ( x e. B , y e. B |-> ( x o. y ) ) = ( x e. B , y e. B |-> ( x o. y ) )
8		mendvscafval.s	. . . . . 6 |- S = (Scalar ` M )
9		mendvscafval.k	. . . . . . 7 |- K = ( Base ` S )
10		eqid 2457	. . . . . . 7 |- B = B
11		mendvscafval.e	. . . . . . . . 9 |- E = ( Base ` M )
12	11	xpeq1i 5028	. . . . . . . 8 |- ( E X. { x } ) = ( ( Base ` M ) X. { x } )
13		eqid 2457	. . . . . . . 8 |- y = y
14		mendvscafval.v	. . . . . . . . 9 |- .x. = ( .s ` M )
15		ofeq 6541	. . . . . . . . 9 |- ( .x. = ( .s ` M ) -> oF .x. = oF ( .s ` M ) )
16	14, 15	ax-mp 5	. . . . . . . 8 |- oF .x. = oF ( .s ` M )
17	12, 13, 16	oveq123i 6310	. . . . . . 7 |- ( ( E X. { x } ) oF .x. y ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y )
18	9, 10, 17	mpt2eq123i 6359	. . . . . 6 |- ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) = ( x e. ( Base ` S ) , y e. B |-> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) )
19	5, 6, 7, 8, 18	mendval 31336	. . . . 5 |- ( M e. _V -> (MEndo ` M ) = ( { <. ( 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 ) , S >. , <. ( .s ` ndx ) , ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) >. } ) )
20	19	fveq2d 5876	. . . 4 |- ( M e. _V -> ( .s ` (MEndo ` M ) ) = ( .s ` ( { <. ( 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 ) , S >. , <. ( .s ` ndx ) , ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) >. } ) ) )
21		fvex 5882	. . . . . . 7 |- ( Base ` S ) e. _V
22	9, 21	eqeltri 2541	. . . . . 6 |- K e. _V
23		fvex 5882	. . . . . . 7 |- ( Base ` A ) e. _V
24	3, 23	eqeltri 2541	. . . . . 6 |- B e. _V
25	22, 24	mpt2ex 6876	. . . . 5 |- ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) e. _V
26		eqid 2457	. . . . . 6 |- ( { <. ( 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 ) , S >. , <. ( .s ` ndx ) , ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) >. } ) = ( { <. ( 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 ) , S >. , <. ( .s ` ndx ) , ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) >. } )
27	26	algvsca 31335	. . . . 5 |- ( ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) e. _V -> ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) = ( .s ` ( { <. ( 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 ) , S >. , <. ( .s ` ndx ) , ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) >. } ) ) )
28	25, 27	mp1i 12	. . . 4 |- ( M e. _V -> ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) = ( .s ` ( { <. ( 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 ) , S >. , <. ( .s ` ndx ) , ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) >. } ) ) )
29	20, 28	eqtr4d 2501	. . 3 |- ( M e. _V -> ( .s ` (MEndo ` M ) ) = ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) )
30		fvprc 5866	. . . . . 6 |- ( -. M e. _V -> (MEndo ` M ) = (/) )
31	30	fveq2d 5876	. . . . 5 |- ( -. M e. _V -> ( .s ` (MEndo ` M ) ) = ( .s ` (/) ) )
32		df-vsca 14729	. . . . . 6 |- .s = Slot 6
33	32	str0 14684	. . . . 5 |- (/) = ( .s ` (/) )
34	31, 33	syl6eqr 2516	. . . 4 |- ( -. M e. _V -> ( .s ` (MEndo ` M ) ) = (/) )
35		fvprc 5866	. . . . . . . . 9 |- ( -. M e. _V -> (Scalar ` M ) = (/) )
36	8, 35	syl5eq 2510	. . . . . . . 8 |- ( -. M e. _V -> S = (/) )
37	36	fveq2d 5876	. . . . . . 7 |- ( -. M e. _V -> ( Base ` S ) = ( Base ` (/) ) )
38		base0 14685	. . . . . . 7 |- (/) = ( Base ` (/) )
39	37, 9, 38	3eqtr4g 2523	. . . . . 6 |- ( -. M e. _V -> K = (/) )
40		mpt2eq12 6356	. . . . . 6 |- ( ( K = (/) /\ B = B ) -> ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) = ( x e. (/) , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) )
41	39, 10, 40	sylancl 662	. . . . 5 |- ( -. M e. _V -> ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) = ( x e. (/) , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) )
42		mpt20 6366	. . . . 5 |- ( x e. (/) , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) = (/)
43	41, 42	syl6eq 2514	. . . 4 |- ( -. M e. _V -> ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) = (/) )
44	34, 43	eqtr4d 2501	. . 3 |- ( -. M e. _V -> ( .s ` (MEndo ` M ) ) = ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) )
45	29, 44	pm2.61i 164	. 2 |- ( .s ` (MEndo ` M ) ) = ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) )
46	2, 45	eqtri 2486	1 |- ( .s ` A ) = ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) )

Syntax hints:   -. wn 3   = wceq 1395   e. wcel 1819   _Vcvv 3109   u. cun 3469   (/)c0 3793   {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   6c6 10610   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-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  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-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-lmhm 17795  df-mend 31329

This theorem is referenced by:  mendvsca  31344