Theorem mendvscafval 36068

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 5873	. 2 |- ( .s ` A ) = ( .s ` (MEndo ` M ) )
3		mendvscafval.b	. . . . . . 7 |- B = ( Base ` A )
4	1	mendbas 36062	. . . . . . 7 |- ( M LMHom M ) = ( Base ` A )
5	3, 4	eqtr4i 2478	. . . . . 6 |- B = ( M LMHom M )
6		eqid 2453	. . . . . 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 2453	. . . . . 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 2453	. . . . . . 7 |- B = B
11		mendvscafval.e	. . . . . . . . 9 |- E = ( Base ` M )
12	11	xpeq1i 4857	. . . . . . . 8 |- ( E X. { x } ) = ( ( Base ` M ) X. { x } )
13		eqid 2453	. . . . . . . 8 |- y = y
14		mendvscafval.v	. . . . . . . . 9 |- .x. = ( .s ` M )
15		ofeq 6538	. . . . . . . . 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 6309	. . . . . . 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 36061	. . . . 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 5874	. . . 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 5880	. . . . . . 7 |- ( Base ` S ) e. _V
22	9, 21	eqeltri 2527	. . . . . 6 |- K e. _V
23		fvex 5880	. . . . . . 7 |- ( Base ` A ) e. _V
24	3, 23	eqeltri 2527	. . . . . 6 |- B e. _V
25	22, 24	mpt2ex 6875	. . . . 5 |- ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) e. _V
26		eqid 2453	. . . . . 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 36060	. . . . 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 13	. . . 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 2490	. . 3 |- ( M e. _V -> ( .s ` (MEndo ` M ) ) = ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) )
30		fvprc 5864	. . . . . 6 |- ( -. M e. _V -> (MEndo ` M ) = (/) )
31	30	fveq2d 5874	. . . . 5 |- ( -. M e. _V -> ( .s ` (MEndo ` M ) ) = ( .s ` (/) ) )
32		df-vsca 15219	. . . . . 6 |- .s = Slot 6
33	32	str0 15173	. . . . 5 |- (/) = ( .s ` (/) )
34	31, 33	syl6eqr 2505	. . . 4 |- ( -. M e. _V -> ( .s ` (MEndo ` M ) ) = (/) )
35		fvprc 5864	. . . . . . . . 9 |- ( -. M e. _V -> (Scalar ` M ) = (/) )
36	8, 35	syl5eq 2499	. . . . . . . 8 |- ( -. M e. _V -> S = (/) )
37	36	fveq2d 5874	. . . . . . 7 |- ( -. M e. _V -> ( Base ` S ) = ( Base ` (/) ) )
38		base0 15174	. . . . . . 7 |- (/) = ( Base ` (/) )
39	37, 9, 38	3eqtr4g 2512	. . . . . 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 669	. . . . 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 2503	. . . 4 |- ( -. M e. _V -> ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) = (/) )
44	34, 43	eqtr4d 2490	. . 3 |- ( -. M e. _V -> ( .s ` (MEndo ` M ) ) = ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) )
45	29, 44	pm2.61i 168	. 2 |- ( .s ` (MEndo ` M ) ) = ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) )
46	2, 45	eqtri 2475	1 |- ( .s ` A ) = ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) )

Syntax hints:   -. wn 3   = wceq 1446   e. wcel 1889   _Vcvv 3047   u. cun 3404   (/)c0 3733   {csn 3970   {cpr 3972   {ctp 3974   <.cop 3976   X. cxp 4835   o. ccom 4841   ` cfv 5585  (class class class)co 6295   |-> cmpt2 6297   oFcof 6534   6c6 10670   ndxcnx 15130   Basecbs 15133   +g cplusg 15202   .rcmulr 15203  Scalarcsca 15205   .scvsca 15206   LMHom clmhm 18254  MEndocmend 36053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-plusg 15215  df-mulr 15216  df-sca 15218  df-vsca 15219  df-lmhm 18257  df-mend 36054

This theorem is referenced by:  mendvsca  36069