Theorem mendvscafval 29552

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 5699	. 2 |- ( .s ` A ) = ( .s ` (MEndo ` M ) )
3		mendvscafval.b	. . . . . . 7 |- B = ( Base ` A )
4	1	mendbas 29546	. . . . . . 7 |- ( M LMHom M ) = ( Base ` A )
5	3, 4	eqtr4i 2466	. . . . . 6 |- B = ( M LMHom M )
6		eqid 2443	. . . . . 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 2443	. . . . . 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 2443	. . . . . . 7 |- B = B
11		mendvscafval.e	. . . . . . . . 9 |- E = ( Base ` M )
12	11	xpeq1i 4865	. . . . . . . 8 |- ( E X. { x } ) = ( ( Base ` M ) X. { x } )
13		eqid 2443	. . . . . . . 8 |- y = y
14		mendvscafval.v	. . . . . . . . 9 |- .x. = ( .s ` M )
15		ofeq 6327	. . . . . . . . 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 6110	. . . . . . 7 |- ( ( E X. { x } ) oF .x. y ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y )
18	9, 10, 17	mpt2eq123i 6154	. . . . . 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 29545	. . . . 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 5700	. . . 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 5706	. . . . . . 7 |- ( Base ` S ) e. _V
22	9, 21	eqeltri 2513	. . . . . 6 |- K e. _V
23		fvex 5706	. . . . . . 7 |- ( Base ` A ) e. _V
24	3, 23	eqeltri 2513	. . . . . 6 |- B e. _V
25	22, 24	mpt2ex 6655	. . . . 5 |- ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) e. _V
26		eqid 2443	. . . . . 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 29544	. . . . 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 2478	. . 3 |- ( M e. _V -> ( .s ` (MEndo ` M ) ) = ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) )
30		fvprc 5690	. . . . . 6 |- ( -. M e. _V -> (MEndo ` M ) = (/) )
31	30	fveq2d 5700	. . . . 5 |- ( -. M e. _V -> ( .s ` (MEndo ` M ) ) = ( .s ` (/) ) )
32		df-vsca 14260	. . . . . 6 |- .s = Slot 6
33	32	str0 14217	. . . . 5 |- (/) = ( .s ` (/) )
34	31, 33	syl6eqr 2493	. . . 4 |- ( -. M e. _V -> ( .s ` (MEndo ` M ) ) = (/) )
35		fvprc 5690	. . . . . . . . 9 |- ( -. M e. _V -> (Scalar ` M ) = (/) )
36	8, 35	syl5eq 2487	. . . . . . . 8 |- ( -. M e. _V -> S = (/) )
37	36	fveq2d 5700	. . . . . . 7 |- ( -. M e. _V -> ( Base ` S ) = ( Base ` (/) ) )
38		base0 14218	. . . . . . 7 |- (/) = ( Base ` (/) )
39	37, 9, 38	3eqtr4g 2500	. . . . . 6 |- ( -. M e. _V -> K = (/) )
40		mpt2eq12 6151	. . . . . 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 6161	. . . . 5 |- ( x e. (/) , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) = (/)
43	41, 42	syl6eq 2491	. . . 4 |- ( -. M e. _V -> ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) ) = (/) )
44	34, 43	eqtr4d 2478	. . 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 2463	1 |- ( .s ` A ) = ( x e. K , y e. B |-> ( ( E X. { x } ) oF .x. y ) )

Syntax hints:   -. wn 3   = wceq 1369   e. wcel 1756   _Vcvv 2977   u. cun 3331   (/)c0 3642   {csn 3882   {cpr 3884   {ctp 3886   <.cop 3888   X. cxp 4843   o. ccom 4849   ` cfv 5423  (class class class)co 6096   e. cmpt2 6098   oFcof 6323   6c6 10380   ndxcnx 14176   Basecbs 14179   +g cplusg 14243   .rcmulr 14244  Scalarcsca 14246   .scvsca 14247   LMHom clmhm 17105  MEndocmend 29537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-lmhm 17108  df-mend 29538

This theorem is referenced by:  mendvsca  29553