Theorem mendbas 27368

Description: Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)

Hypothesis

Ref	Expression
mendbas.a	|- A = (MEndo ` M )

Assertion

Ref	Expression
mendbas	|- ( M LMHom M ) = ( Base ` A )

Proof of Theorem mendbas

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6073	. . . 4 |- ( M LMHom M ) e. _V
2		eqid 2412	. . . . 5 |- ( { <. ( Base ` ndx ) , ( M LMHom M ) >. , <. ( +g ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o F ( +g ` M ) y ) ) >. , <. ( .r ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) >. } u. { <. (Scalar ` ndx ) , (Scalar ` M ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) o F ( .s ` M ) y ) ) >. } ) = ( { <. ( Base ` ndx ) , ( M LMHom M ) >. , <. ( +g ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o F ( +g ` M ) y ) ) >. , <. ( .r ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) >. } u. { <. (Scalar ` ndx ) , (Scalar ` M ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) o F ( .s ` M ) y ) ) >. } )
3	2	algbase 13562	. . . 4 |- ( ( M LMHom M ) e. _V -> ( M LMHom M ) = ( Base ` ( { <. ( Base ` ndx ) , ( M LMHom M ) >. , <. ( +g ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o F ( +g ` M ) y ) ) >. , <. ( .r ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) >. } u. { <. (Scalar ` ndx ) , (Scalar ` M ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) o F ( .s ` M ) y ) ) >. } ) ) )
4	1, 3	mp1i 12	. . 3 |- ( M e. _V -> ( M LMHom M ) = ( Base ` ( { <. ( Base ` ndx ) , ( M LMHom M ) >. , <. ( +g ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o F ( +g ` M ) y ) ) >. , <. ( .r ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) >. } u. { <. (Scalar ` ndx ) , (Scalar ` M ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) o F ( .s ` M ) y ) ) >. } ) ) )
5		mendbas.a	. . . . 5 |- A = (MEndo ` M )
6		eqid 2412	. . . . . 6 |- ( M LMHom M ) = ( M LMHom M )
7		eqid 2412	. . . . . 6 |- ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o F ( +g ` M ) y ) ) = ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o F ( +g ` M ) y ) )
8		eqid 2412	. . . . . 6 |- ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) = ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) )
9		eqid 2412	. . . . . 6 |- (Scalar ` M ) = (Scalar ` M )
10		eqid 2412	. . . . . 6 |- ( x e. ( Base ` (Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) o F ( .s ` M ) y ) ) = ( x e. ( Base ` (Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) o F ( .s ` M ) y ) )
11	6, 7, 8, 9, 10	mendval 27367	. . . . 5 |- ( M e. _V -> (MEndo ` M ) = ( { <. ( Base ` ndx ) , ( M LMHom M ) >. , <. ( +g ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o F ( +g ` M ) y ) ) >. , <. ( .r ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) >. } u. { <. (Scalar ` ndx ) , (Scalar ` M ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) o F ( .s ` M ) y ) ) >. } ) )
12	5, 11	syl5eq 2456	. . . 4 |- ( M e. _V -> A = ( { <. ( Base ` ndx ) , ( M LMHom M ) >. , <. ( +g ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o F ( +g ` M ) y ) ) >. , <. ( .r ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) >. } u. { <. (Scalar ` ndx ) , (Scalar ` M ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) o F ( .s ` M ) y ) ) >. } ) )
13	12	fveq2d 5699	. . 3 |- ( M e. _V -> ( Base ` A ) = ( Base ` ( { <. ( Base ` ndx ) , ( M LMHom M ) >. , <. ( +g ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o F ( +g ` M ) y ) ) >. , <. ( .r ` ndx ) , ( x e. ( M LMHom M ) , y e. ( M LMHom M ) |-> ( x o. y ) ) >. } u. { <. (Scalar ` ndx ) , (Scalar ` M ) >. , <. ( .s ` ndx ) , ( x e. ( Base ` (Scalar ` M ) ) , y e. ( M LMHom M ) |-> ( ( ( Base ` M ) X. { x } ) o F ( .s ` M ) y ) ) >. } ) ) )
14	4, 13	eqtr4d 2447	. 2 |- ( M e. _V -> ( M LMHom M ) = ( Base ` A ) )
15		base0 13469	. . 3 |- (/) = ( Base ` (/) )
16		reldmlmhm 16064	. . . 4 |- Rel dom LMHom
17	16	ovprc1 6076	. . 3 |- ( -. M e. _V -> ( M LMHom M ) = (/) )
18		fvprc 5689	. . . . 5 |- ( -. M e. _V -> (MEndo ` M ) = (/) )
19	5, 18	syl5eq 2456	. . . 4 |- ( -. M e. _V -> A = (/) )
20	19	fveq2d 5699	. . 3 |- ( -. M e. _V -> ( Base ` A ) = ( Base ` (/) ) )
21	15, 17, 20	3eqtr4a 2470	. 2 |- ( -. M e. _V -> ( M LMHom M ) = ( Base ` A ) )
22	14, 21	pm2.61i 158	1 |- ( M LMHom M ) = ( Base ` A )

Syntax hints:   -. wn 3   = wceq 1649   e. wcel 1721   _Vcvv 2924   u. cun 3286   (/)c0 3596   {csn 3782   {cpr 3783   {ctp 3784   <.cop 3785   X. cxp 4843   o. ccom 4849   ` cfv 5421  (class class class)co 6048   e. cmpt2 6050   o Fcof 6270   ndxcnx 13429   Basecbs 13432   +g cplusg 13492   .rcmulr 13493  Scalarcsca 13495   .scvsca 13496   LMHom clmhm 16058  MEndocmend 27365

This theorem is referenced by:  mendplusgfval  27369  mendmulrfval  27371  mendvscafval  27374  mendrng  27376  mendlmod  27377  mendassa  27378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-plusg 13505  df-mulr 13506  df-sca 13508  df-vsca 13509  df-lmhm 16061  df-mend 27366