Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ellcoellss Structured version   Visualization version   Unicode version

Theorem ellcoellss 40736
Description: Every linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Assertion
Ref Expression
ellcoellss  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  A. x  e.  ( M LinCo  V ) x  e.  S )
Distinct variable groups:    x, M    x, S    x, V

Proof of Theorem ellcoellss
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 simp1 1030 . . . 4  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  M  e.  LMod )
2 eqid 2471 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2471 . . . . . . 7  |-  ( LSubSp `  M )  =  (
LSubSp `  M )
42, 3lssss 18238 . . . . . 6  |-  ( S  e.  ( LSubSp `  M
)  ->  S  C_  ( Base `  M ) )
543ad2ant2 1052 . . . . 5  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  S  C_  ( Base `  M ) )
6 sstr 3426 . . . . . . . 8  |-  ( ( V  C_  S  /\  S  C_  ( Base `  M
) )  ->  V  C_  ( Base `  M
) )
7 fvex 5889 . . . . . . . . . 10  |-  ( Base `  M )  e.  _V
87ssex 4540 . . . . . . . . 9  |-  ( V 
C_  ( Base `  M
)  ->  V  e.  _V )
9 elpwg 3950 . . . . . . . . . 10  |-  ( V  e.  _V  ->  ( V  e.  ~P ( Base `  M )  <->  V  C_  ( Base `  M ) ) )
109biimprd 231 . . . . . . . . 9  |-  ( V  e.  _V  ->  ( V  C_  ( Base `  M
)  ->  V  e.  ~P ( Base `  M
) ) )
118, 10mpcom 36 . . . . . . . 8  |-  ( V 
C_  ( Base `  M
)  ->  V  e.  ~P ( Base `  M
) )
126, 11syl 17 . . . . . . 7  |-  ( ( V  C_  S  /\  S  C_  ( Base `  M
) )  ->  V  e.  ~P ( Base `  M
) )
1312ex 441 . . . . . 6  |-  ( V 
C_  S  ->  ( S  C_  ( Base `  M
)  ->  V  e.  ~P ( Base `  M
) ) )
14133ad2ant3 1053 . . . . 5  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( S  C_  ( Base `  M )  ->  V  e.  ~P ( Base `  M ) ) )
155, 14mpd 15 . . . 4  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  V  e.  ~P ( Base `  M )
)
16 eqid 2471 . . . . 5  |-  (Scalar `  M )  =  (Scalar `  M )
17 eqid 2471 . . . . 5  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
182, 16, 17lcoval 40713 . . . 4  |-  ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M
) )  ->  (
x  e.  ( M LinCo 
V )  <->  ( x  e.  ( Base `  M
)  /\  E. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  x  =  ( f
( linC  `  M ) V ) ) ) ) )
191, 15, 18syl2anc 673 . . 3  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( x  e.  ( M LinCo  V )  <-> 
( x  e.  (
Base `  M )  /\  E. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  x  =  ( f
( linC  `  M ) V ) ) ) ) )
20 lincellss 40727 . . . . . . . . . . . . 13  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( ( f  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )  /\  f finSupp  ( 0g `  (Scalar `  M ) ) )  ->  ( f
( linC  `  M ) V )  e.  S
) )
2120imp 436 . . . . . . . . . . . 12  |-  ( ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  (
f  e.  ( (
Base `  (Scalar `  M
) )  ^m  V
)  /\  f finSupp  ( 0g
`  (Scalar `  M )
) ) )  -> 
( f ( linC  `  M ) V )  e.  S )
22 eleq1 2537 . . . . . . . . . . . 12  |-  ( x  =  ( f ( linC  `  M ) V )  ->  ( x  e.  S  <->  ( f ( linC  `  M ) V )  e.  S ) )
2321, 22syl5ibr 229 . . . . . . . . . . 11  |-  ( x  =  ( f ( linC  `  M ) V )  ->  ( ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  ( f  e.  ( ( Base `  (Scalar `  M ) )  ^m  V )  /\  f finSupp  ( 0g `  (Scalar `  M ) ) ) )  ->  x  e.  S ) )
2423expd 443 . . . . . . . . . 10  |-  ( x  =  ( f ( linC  `  M ) V )  ->  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( ( f  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )  /\  f finSupp  ( 0g `  (Scalar `  M ) ) )  ->  x  e.  S ) ) )
2524com12 31 . . . . . . . . 9  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( x  =  ( f ( linC  `  M ) V )  ->  ( ( f  e.  ( ( Base `  (Scalar `  M )
)  ^m  V )  /\  f finSupp  ( 0g `  (Scalar `  M ) ) )  ->  x  e.  S ) ) )
2625adantr 472 . . . . . . . 8  |-  ( ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  x  e.  ( Base `  M
) )  ->  (
x  =  ( f ( linC  `  M ) V )  ->  (
( f  e.  ( ( Base `  (Scalar `  M ) )  ^m  V )  /\  f finSupp  ( 0g `  (Scalar `  M ) ) )  ->  x  e.  S
) ) )
2726com13 82 . . . . . . 7  |-  ( ( f  e.  ( (
Base `  (Scalar `  M
) )  ^m  V
)  /\  f finSupp  ( 0g
`  (Scalar `  M )
) )  ->  (
x  =  ( f ( linC  `  M ) V )  ->  (
( ( M  e. 
LMod  /\  S  e.  (
LSubSp `  M )  /\  V  C_  S )  /\  x  e.  ( Base `  M ) )  ->  x  e.  S )
) )
2827impr 631 . . . . . 6  |-  ( ( f  e.  ( (
Base `  (Scalar `  M
) )  ^m  V
)  /\  ( f finSupp  ( 0g `  (Scalar `  M ) )  /\  x  =  ( f
( linC  `  M ) V ) ) )  ->  ( ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  x  e.  (
Base `  M )
)  ->  x  e.  S ) )
2928rexlimiva 2868 . . . . 5  |-  ( E. f  e.  ( (
Base `  (Scalar `  M
) )  ^m  V
) ( f finSupp  ( 0g `  (Scalar `  M
) )  /\  x  =  ( f ( linC  `  M ) V ) )  ->  ( (
( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  x  e.  ( Base `  M
) )  ->  x  e.  S ) )
3029com12 31 . . . 4  |-  ( ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  /\  x  e.  ( Base `  M
) )  ->  ( E. f  e.  (
( Base `  (Scalar `  M
) )  ^m  V
) ( f finSupp  ( 0g `  (Scalar `  M
) )  /\  x  =  ( f ( linC  `  M ) V ) )  ->  x  e.  S ) )
3130expimpd 614 . . 3  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( ( x  e.  ( Base `  M
)  /\  E. f  e.  ( ( Base `  (Scalar `  M ) )  ^m  V ) ( f finSupp 
( 0g `  (Scalar `  M ) )  /\  x  =  ( f
( linC  `  M ) V ) ) )  ->  x  e.  S
) )
3219, 31sylbid 223 . 2  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( x  e.  ( M LinCo  V )  ->  x  e.  S
) )
3332ralrimiv 2808 1  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  A. x  e.  ( M LinCo  V ) x  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   _Vcvv 3031    C_ wss 3390   ~Pcpw 3942   class class class wbr 4395   ` cfv 5589  (class class class)co 6308    ^m cmap 7490   finSupp cfsupp 7901   Basecbs 15199  Scalarcsca 15271   0gc0g 15416   LModclmod 18169   LSubSpclss 18233   linC clinc 40705   LinCo clinco 40706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-gsum 15419  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-subg 16892  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-lmod 18171  df-lss 18234  df-linc 40707  df-lco 40708
This theorem is referenced by:  lcosslsp  40739
  Copyright terms: Public domain W3C validator