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Theorem ellcoellss 33290
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 994 . . . 4  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  M  e.  LMod )
2 eqid 2454 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2454 . . . . . . 7  |-  ( LSubSp `  M )  =  (
LSubSp `  M )
42, 3lssss 17778 . . . . . 6  |-  ( S  e.  ( LSubSp `  M
)  ->  S  C_  ( Base `  M ) )
543ad2ant2 1016 . . . . 5  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  S  C_  ( Base `  M ) )
6 sstr 3497 . . . . . . . 8  |-  ( ( V  C_  S  /\  S  C_  ( Base `  M
) )  ->  V  C_  ( Base `  M
) )
7 fvex 5858 . . . . . . . . . 10  |-  ( Base `  M )  e.  _V
87ssex 4581 . . . . . . . . 9  |-  ( V 
C_  ( Base `  M
)  ->  V  e.  _V )
9 elpwg 4007 . . . . . . . . . 10  |-  ( V  e.  _V  ->  ( V  e.  ~P ( Base `  M )  <->  V  C_  ( Base `  M ) ) )
109biimprd 223 . . . . . . . . 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 16 . . . . . . 7  |-  ( ( V  C_  S  /\  S  C_  ( Base `  M
) )  ->  V  e.  ~P ( Base `  M
) )
1312ex 432 . . . . . 6  |-  ( V 
C_  S  ->  ( S  C_  ( Base `  M
)  ->  V  e.  ~P ( Base `  M
) ) )
14133ad2ant3 1017 . . . . 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 2454 . . . . 5  |-  (Scalar `  M )  =  (Scalar `  M )
17 eqid 2454 . . . . 5  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
182, 16, 17lcoval 33267 . . . 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 659 . . 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 33281 . . . . . . . . . . . . 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 427 . . . . . . . . . . . 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 2526 . . . . . . . . . . . 12  |-  ( x  =  ( f ( linC  `  M ) V )  ->  ( x  e.  S  <->  ( f ( linC  `  M ) V )  e.  S ) )
2321, 22syl5ibr 221 . . . . . . . . . . 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 434 . . . . . . . . . 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 463 . . . . . . . 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 80 . . . . . . 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 617 . . . . . 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 2942 . . . . 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 601 . . 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 215 . 2  |-  ( ( M  e.  LMod  /\  S  e.  ( LSubSp `  M )  /\  V  C_  S )  ->  ( x  e.  ( M LinCo  V )  ->  x  e.  S
) )
3332ralrimiv 2866 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 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   class class class wbr 4439   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   finSupp cfsupp 7821   Basecbs 14716  Scalarcsca 14787   0gc0g 14929   LModclmod 17707   LSubSpclss 17773   linC clinc 33259   LinCo clinco 33260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-gsum 14932  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-cntz 16554  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-lmod 17709  df-lss 17774  df-linc 33261  df-lco 33262
This theorem is referenced by:  lcosslsp  33293
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