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Theorem lssintcl 17393
Description: The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypothesis
Ref Expression
lssintcl.s  |-  S  =  ( LSubSp `  W )
Assertion
Ref Expression
lssintcl  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  e.  S )

Proof of Theorem lssintcl
Dummy variables  a 
b  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2468 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  (Scalar `  W )  =  (Scalar `  W ) )
2 eqidd 2468 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
) )
3 eqidd 2468 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( Base `  W )  =  (
Base `  W )
)
4 eqidd 2468 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( +g  `  W )  =  ( +g  `  W ) )
5 eqidd 2468 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( .s
`  W )  =  ( .s `  W
) )
6 lssintcl.s . . 3  |-  S  =  ( LSubSp `  W )
76a1i 11 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  S  =  ( LSubSp `  W )
)
8 intssuni2 4307 . . . 4  |-  ( ( A  C_  S  /\  A  =/=  (/) )  ->  |^| A  C_ 
U. S )
983adant1 1014 . . 3  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  C_ 
U. S )
10 eqid 2467 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
1110, 6lssss 17366 . . . . . 6  |-  ( y  e.  S  ->  y  C_  ( Base `  W
) )
12 selpw 4017 . . . . . 6  |-  ( y  e.  ~P ( Base `  W )  <->  y  C_  ( Base `  W )
)
1311, 12sylibr 212 . . . . 5  |-  ( y  e.  S  ->  y  e.  ~P ( Base `  W
) )
1413ssriv 3508 . . . 4  |-  S  C_  ~P ( Base `  W
)
15 sspwuni 4411 . . . 4  |-  ( S 
C_  ~P ( Base `  W
)  <->  U. S  C_  ( Base `  W ) )
1614, 15mpbi 208 . . 3  |-  U. S  C_  ( Base `  W
)
179, 16syl6ss 3516 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  C_  ( Base `  W
) )
18 simpl1 999 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  y  e.  A )  ->  W  e.  LMod )
19 simp2 997 . . . . . . 7  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  A  C_  S )
2019sselda 3504 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  y  e.  A )  ->  y  e.  S )
21 eqid 2467 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
2221, 6lss0cl 17376 . . . . . 6  |-  ( ( W  e.  LMod  /\  y  e.  S )  ->  ( 0g `  W )  e.  y )
2318, 20, 22syl2anc 661 . . . . 5  |-  ( ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  y  e.  A )  ->  ( 0g `  W )  e.  y )
2423ralrimiva 2878 . . . 4  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  A. y  e.  A  ( 0g `  W )  e.  y )
25 fvex 5874 . . . . 5  |-  ( 0g
`  W )  e. 
_V
2625elint2 4289 . . . 4  |-  ( ( 0g `  W )  e.  |^| A  <->  A. y  e.  A  ( 0g `  W )  e.  y )
2724, 26sylibr 212 . . 3  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( 0g
`  W )  e. 
|^| A )
28 ne0i 3791 . . 3  |-  ( ( 0g `  W )  e.  |^| A  ->  |^| A  =/=  (/) )
2927, 28syl 16 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  =/=  (/) )
3020adantlr 714 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  a  e.  |^| A  /\  b  e.  |^| A ) )  /\  y  e.  A
)  ->  y  e.  S )
31 simplr1 1038 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  a  e.  |^| A  /\  b  e.  |^| A ) )  /\  y  e.  A
)  ->  x  e.  ( Base `  (Scalar `  W
) ) )
32 simplr2 1039 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  a  e.  |^| A  /\  b  e.  |^| A ) )  /\  y  e.  A
)  ->  a  e.  |^| A )
33 simpr 461 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  a  e.  |^| A  /\  b  e.  |^| A ) )  /\  y  e.  A
)  ->  y  e.  A )
34 elinti 4291 . . . . . 6  |-  ( a  e.  |^| A  ->  (
y  e.  A  -> 
a  e.  y ) )
3532, 33, 34sylc 60 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  a  e.  |^| A  /\  b  e.  |^| A ) )  /\  y  e.  A
)  ->  a  e.  y )
36 simplr3 1040 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  a  e.  |^| A  /\  b  e.  |^| A ) )  /\  y  e.  A
)  ->  b  e.  |^| A )
37 elinti 4291 . . . . . 6  |-  ( b  e.  |^| A  ->  (
y  e.  A  -> 
b  e.  y ) )
3836, 33, 37sylc 60 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  a  e.  |^| A  /\  b  e.  |^| A ) )  /\  y  e.  A
)  ->  b  e.  y )
39 eqid 2467 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
40 eqid 2467 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
41 eqid 2467 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
42 eqid 2467 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
4339, 40, 41, 42, 6lsscl 17372 . . . . 5  |-  ( ( y  e.  S  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  a  e.  y  /\  b  e.  y ) )  -> 
( ( x ( .s `  W ) a ) ( +g  `  W ) b )  e.  y )
4430, 31, 35, 38, 43syl13anc 1230 . . . 4  |-  ( ( ( ( W  e. 
LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  a  e.  |^| A  /\  b  e.  |^| A ) )  /\  y  e.  A
)  ->  ( (
x ( .s `  W ) a ) ( +g  `  W
) b )  e.  y )
4544ralrimiva 2878 . . 3  |-  ( ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  |^| A  /\  b  e. 
|^| A ) )  ->  A. y  e.  A  ( ( x ( .s `  W ) a ) ( +g  `  W ) b )  e.  y )
46 ovex 6307 . . . 4  |-  ( ( x ( .s `  W ) a ) ( +g  `  W
) b )  e. 
_V
4746elint2 4289 . . 3  |-  ( ( ( x ( .s
`  W ) a ) ( +g  `  W
) b )  e. 
|^| A  <->  A. y  e.  A  ( (
x ( .s `  W ) a ) ( +g  `  W
) b )  e.  y )
4845, 47sylibr 212 . 2  |-  ( ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  |^| A  /\  b  e. 
|^| A ) )  ->  ( ( x ( .s `  W
) a ) ( +g  `  W ) b )  e.  |^| A )
491, 2, 3, 4, 5, 7, 17, 29, 48islssd 17365 1  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   U.cuni 4245   |^|cint 4282   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551  Scalarcsca 14554   .scvsca 14555   0gc0g 14691   LModclmod 17295   LSubSpclss 17361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-plusg 14564  df-0g 14693  df-mnd 15728  df-grp 15858  df-minusg 15859  df-sbg 15860  df-mgp 16932  df-ur 16944  df-rng 16988  df-lmod 17297  df-lss 17362
This theorem is referenced by:  lssincl  17394  lssmre  17395  lspf  17403  asplss  17749  dihglblem5  36095
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