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Theorem lssintcl 17481
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 2442 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  (Scalar `  W )  =  (Scalar `  W ) )
2 eqidd 2442 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
) )
3 eqidd 2442 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( Base `  W )  =  (
Base `  W )
)
4 eqidd 2442 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( +g  `  W )  =  ( +g  `  W ) )
5 eqidd 2442 . 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 4294 . . . 4  |-  ( ( A  C_  S  /\  A  =/=  (/) )  ->  |^| A  C_ 
U. S )
983adant1 1013 . . 3  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  C_ 
U. S )
10 eqid 2441 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
1110, 6lssss 17454 . . . . . 6  |-  ( y  e.  S  ->  y  C_  ( Base `  W
) )
12 selpw 4001 . . . . . 6  |-  ( y  e.  ~P ( Base `  W )  <->  y  C_  ( Base `  W )
)
1311, 12sylibr 212 . . . . 5  |-  ( y  e.  S  ->  y  e.  ~P ( Base `  W
) )
1413ssriv 3491 . . . 4  |-  S  C_  ~P ( Base `  W
)
15 sspwuni 4398 . . . 4  |-  ( S 
C_  ~P ( Base `  W
)  <->  U. S  C_  ( Base `  W ) )
1614, 15mpbi 208 . . 3  |-  U. S  C_  ( Base `  W
)
179, 16syl6ss 3499 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  C_  ( Base `  W
) )
18 simpl1 998 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  y  e.  A )  ->  W  e.  LMod )
19 simp2 996 . . . . . . 7  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  A  C_  S )
2019sselda 3487 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  y  e.  A )  ->  y  e.  S )
21 eqid 2441 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
2221, 6lss0cl 17464 . . . . . 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 2855 . . . 4  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  A. y  e.  A  ( 0g `  W )  e.  y )
25 fvex 5863 . . . . 5  |-  ( 0g
`  W )  e. 
_V
2625elint2 4275 . . . 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 3774 . . 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 1037 . . . . 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 1038 . . . . . 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 4277 . . . . . 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 1039 . . . . . 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 4277 . . . . . 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 2441 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
40 eqid 2441 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
41 eqid 2441 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
42 eqid 2441 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
4339, 40, 41, 42, 6lsscl 17460 . . . . 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 1229 . . . 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 2855 . . 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 6306 . . . 4  |-  ( ( x ( .s `  W ) a ) ( +g  `  W
) b )  e. 
_V
4746elint2 4275 . . 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 17453 1  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791    C_ wss 3459   (/)c0 3768   ~Pcpw 3994   U.cuni 4231   |^|cint 4268   ` cfv 5575  (class class class)co 6278   Basecbs 14506   +g cplusg 14571  Scalarcsca 14574   .scvsca 14575   0gc0g 14711   LModclmod 17383   LSubSpclss 17449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-recs 7041  df-rdg 7075  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-nn 10540  df-2 10597  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-plusg 14584  df-0g 14713  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-grp 15928  df-minusg 15929  df-sbg 15930  df-mgp 17013  df-ur 17025  df-ring 17071  df-lmod 17385  df-lss 17450
This theorem is referenced by:  lssincl  17482  lssmre  17483  lspf  17491  asplss  17849  dihglblem5  36748
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