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Theorem lssintcl 15721
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 2284 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  (Scalar `  W )  =  (Scalar `  W ) )
2 eqidd 2284 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
) )
3 eqidd 2284 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( Base `  W )  =  (
Base `  W )
)
4 eqidd 2284 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( +g  `  W )  =  ( +g  `  W ) )
5 eqidd 2284 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( .s
`  W )  =  ( .s `  W
) )
6 lssintcl.s . . 3  |-  S  =  ( LSubSp `  W )
76a1i 10 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  S  =  ( LSubSp `  W )
)
8 intssuni2 3887 . . . 4  |-  ( ( A  C_  S  /\  A  =/=  (/) )  ->  |^| A  C_ 
U. S )
983adant1 973 . . 3  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  C_ 
U. S )
10 eqid 2283 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
1110, 6lssss 15694 . . . . . 6  |-  ( y  e.  S  ->  y  C_  ( Base `  W
) )
12 vex 2791 . . . . . . 7  |-  y  e. 
_V
1312elpw 3631 . . . . . 6  |-  ( y  e.  ~P ( Base `  W )  <->  y  C_  ( Base `  W )
)
1411, 13sylibr 203 . . . . 5  |-  ( y  e.  S  ->  y  e.  ~P ( Base `  W
) )
1514ssriv 3184 . . . 4  |-  S  C_  ~P ( Base `  W
)
16 sspwuni 3987 . . . 4  |-  ( S 
C_  ~P ( Base `  W
)  <->  U. S  C_  ( Base `  W ) )
1715, 16mpbi 199 . . 3  |-  U. S  C_  ( Base `  W
)
189, 17syl6ss 3191 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  C_  ( Base `  W
) )
19 simpl1 958 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  y  e.  A )  ->  W  e.  LMod )
20 simp2 956 . . . . . . 7  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  A  C_  S )
2120sselda 3180 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  y  e.  A )  ->  y  e.  S )
22 eqid 2283 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
2322, 6lss0cl 15704 . . . . . 6  |-  ( ( W  e.  LMod  /\  y  e.  S )  ->  ( 0g `  W )  e.  y )
2419, 21, 23syl2anc 642 . . . . 5  |-  ( ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  y  e.  A )  ->  ( 0g `  W )  e.  y )
2524ralrimiva 2626 . . . 4  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  A. y  e.  A  ( 0g `  W )  e.  y )
26 fvex 5539 . . . . 5  |-  ( 0g
`  W )  e. 
_V
2726elint2 3869 . . . 4  |-  ( ( 0g `  W )  e.  |^| A  <->  A. y  e.  A  ( 0g `  W )  e.  y )
2825, 27sylibr 203 . . 3  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( 0g
`  W )  e. 
|^| A )
29 ne0i 3461 . . 3  |-  ( ( 0g `  W )  e.  |^| A  ->  |^| A  =/=  (/) )
3028, 29syl 15 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  =/=  (/) )
3121adantlr 695 . . . . 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 )
32 simplr1 997 . . . . 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
) ) )
33 simplr2 998 . . . . . 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 )
34 simpr 447 . . . . . 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 )
35 elinti 3871 . . . . . 6  |-  ( a  e.  |^| A  ->  (
y  e.  A  -> 
a  e.  y ) )
3633, 34, 35sylc 56 . . . . 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 )
37 simplr3 999 . . . . . 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 )
38 elinti 3871 . . . . . 6  |-  ( b  e.  |^| A  ->  (
y  e.  A  -> 
b  e.  y ) )
3937, 34, 38sylc 56 . . . . 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 )
40 eqid 2283 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
41 eqid 2283 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
42 eqid 2283 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
43 eqid 2283 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
4440, 41, 42, 43, 6lsscl 15700 . . . . 5  |-  ( ( y  e.  S  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  a  e.  y  /\  b  e.  y ) )  -> 
( ( x ( .s `  W ) a ) ( +g  `  W ) b )  e.  y )
4531, 32, 36, 39, 44syl13anc 1184 . . . 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 )
4645ralrimiva 2626 . . 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 )
47 ovex 5883 . . . 4  |-  ( ( x ( .s `  W ) a ) ( +g  `  W
) b )  e. 
_V
4847elint2 3869 . . 3  |-  ( ( ( x ( .s
`  W ) a ) ( +g  `  W
) b )  e. 
|^| A  <->  A. y  e.  A  ( (
x ( .s `  W ) a ) ( +g  `  W
) b )  e.  y )
4946, 48sylibr 203 . 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 )
501, 2, 3, 4, 5, 7, 18, 30, 49islssd 15693 1  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   |^|cint 3862   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   LModclmod 15627   LSubSpclss 15689
This theorem is referenced by:  lssincl  15722  lssmre  15723  lspf  15731  asplss  16069  dihglblem5  31488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690
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