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Theorem lssintcl 15737
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 2297 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  (Scalar `  W )  =  (Scalar `  W ) )
2 eqidd 2297 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
) )
3 eqidd 2297 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( Base `  W )  =  (
Base `  W )
)
4 eqidd 2297 . 2  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  ( +g  `  W )  =  ( +g  `  W ) )
5 eqidd 2297 . 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 3903 . . . 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 2296 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
1110, 6lssss 15710 . . . . . 6  |-  ( y  e.  S  ->  y  C_  ( Base `  W
) )
12 vex 2804 . . . . . . 7  |-  y  e. 
_V
1312elpw 3644 . . . . . 6  |-  ( y  e.  ~P ( Base `  W )  <->  y  C_  ( Base `  W )
)
1411, 13sylibr 203 . . . . 5  |-  ( y  e.  S  ->  y  e.  ~P ( Base `  W
) )
1514ssriv 3197 . . . 4  |-  S  C_  ~P ( Base `  W
)
16 sspwuni 4003 . . . 4  |-  ( S 
C_  ~P ( Base `  W
)  <->  U. S  C_  ( Base `  W ) )
1715, 16mpbi 199 . . 3  |-  U. S  C_  ( Base `  W
)
189, 17syl6ss 3204 . 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 3193 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  /\  y  e.  A )  ->  y  e.  S )
22 eqid 2296 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
2322, 6lss0cl 15720 . . . . . 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 2639 . . . 4  |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  A. y  e.  A  ( 0g `  W )  e.  y )
26 fvex 5555 . . . . 5  |-  ( 0g
`  W )  e. 
_V
2726elint2 3885 . . . 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 3474 . . 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 3887 . . . . . 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 3887 . . . . . 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 2296 . . . . . 6  |-  (Scalar `  W )  =  (Scalar `  W )
41 eqid 2296 . . . . . 6  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
42 eqid 2296 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
43 eqid 2296 . . . . . 6  |-  ( .s
`  W )  =  ( .s `  W
)
4440, 41, 42, 43, 6lsscl 15716 . . . . 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 2639 . . 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 5899 . . . 4  |-  ( ( x ( .s `  W ) a ) ( +g  `  W
) b )  e. 
_V
4847elint2 3885 . . 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 15709 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   U.cuni 3843   |^|cint 3878   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   LModclmod 15643   LSubSpclss 15705
This theorem is referenced by:  lssincl  15738  lssmre  15739  lspf  15747  asplss  16085  dihglblem5  32110
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706
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