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Theorem islssd 18094
Description: Properties that determine a subspace of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
Hypotheses
Ref Expression
islssd.f  |-  ( ph  ->  F  =  (Scalar `  W ) )
islssd.b  |-  ( ph  ->  B  =  ( Base `  F ) )
islssd.v  |-  ( ph  ->  V  =  ( Base `  W ) )
islssd.p  |-  ( ph  ->  .+  =  ( +g  `  W ) )
islssd.t  |-  ( ph  ->  .x.  =  ( .s
`  W ) )
islssd.s  |-  ( ph  ->  S  =  ( LSubSp `  W ) )
islssd.u  |-  ( ph  ->  U  C_  V )
islssd.z  |-  ( ph  ->  U  =/=  (/) )
islssd.c  |-  ( (
ph  /\  ( x  e.  B  /\  a  e.  U  /\  b  e.  U ) )  -> 
( ( x  .x.  a )  .+  b
)  e.  U )
Assertion
Ref Expression
islssd  |-  ( ph  ->  U  e.  S )
Distinct variable groups:    a, b, x, ph    U, a, b, x    W, a, b, x    B, a, b
Allowed substitution hints:    B( x)    .+ ( x, a, b)    S( x, a, b)    .x. ( x, a, b)    F( x, a, b)    V( x, a, b)

Proof of Theorem islssd
StepHypRef Expression
1 islssd.u . . . 4  |-  ( ph  ->  U  C_  V )
2 islssd.v . . . 4  |-  ( ph  ->  V  =  ( Base `  W ) )
31, 2sseqtrd 3506 . . 3  |-  ( ph  ->  U  C_  ( Base `  W ) )
4 islssd.z . . 3  |-  ( ph  ->  U  =/=  (/) )
5 islssd.c . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  B  /\  a  e.  U  /\  b  e.  U ) )  -> 
( ( x  .x.  a )  .+  b
)  e.  U )
653exp2 1223 . . . . . . . 8  |-  ( ph  ->  ( x  e.  B  ->  ( a  e.  U  ->  ( b  e.  U  ->  ( ( x  .x.  a )  .+  b
)  e.  U ) ) ) )
76imp43 598 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  (
a  e.  U  /\  b  e.  U )
)  ->  ( (
x  .x.  a )  .+  b )  e.  U
)
87ralrimivva 2853 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  A. a  e.  U  A. b  e.  U  ( (
x  .x.  a )  .+  b )  e.  U
)
98ex 435 . . . . 5  |-  ( ph  ->  ( x  e.  B  ->  A. a  e.  U  A. b  e.  U  ( ( x  .x.  a )  .+  b
)  e.  U ) )
10 islssd.b . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  F ) )
11 islssd.f . . . . . . . 8  |-  ( ph  ->  F  =  (Scalar `  W ) )
1211fveq2d 5885 . . . . . . 7  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (Scalar `  W )
) )
1310, 12eqtrd 2470 . . . . . 6  |-  ( ph  ->  B  =  ( Base `  (Scalar `  W )
) )
1413eleq2d 2499 . . . . 5  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  (Scalar `  W ) ) ) )
15 islssd.p . . . . . . . . 9  |-  ( ph  ->  .+  =  ( +g  `  W ) )
1615oveqd 6322 . . . . . . . 8  |-  ( ph  ->  ( ( x  .x.  a )  .+  b
)  =  ( ( x  .x.  a ) ( +g  `  W
) b ) )
17 islssd.t . . . . . . . . . 10  |-  ( ph  ->  .x.  =  ( .s
`  W ) )
1817oveqd 6322 . . . . . . . . 9  |-  ( ph  ->  ( x  .x.  a
)  =  ( x ( .s `  W
) a ) )
1918oveq1d 6320 . . . . . . . 8  |-  ( ph  ->  ( ( x  .x.  a ) ( +g  `  W ) b )  =  ( ( x ( .s `  W
) a ) ( +g  `  W ) b ) )
2016, 19eqtrd 2470 . . . . . . 7  |-  ( ph  ->  ( ( x  .x.  a )  .+  b
)  =  ( ( x ( .s `  W ) a ) ( +g  `  W
) b ) )
2120eleq1d 2498 . . . . . 6  |-  ( ph  ->  ( ( ( x 
.x.  a )  .+  b )  e.  U  <->  ( ( x ( .s
`  W ) a ) ( +g  `  W
) b )  e.  U ) )
22212ralbidv 2876 . . . . 5  |-  ( ph  ->  ( A. a  e.  U  A. b  e.  U  ( ( x 
.x.  a )  .+  b )  e.  U  <->  A. a  e.  U  A. b  e.  U  (
( x ( .s
`  W ) a ) ( +g  `  W
) b )  e.  U ) )
239, 14, 223imtr3d 270 . . . 4  |-  ( ph  ->  ( x  e.  (
Base `  (Scalar `  W
) )  ->  A. a  e.  U  A. b  e.  U  ( (
x ( .s `  W ) a ) ( +g  `  W
) b )  e.  U ) )
2423ralrimiv 2844 . . 3  |-  ( ph  ->  A. x  e.  (
Base `  (Scalar `  W
) ) A. a  e.  U  A. b  e.  U  ( (
x ( .s `  W ) a ) ( +g  `  W
) b )  e.  U )
25 eqid 2429 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
26 eqid 2429 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
27 eqid 2429 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
28 eqid 2429 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
29 eqid 2429 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
30 eqid 2429 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
3125, 26, 27, 28, 29, 30islss 18093 . . 3  |-  ( U  e.  ( LSubSp `  W
)  <->  ( U  C_  ( Base `  W )  /\  U  =/=  (/)  /\  A. x  e.  ( Base `  (Scalar `  W )
) A. a  e.  U  A. b  e.  U  ( ( x ( .s `  W
) a ) ( +g  `  W ) b )  e.  U
) )
323, 4, 24, 31syl3anbrc 1189 . 2  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
33 islssd.s . 2  |-  ( ph  ->  S  =  ( LSubSp `  W ) )
3432, 33eleqtrrd 2520 1  |-  ( ph  ->  U  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782    C_ wss 3442   (/)c0 3767   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152  Scalarcsca 15155   .scvsca 15156   LSubSpclss 18090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-lss 18091
This theorem is referenced by:  lss1  18097  lsssn0  18106  islss3  18117  lss1d  18121  lssintcl  18122  lspsolvlem  18300  lbsextlem2  18317  mpllsslem  18594  scmatlss  19481  dialss  34322  diblss  34446  diclss  34469  lincolss  38986
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