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Theorem islssd 15967
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 3344 . . 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 1171 . . . . . . . 8  |-  ( ph  ->  ( x  e.  B  ->  ( a  e.  U  ->  ( b  e.  U  ->  ( ( x  .x.  a )  .+  b
)  e.  U ) ) ) )
76imp43 579 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  (
a  e.  U  /\  b  e.  U )
)  ->  ( (
x  .x.  a )  .+  b )  e.  U
)
87ralrimivva 2758 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  A. a  e.  U  A. b  e.  U  ( (
x  .x.  a )  .+  b )  e.  U
)
98ex 424 . . . . 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 5691 . . . . . . 7  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (Scalar `  W )
) )
1310, 12eqtrd 2436 . . . . . 6  |-  ( ph  ->  B  =  ( Base `  (Scalar `  W )
) )
1413eleq2d 2471 . . . . 5  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  (Scalar `  W ) ) ) )
15 islssd.p . . . . . . . . 9  |-  ( ph  ->  .+  =  ( +g  `  W ) )
1615oveqd 6057 . . . . . . . 8  |-  ( ph  ->  ( ( x  .x.  a )  .+  b
)  =  ( ( x  .x.  a ) ( +g  `  W
) b ) )
17 islssd.t . . . . . . . . . 10  |-  ( ph  ->  .x.  =  ( .s
`  W ) )
1817oveqd 6057 . . . . . . . . 9  |-  ( ph  ->  ( x  .x.  a
)  =  ( x ( .s `  W
) a ) )
1918oveq1d 6055 . . . . . . . 8  |-  ( ph  ->  ( ( x  .x.  a ) ( +g  `  W ) b )  =  ( ( x ( .s `  W
) a ) ( +g  `  W ) b ) )
2016, 19eqtrd 2436 . . . . . . 7  |-  ( ph  ->  ( ( x  .x.  a )  .+  b
)  =  ( ( x ( .s `  W ) a ) ( +g  `  W
) b ) )
2120eleq1d 2470 . . . . . 6  |-  ( ph  ->  ( ( ( x 
.x.  a )  .+  b )  e.  U  <->  ( ( x ( .s
`  W ) a ) ( +g  `  W
) b )  e.  U ) )
22212ralbidv 2708 . . . . 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 259 . . . 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 2748 . . 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 2404 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
26 eqid 2404 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
27 eqid 2404 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
28 eqid 2404 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
29 eqid 2404 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
30 eqid 2404 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
3125, 26, 27, 28, 29, 30islss 15966 . . 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 1138 . 2  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
33 islssd.s . 2  |-  ( ph  ->  S  =  ( LSubSp `  W ) )
3432, 33eleqtrrd 2481 1  |-  ( ph  ->  U  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666    C_ wss 3280   (/)c0 3588   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484  Scalarcsca 13487   .scvsca 13488   LSubSpclss 15963
This theorem is referenced by:  lss1  15970  lsssn0  15979  islss3  15990  lss1d  15994  lssintcl  15995  lspsolvlem  16169  lbsextlem2  16186  mpllsslem  16454  dialss  31529  diblss  31653  diclss  31676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-lss 15964
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