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Theorem islssd 15693
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 3214 . . 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 1169 . . . . . . . 8  |-  ( ph  ->  ( x  e.  B  ->  ( a  e.  U  ->  ( b  e.  U  ->  ( ( x  .x.  a )  .+  b
)  e.  U ) ) ) )
76imp43 578 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  (
a  e.  U  /\  b  e.  U )
)  ->  ( (
x  .x.  a )  .+  b )  e.  U
)
87ralrimivva 2635 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  A. a  e.  U  A. b  e.  U  ( (
x  .x.  a )  .+  b )  e.  U
)
98ex 423 . . . . 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 5529 . . . . . . 7  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (Scalar `  W )
) )
1310, 12eqtrd 2315 . . . . . 6  |-  ( ph  ->  B  =  ( Base `  (Scalar `  W )
) )
1413eleq2d 2350 . . . . 5  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  (Scalar `  W ) ) ) )
15 islssd.p . . . . . . . . 9  |-  ( ph  ->  .+  =  ( +g  `  W ) )
1615oveqd 5875 . . . . . . . 8  |-  ( ph  ->  ( ( x  .x.  a )  .+  b
)  =  ( ( x  .x.  a ) ( +g  `  W
) b ) )
17 islssd.t . . . . . . . . . 10  |-  ( ph  ->  .x.  =  ( .s
`  W ) )
1817oveqd 5875 . . . . . . . . 9  |-  ( ph  ->  ( x  .x.  a
)  =  ( x ( .s `  W
) a ) )
1918oveq1d 5873 . . . . . . . 8  |-  ( ph  ->  ( ( x  .x.  a ) ( +g  `  W ) b )  =  ( ( x ( .s `  W
) a ) ( +g  `  W ) b ) )
2016, 19eqtrd 2315 . . . . . . 7  |-  ( ph  ->  ( ( x  .x.  a )  .+  b
)  =  ( ( x ( .s `  W ) a ) ( +g  `  W
) b ) )
2120eleq1d 2349 . . . . . 6  |-  ( ph  ->  ( ( ( x 
.x.  a )  .+  b )  e.  U  <->  ( ( x ( .s
`  W ) a ) ( +g  `  W
) b )  e.  U ) )
22212ralbidv 2585 . . . . 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 258 . . . 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 2625 . . 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 2283 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
26 eqid 2283 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
27 eqid 2283 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
28 eqid 2283 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
29 eqid 2283 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
30 eqid 2283 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
3125, 26, 27, 28, 29, 30islss 15692 . . 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 1136 . 2  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
33 islssd.s . 2  |-  ( ph  ->  S  =  ( LSubSp `  W ) )
3432, 33eleqtrrd 2360 1  |-  ( ph  ->  U  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   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   LSubSpclss 15689
This theorem is referenced by:  lss1  15696  lsssn0  15705  islss3  15716  lss1d  15720  lssintcl  15721  lspsolvlem  15895  lbsextlem2  15912  mpllsslem  16180  dialss  31236  diblss  31360  diclss  31383
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-lss 15690
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