MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  islssd Structured version   Unicode version

Theorem islssd 17382
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 3540 . . 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 1214 . . . . . . . 8  |-  ( ph  ->  ( x  e.  B  ->  ( a  e.  U  ->  ( b  e.  U  ->  ( ( x  .x.  a )  .+  b
)  e.  U ) ) ) )
76imp43 595 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  (
a  e.  U  /\  b  e.  U )
)  ->  ( (
x  .x.  a )  .+  b )  e.  U
)
87ralrimivva 2885 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  A. a  e.  U  A. b  e.  U  ( (
x  .x.  a )  .+  b )  e.  U
)
98ex 434 . . . . 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 5870 . . . . . . 7  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (Scalar `  W )
) )
1310, 12eqtrd 2508 . . . . . 6  |-  ( ph  ->  B  =  ( Base `  (Scalar `  W )
) )
1413eleq2d 2537 . . . . 5  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  (Scalar `  W ) ) ) )
15 islssd.p . . . . . . . . 9  |-  ( ph  ->  .+  =  ( +g  `  W ) )
1615oveqd 6301 . . . . . . . 8  |-  ( ph  ->  ( ( x  .x.  a )  .+  b
)  =  ( ( x  .x.  a ) ( +g  `  W
) b ) )
17 islssd.t . . . . . . . . . 10  |-  ( ph  ->  .x.  =  ( .s
`  W ) )
1817oveqd 6301 . . . . . . . . 9  |-  ( ph  ->  ( x  .x.  a
)  =  ( x ( .s `  W
) a ) )
1918oveq1d 6299 . . . . . . . 8  |-  ( ph  ->  ( ( x  .x.  a ) ( +g  `  W ) b )  =  ( ( x ( .s `  W
) a ) ( +g  `  W ) b ) )
2016, 19eqtrd 2508 . . . . . . 7  |-  ( ph  ->  ( ( x  .x.  a )  .+  b
)  =  ( ( x ( .s `  W ) a ) ( +g  `  W
) b ) )
2120eleq1d 2536 . . . . . 6  |-  ( ph  ->  ( ( ( x 
.x.  a )  .+  b )  e.  U  <->  ( ( x ( .s
`  W ) a ) ( +g  `  W
) b )  e.  U ) )
22212ralbidv 2908 . . . . 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 267 . . . 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 2876 . . 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 2467 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
26 eqid 2467 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
27 eqid 2467 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
28 eqid 2467 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
29 eqid 2467 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
30 eqid 2467 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
3125, 26, 27, 28, 29, 30islss 17381 . . 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 1180 . 2  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
33 islssd.s . 2  |-  ( ph  ->  S  =  ( LSubSp `  W ) )
3432, 33eleqtrrd 2558 1  |-  ( ph  ->  U  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    C_ wss 3476   (/)c0 3785   ` cfv 5588  (class class class)co 6284   Basecbs 14490   +g cplusg 14555  Scalarcsca 14558   .scvsca 14559   LSubSpclss 17378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-lss 17379
This theorem is referenced by:  lss1  17385  lsssn0  17394  islss3  17405  lss1d  17409  lssintcl  17410  lspsolvlem  17588  lbsextlem2  17605  mpllsslem  17893  mpllsslemOLD  17895  scmatlss  18822  lincolss  32134  dialss  35861  diblss  35985  diclss  36008
  Copyright terms: Public domain W3C validator