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

Theorem lssuni 15697
Description: The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.)
Hypotheses
Ref Expression
lssss.v  |-  V  =  ( Base `  W
)
lssss.s  |-  S  =  ( LSubSp `  W )
lssuni.w  |-  ( ph  ->  W  e.  LMod )
Assertion
Ref Expression
lssuni  |-  ( ph  ->  U. S  =  V )

Proof of Theorem lssuni
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rabid2 2717 . . . 4  |-  ( S  =  { x  e.  S  |  x  C_  V }  <->  A. x  e.  S  x  C_  V )
2 lssss.v . . . . 5  |-  V  =  ( Base `  W
)
3 lssss.s . . . . 5  |-  S  =  ( LSubSp `  W )
42, 3lssss 15694 . . . 4  |-  ( x  e.  S  ->  x  C_  V )
51, 4mprgbir 2613 . . 3  |-  S  =  { x  e.  S  |  x  C_  V }
65unieqi 3837 . 2  |-  U. S  =  U. { x  e.  S  |  x  C_  V }
7 lssuni.w . . 3  |-  ( ph  ->  W  e.  LMod )
82, 3lss1 15696 . . 3  |-  ( W  e.  LMod  ->  V  e.  S )
9 unimax 3861 . . 3  |-  ( V  e.  S  ->  U. {
x  e.  S  |  x  C_  V }  =  V )
107, 8, 93syl 18 . 2  |-  ( ph  ->  U. { x  e.  S  |  x  C_  V }  =  V
)
116, 10syl5eq 2327 1  |-  ( ph  ->  U. S  =  V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   U.cuni 3827   ` cfv 5255   Basecbs 13148   LModclmod 15627   LSubSpclss 15689
This theorem is referenced by:  mapdunirnN  31840
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-reu 2550  df-rmo 2551  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-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-lmod 15629  df-lss 15690
  Copyright terms: Public domain W3C validator