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

Theorem lssuni 16008
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 2877 . . . 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 16005 . . . 4  |-  ( x  e.  S  ->  x  C_  V )
51, 4mprgbir 2768 . . 3  |-  S  =  { x  e.  S  |  x  C_  V }
65unieqi 4017 . 2  |-  U. S  =  U. { x  e.  S  |  x  C_  V }
7 lssuni.w . . 3  |-  ( ph  ->  W  e.  LMod )
82, 3lss1 16007 . . 3  |-  ( W  e.  LMod  ->  V  e.  S )
9 unimax 4041 . . 3  |-  ( V  e.  S  ->  U. {
x  e.  S  |  x  C_  V }  =  V )
107, 8, 93syl 19 . 2  |-  ( ph  ->  U. { x  e.  S  |  x  C_  V }  =  V
)
116, 10syl5eq 2479 1  |-  ( ph  ->  U. S  =  V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {crab 2701    C_ wss 3312   U.cuni 4007   ` cfv 5446   Basecbs 13461   LModclmod 15942   LSubSpclss 16000
This theorem is referenced by:  mapdunirnN  32385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-riota 6541  df-0g 13719  df-mnd 14682  df-grp 14804  df-lmod 15944  df-lss 16001
  Copyright terms: Public domain W3C validator