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Theorem lssuni 15936
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 2821 . . . 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 15933 . . . 4  |-  ( x  e.  S  ->  x  C_  V )
51, 4mprgbir 2712 . . 3  |-  S  =  { x  e.  S  |  x  C_  V }
65unieqi 3960 . 2  |-  U. S  =  U. { x  e.  S  |  x  C_  V }
7 lssuni.w . . 3  |-  ( ph  ->  W  e.  LMod )
82, 3lss1 15935 . . 3  |-  ( W  e.  LMod  ->  V  e.  S )
9 unimax 3984 . . 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 2424 1  |-  ( ph  ->  U. S  =  V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   {crab 2646    C_ wss 3256   U.cuni 3950   ` cfv 5387   Basecbs 13389   LModclmod 15870   LSubSpclss 15928
This theorem is referenced by:  mapdunirnN  31816
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-riota 6478  df-0g 13647  df-mnd 14610  df-grp 14732  df-lmod 15872  df-lss 15929
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