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Theorem lssuni 20823
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 (𝜑𝑊 ∈ LMod)
Assertion
Ref Expression
lssuni (𝜑 𝑆 = 𝑉)

Proof of Theorem lssuni
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rabid2 3461 . . . 4 (𝑆 = {𝑥𝑆𝑥𝑉} ↔ ∀𝑥𝑆 𝑥𝑉)
2 lssss.v . . . . 5 𝑉 = (Base‘𝑊)
3 lssss.s . . . . 5 𝑆 = (LSubSp‘𝑊)
42, 3lssss 20820 . . . 4 (𝑥𝑆𝑥𝑉)
51, 4mprgbir 3065 . . 3 𝑆 = {𝑥𝑆𝑥𝑉}
65unieqi 4920 . 2 𝑆 = {𝑥𝑆𝑥𝑉}
7 lssuni.w . . 3 (𝜑𝑊 ∈ LMod)
82, 3lss1 20822 . . 3 (𝑊 ∈ LMod → 𝑉𝑆)
9 unimax 4947 . . 3 (𝑉𝑆 {𝑥𝑆𝑥𝑉} = 𝑉)
107, 8, 93syl 18 . 2 (𝜑 {𝑥𝑆𝑥𝑉} = 𝑉)
116, 10eqtrid 2780 1 (𝜑 𝑆 = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  {crab 3429  wss 3947   cuni 4908  cfv 6548  Basecbs 17180  LModclmod 20743  LSubSpclss 20815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-riota 7376  df-ov 7423  df-0g 17423  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-grp 18893  df-lmod 20745  df-lss 20816
This theorem is referenced by:  mapdunirnN  41123
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