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Theorem lssuni 20937
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 3470 . . . 4 (𝑆 = {𝑥𝑆𝑥𝑉} ↔ ∀𝑥𝑆 𝑥𝑉)
2 lssss.v . . . . 5 𝑉 = (Base‘𝑊)
3 lssss.s . . . . 5 𝑆 = (LSubSp‘𝑊)
42, 3lssss 20934 . . . 4 (𝑥𝑆𝑥𝑉)
51, 4mprgbir 3068 . . 3 𝑆 = {𝑥𝑆𝑥𝑉}
65unieqi 4919 . 2 𝑆 = {𝑥𝑆𝑥𝑉}
7 lssuni.w . . 3 (𝜑𝑊 ∈ LMod)
82, 3lss1 20936 . . 3 (𝑊 ∈ LMod → 𝑉𝑆)
9 unimax 4944 . . 3 (𝑉𝑆 {𝑥𝑆𝑥𝑉} = 𝑉)
107, 8, 93syl 18 . 2 (𝜑 {𝑥𝑆𝑥𝑉} = 𝑉)
116, 10eqtrid 2789 1 (𝜑 𝑆 = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {crab 3436  wss 3951   cuni 4907  cfv 6561  Basecbs 17247  LModclmod 20858  LSubSpclss 20929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-riota 7388  df-ov 7434  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-lmod 20860  df-lss 20930
This theorem is referenced by:  mapdunirnN  41652
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