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Theorem lssuni 20955
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 3468 . . . 4 (𝑆 = {𝑥𝑆𝑥𝑉} ↔ ∀𝑥𝑆 𝑥𝑉)
2 lssss.v . . . . 5 𝑉 = (Base‘𝑊)
3 lssss.s . . . . 5 𝑆 = (LSubSp‘𝑊)
42, 3lssss 20952 . . . 4 (𝑥𝑆𝑥𝑉)
51, 4mprgbir 3066 . . 3 𝑆 = {𝑥𝑆𝑥𝑉}
65unieqi 4924 . 2 𝑆 = {𝑥𝑆𝑥𝑉}
7 lssuni.w . . 3 (𝜑𝑊 ∈ LMod)
82, 3lss1 20954 . . 3 (𝑊 ∈ LMod → 𝑉𝑆)
9 unimax 4949 . . 3 (𝑉𝑆 {𝑥𝑆𝑥𝑉} = 𝑉)
107, 8, 93syl 18 . 2 (𝜑 {𝑥𝑆𝑥𝑉} = 𝑉)
116, 10eqtrid 2787 1 (𝜑 𝑆 = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  wss 3963   cuni 4912  cfv 6563  Basecbs 17245  LModclmod 20875  LSubSpclss 20947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-lmod 20877  df-lss 20948
This theorem is referenced by:  mapdunirnN  41633
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