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Theorem lssuni 19332
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 3304 . . . 4 (𝑆 = {𝑥𝑆𝑥𝑉} ↔ ∀𝑥𝑆 𝑥𝑉)
2 lssss.v . . . . 5 𝑉 = (Base‘𝑊)
3 lssss.s . . . . 5 𝑆 = (LSubSp‘𝑊)
42, 3lssss 19329 . . . 4 (𝑥𝑆𝑥𝑉)
51, 4mprgbir 3108 . . 3 𝑆 = {𝑥𝑆𝑥𝑉}
65unieqi 4680 . 2 𝑆 = {𝑥𝑆𝑥𝑉}
7 lssuni.w . . 3 (𝜑𝑊 ∈ LMod)
82, 3lss1 19331 . . 3 (𝑊 ∈ LMod → 𝑉𝑆)
9 unimax 4708 . . 3 (𝑉𝑆 {𝑥𝑆𝑥𝑉} = 𝑉)
107, 8, 93syl 18 . 2 (𝜑 {𝑥𝑆𝑥𝑉} = 𝑉)
116, 10syl5eq 2825 1 (𝜑 𝑆 = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2106  {crab 3093  wss 3791   cuni 4671  cfv 6135  Basecbs 16255  LModclmod 19255  LSubSpclss 19324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-riota 6883  df-ov 6925  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-grp 17812  df-lmod 19257  df-lss 19325
This theorem is referenced by:  mapdunirnN  37788
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