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Theorem lssuni 18707
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 3095 . . . 4 (𝑆 = {𝑥𝑆𝑥𝑉} ↔ ∀𝑥𝑆 𝑥𝑉)
2 lssss.v . . . . 5 𝑉 = (Base‘𝑊)
3 lssss.s . . . . 5 𝑆 = (LSubSp‘𝑊)
42, 3lssss 18704 . . . 4 (𝑥𝑆𝑥𝑉)
51, 4mprgbir 2910 . . 3 𝑆 = {𝑥𝑆𝑥𝑉}
65unieqi 4375 . 2 𝑆 = {𝑥𝑆𝑥𝑉}
7 lssuni.w . . 3 (𝜑𝑊 ∈ LMod)
82, 3lss1 18706 . . 3 (𝑊 ∈ LMod → 𝑉𝑆)
9 unimax 4403 . . 3 (𝑉𝑆 {𝑥𝑆𝑥𝑉} = 𝑉)
107, 8, 93syl 18 . 2 (𝜑 {𝑥𝑆𝑥𝑉} = 𝑉)
116, 10syl5eq 2655 1 (𝜑 𝑆 = 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  {crab 2899  wss 3539   cuni 4366  cfv 5790  Basecbs 15641  LModclmod 18632  LSubSpclss 18699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-riota 6489  df-ov 6530  df-0g 15871  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-grp 17194  df-lmod 18634  df-lss 18700
This theorem is referenced by:  mapdunirnN  35753
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