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Theorem lspsnel3 19763
 Description: A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 29358 analog.) (Contributed by NM, 4-Jul-2014.)
Hypotheses
Ref Expression
lspsnss.s 𝑆 = (LSubSp‘𝑊)
lspsnss.n 𝑁 = (LSpan‘𝑊)
lspsnel3.w (𝜑𝑊 ∈ LMod)
lspsnel3.u (𝜑𝑈𝑆)
lspsnel3.x (𝜑𝑋𝑈)
lspsnel3.y (𝜑𝑌 ∈ (𝑁‘{𝑋}))
Assertion
Ref Expression
lspsnel3 (𝜑𝑌𝑈)

Proof of Theorem lspsnel3
StepHypRef Expression
1 lspsnel3.w . . 3 (𝜑𝑊 ∈ LMod)
2 lspsnel3.u . . 3 (𝜑𝑈𝑆)
3 lspsnel3.x . . 3 (𝜑𝑋𝑈)
4 lspsnss.s . . . 4 𝑆 = (LSubSp‘𝑊)
5 lspsnss.n . . . 4 𝑁 = (LSpan‘𝑊)
64, 5lspsnss 19762 . . 3 ((𝑊 ∈ LMod ∧ 𝑈𝑆𝑋𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈)
71, 2, 3, 6syl3anc 1368 . 2 (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈)
8 lspsnel3.y . 2 (𝜑𝑌 ∈ (𝑁‘{𝑋}))
97, 8sseldd 3954 1 (𝜑𝑌𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115   ⊆ wss 3919  {csn 4550  ‘cfv 6343  LModclmod 19634  LSubSpclss 19703  LSpanclspn 19743 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-lmod 19636  df-lss 19704  df-lsp 19744 This theorem is referenced by:  lspsnel4  19896
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