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Theorem lspsnel3 20882
Description: A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 31402 analog.) (Contributed by NM, 4-Jul-2014.)
Hypotheses
Ref Expression
lspsnss.s 𝑆 = (LSubSpβ€˜π‘Š)
lspsnss.n 𝑁 = (LSpanβ€˜π‘Š)
lspsnel3.w (πœ‘ β†’ π‘Š ∈ LMod)
lspsnel3.u (πœ‘ β†’ π‘ˆ ∈ 𝑆)
lspsnel3.x (πœ‘ β†’ 𝑋 ∈ π‘ˆ)
lspsnel3.y (πœ‘ β†’ π‘Œ ∈ (π‘β€˜{𝑋}))
Assertion
Ref Expression
lspsnel3 (πœ‘ β†’ π‘Œ ∈ π‘ˆ)

Proof of Theorem lspsnel3
StepHypRef Expression
1 lspsnel3.w . . 3 (πœ‘ β†’ π‘Š ∈ LMod)
2 lspsnel3.u . . 3 (πœ‘ β†’ π‘ˆ ∈ 𝑆)
3 lspsnel3.x . . 3 (πœ‘ β†’ 𝑋 ∈ π‘ˆ)
4 lspsnss.s . . . 4 𝑆 = (LSubSpβ€˜π‘Š)
5 lspsnss.n . . . 4 𝑁 = (LSpanβ€˜π‘Š)
64, 5lspsnss 20881 . . 3 ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆 ∧ 𝑋 ∈ π‘ˆ) β†’ (π‘β€˜{𝑋}) βŠ† π‘ˆ)
71, 2, 3, 6syl3anc 1368 . 2 (πœ‘ β†’ (π‘β€˜{𝑋}) βŠ† π‘ˆ)
8 lspsnel3.y . 2 (πœ‘ β†’ π‘Œ ∈ (π‘β€˜{𝑋}))
97, 8sseldd 3983 1 (πœ‘ β†’ π‘Œ ∈ π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949  {csn 4632  β€˜cfv 6553  LModclmod 20750  LSubSpclss 20822  LSpanclspn 20862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-lmod 20752  df-lss 20823  df-lsp 20863
This theorem is referenced by:  lspsnel4  21019
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