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Theorem lspsnss 18971
Description: The span of the singleton of a subspace member is included in the subspace. (spansnss 28400 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.)
Hypotheses
Ref Expression
lspsnss.s 𝑆 = (LSubSp‘𝑊)
lspsnss.n 𝑁 = (LSpan‘𝑊)
Assertion
Ref Expression
lspsnss ((𝑊 ∈ LMod ∧ 𝑈𝑆𝑋𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈)

Proof of Theorem lspsnss
StepHypRef Expression
1 snssi 4330 . 2 (𝑋𝑈 → {𝑋} ⊆ 𝑈)
2 lspsnss.s . . 3 𝑆 = (LSubSp‘𝑊)
3 lspsnss.n . . 3 𝑁 = (LSpan‘𝑊)
42, 3lspssp 18969 . 2 ((𝑊 ∈ LMod ∧ 𝑈𝑆 ∧ {𝑋} ⊆ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈)
51, 4syl3an3 1359 1 ((𝑊 ∈ LMod ∧ 𝑈𝑆𝑋𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1481  wcel 1988  wss 3567  {csn 4168  cfv 5876  LModclmod 18844  LSubSpclss 18913  LSpanclspn 18952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-0g 16083  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-lmod 18846  df-lss 18914  df-lsp 18953
This theorem is referenced by:  lspsnel3  18972  lspsnel6  18975
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