MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lspsnss Structured version   Visualization version   GIF version

Theorem lspsnss 19757
Description: The span of the singleton of a subspace member is included in the subspace. (spansnss 29352 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 4726 . 2 (𝑋𝑈 → {𝑋} ⊆ 𝑈)
2 lspsnss.s . . 3 𝑆 = (LSubSp‘𝑊)
3 lspsnss.n . . 3 𝑁 = (LSpan‘𝑊)
42, 3lspssp 19755 . 2 ((𝑊 ∈ LMod ∧ 𝑈𝑆 ∧ {𝑋} ⊆ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈)
51, 4syl3an3 1162 1 ((𝑊 ∈ LMod ∧ 𝑈𝑆𝑋𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  wss 3919  {csn 4550  cfv 6344  LModclmod 19629  LSubSpclss 19698  LSpanclspn 19738
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 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
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 4826  df-int 4864  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7104  df-ov 7149  df-0g 16713  df-mgm 17850  df-sgrp 17899  df-mnd 17910  df-grp 18104  df-lmod 19631  df-lss 19699  df-lsp 19739
This theorem is referenced by:  lspsnel3  19758  lspsnel6  19761
  Copyright terms: Public domain W3C validator