Theorem lshplss 37021

Description: A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)

Hypotheses

Ref	Expression
lshplss.s	⊢ 𝑆 = (LSubSp‘𝑊)
lshplss.h	⊢ 𝐻 = (LSHyp‘𝑊)
lshplss.w	⊢ (𝜑 → 𝑊 ∈ LMod)
lshplss.u	⊢ (𝜑 → 𝑈 ∈ 𝐻)

Assertion

Ref	Expression
lshplss	⊢ (𝜑 → 𝑈 ∈ 𝑆)

Proof of Theorem lshplss

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lshplss.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐻)
2		lshplss.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod)
3		eqid 2733	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		eqid 2733	. . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
5		lshplss.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊)
6		lshplss.h	. . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊)
7	3, 4, 5, 6	islshp 37019	. . . 4 ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊))))
8	2, 7	syl 17	. . 3 ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊))))
9	1, 8	mpbid 231	. 2 ⊢ (𝜑 → (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊)))
10	9	simp1d 1140	1 ⊢ (𝜑 → 𝑈 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085   = wceq 1537   ∈ wcel 2101   ≠ wne 2938  ∃wrex 3068   ∪ cun 3887  {csn 4564  ‘cfv 6447  Basecbs 16940  LModclmod 20151  LSubSpclss 20221  LSpanclspn 20261  LSHypclsh 37015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-iota 6399  df-fun 6449  df-fv 6455  df-lshyp 37017

This theorem is referenced by:  lshpnel  37023  lshpnelb  37024  lshpne0  37026  lshpdisj  37027  lshpcmp  37028  lshpsmreu  37149  lshpkrlem1  37150  lshpkrlem5  37154  lshpkr  37157  dochshpncl  39424  dochshpsat  39494  lclkrlem2f  39552