Theorem lshplss 39241

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 2736	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		eqid 2736	. . . . 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 39239	. . . 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 232	. 2 ⊢ (𝜑 → (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊)))
10	9	simp1d 1142	1 ⊢ (𝜑 → 𝑈 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∃wrex 3060   ∪ cun 3899  {csn 4580  ‘cfv 6492  Basecbs 17136  LModclmod 20811  LSubSpclss 20882  LSpanclspn 20922  LSHypclsh 39235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-lshyp 39237

This theorem is referenced by:  lshpnel  39243  lshpnelb  39244  lshpne0  39246  lshpdisj  39247  lshpcmp  39248  lshpsmreu  39369  lshpkrlem1  39370  lshpkrlem5  39374  lshpkr  39377  dochshpncl  41644  dochshpsat  41714  lclkrlem2f  41772