Theorem lshplss 38979

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 2729	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		eqid 2729	. . . . 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 38977	. . . 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 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   ∪ cun 3903  {csn 4579  ‘cfv 6486  Basecbs 17139  LModclmod 20782  LSubSpclss 20853  LSpanclspn 20893  LSHypclsh 38973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-lshyp 38975

This theorem is referenced by:  lshpnel  38981  lshpnelb  38982  lshpne0  38984  lshpdisj  38985  lshpcmp  38986  lshpsmreu  39107  lshpkrlem1  39108  lshpkrlem5  39112  lshpkr  39115  dochshpncl  41383  dochshpsat  41453  lclkrlem2f  41511