Theorem lshpne 39784

Description: A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)

Hypotheses

Ref	Expression
lshpne.v	⊢ 𝑉 = (Base‘𝑊)
lshpne.h	⊢ 𝐻 = (LSHyp‘𝑊)
lshpne.w	⊢ (𝜑 → 𝑊 ∈ LMod)
lshpne.u	⊢ (𝜑 → 𝑈 ∈ 𝐻)

Assertion

Ref	Expression
lshpne	⊢ (𝜑 → 𝑈 ≠ 𝑉)

Proof of Theorem lshpne

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lshpne.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐻)
2		lshpne.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod)
3		lshpne.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
4		eqid 2762	. . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
5		eqid 2762	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
6		lshpne.h	. . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊)
7	3, 4, 5, 6	islshp 39781	. . . 4 ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉)))
8	2, 7	syl 18	. . 3 ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉)))
9	1, 8	mpbid 235	. 2 ⊢ (𝜑 → (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉))
10	9	simp2d 1160	1 ⊢ (𝜑 → 𝑈 ≠ 𝑉)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3088   ∪ cun 3902  {csn 4588  ‘cfv 6536  Basecbs 17275  LModclmod 20992  LSubSpclss 21063  LSpanclspn 21103  LSHypclsh 39777

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-lshyp 39779

This theorem is used by:  lshpnel  39785  lshpcmp  39790  lkrshp3  39908  lkrshp4  39910  dochshpncl  42186  dochlkr  42187  dochkrshp  42188  dochsatshpb  42254