Theorem lshpne 39702

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 2761	. . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
5		eqid 2761	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
6		lshpne.h	. . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊)
7	3, 4, 5, 6	islshp 39699	. . . 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 1159	1 ⊢ (𝜑 → 𝑈 ≠ 𝑉)

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3087   ∪ cun 3902  {csn 4588  ‘cfv 6536  Basecbs 17268  LModclmod 20960  LSubSpclss 21031  LSpanclspn 21071  LSHypclsh 39695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  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 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-lshyp 39697

This theorem is referenced by:  lshpnel  39703  lshpcmp  39708  lkrshp3  39826  lkrshp4  39828  dochshpncl  42104  dochlkr  42105  dochkrshp  42106  dochsatshpb  42172