Theorem lshpne 38983

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 2737	. . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
5		eqid 2737	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
6		lshpne.h	. . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊)
7	3, 4, 5, 6	islshp 38980	. . . 4 ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉)))
8	2, 7	syl 17	. . 3 ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉)))
9	1, 8	mpbid 232	. 2 ⊢ (𝜑 → (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉))
10	9	simp2d 1144	1 ⊢ (𝜑 → 𝑈 ≠ 𝑉)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070   ∪ cun 3949  {csn 4626  ‘cfv 6561  Basecbs 17247  LModclmod 20858  LSubSpclss 20929  LSpanclspn 20969  LSHypclsh 38976

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-lshyp 38978

This theorem is referenced by:  lshpnel  38984  lshpcmp  38989  lkrshp3  39107  lkrshp4  39109  dochshpncl  41386  dochlkr  41387  dochkrshp  41388  dochsatshpb  41454