Theorem lshpne 40231

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 2763	. . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
5		eqid 2763	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
6		lshpne.h	. . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊)
7	3, 4, 5, 6	islshp 40228	. . . 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 1161	1 ⊢ (𝜑 → 𝑈 ≠ 𝑉)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090   ∪ cun 4152  {csn 4588  ‘cfv 6558  Basecbs 17416  LModclmod 21165  LSubSpclss 21236  LSpanclspn 21276  LSHypclsh 40224

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fv 6566  df-lshyp 40226

This theorem is used by:  lshpnel  40232  lshpcmp  40237  lkrshp3  40355  lkrshp4  40357  dochshpncl  42633  dochlkr  42634  dochkrshp  42635  dochsatshpb  42701