Theorem lshpne 39087

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 2731	. . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
5		eqid 2731	. . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
6		lshpne.h	. . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊)
7	3, 4, 5, 6	islshp 39084	. . . 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 1143	1 ⊢ (𝜑 → 𝑈 ≠ 𝑉)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056   ∪ cun 3895  {csn 4575  ‘cfv 6487  Basecbs 17126  LModclmod 20799  LSubSpclss 20870  LSpanclspn 20910  LSHypclsh 39080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6443  df-fun 6489  df-fv 6495  df-lshyp 39082

This theorem is referenced by:  lshpnel  39088  lshpcmp  39093  lkrshp3  39211  lkrshp4  39213  dochshpncl  41489  dochlkr  41490  dochkrshp  41491  dochsatshpb  41557