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Theorem lshpne 37473
Description: A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)
Hypotheses
Ref Expression
lshpne.v 𝑉 = (Baseβ€˜π‘Š)
lshpne.h 𝐻 = (LSHypβ€˜π‘Š)
lshpne.w (πœ‘ β†’ π‘Š ∈ LMod)
lshpne.u (πœ‘ β†’ π‘ˆ ∈ 𝐻)
Assertion
Ref Expression
lshpne (πœ‘ β†’ π‘ˆ β‰  𝑉)

Proof of Theorem lshpne
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 lshpne.u . . 3 (πœ‘ β†’ π‘ˆ ∈ 𝐻)
2 lshpne.w . . . 4 (πœ‘ β†’ π‘Š ∈ LMod)
3 lshpne.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
4 eqid 2737 . . . . 5 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
5 eqid 2737 . . . . 5 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
6 lshpne.h . . . . 5 𝐻 = (LSHypβ€˜π‘Š)
73, 4, 5, 6islshp 37470 . . . 4 (π‘Š ∈ LMod β†’ (π‘ˆ ∈ 𝐻 ↔ (π‘ˆ ∈ (LSubSpβ€˜π‘Š) ∧ π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 ((LSpanβ€˜π‘Š)β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉)))
82, 7syl 17 . . 3 (πœ‘ β†’ (π‘ˆ ∈ 𝐻 ↔ (π‘ˆ ∈ (LSubSpβ€˜π‘Š) ∧ π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 ((LSpanβ€˜π‘Š)β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉)))
91, 8mpbid 231 . 2 (πœ‘ β†’ (π‘ˆ ∈ (LSubSpβ€˜π‘Š) ∧ π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 ((LSpanβ€˜π‘Š)β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉))
109simp2d 1144 1 (πœ‘ β†’ π‘ˆ β‰  𝑉)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074   βˆͺ cun 3913  {csn 4591  β€˜cfv 6501  Basecbs 17090  LModclmod 20338  LSubSpclss 20408  LSpanclspn 20448  LSHypclsh 37466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-lshyp 37468
This theorem is referenced by:  lshpnel  37474  lshpcmp  37479  lkrshp3  37597  lkrshp4  37599  dochshpncl  39876  dochlkr  39877  dochkrshp  39878  dochsatshpb  39944
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