Theorem lshplss 36121

Description: A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)

Hypotheses

Ref	Expression
lshplss.s	⊢ 𝑆 = (LSubSp‘𝑊)
lshplss.h	⊢ 𝐻 = (LSHyp‘𝑊)
lshplss.w	⊢ (𝜑 → 𝑊 ∈ LMod)
lshplss.u	⊢ (𝜑 → 𝑈 ∈ 𝐻)

Assertion

Ref	Expression
lshplss	⊢ (𝜑 → 𝑈 ∈ 𝑆)

Proof of Theorem lshplss

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lshplss.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐻)
2		lshplss.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod)
3		eqid 2824	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		eqid 2824	. . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
5		lshplss.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊)
6		lshplss.h	. . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊)
7	3, 4, 5, 6	islshp 36119	. . . 4 ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊))))
8	2, 7	syl 17	. . 3 ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊))))
9	1, 8	mpbid 234	. 2 ⊢ (𝜑 → (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊)))
10	9	simp1d 1138	1 ⊢ (𝜑 → 𝑈 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113   ≠ wne 3019  ∃wrex 3142   ∪ cun 3937  {csn 4570  ‘cfv 6358  Basecbs 16486  LModclmod 19637  LSubSpclss 19706  LSpanclspn 19746  LSHypclsh 36115

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fv 6366  df-lshyp 36117

This theorem is referenced by:  lshpnel  36123  lshpnelb  36124  lshpne0  36126  lshpdisj  36127  lshpcmp  36128  lshpsmreu  36249  lshpkrlem1  36250  lshpkrlem5  36254  lshpkr  36257  dochshpncl  38524  dochshpsat  38594  lclkrlem2f  38652