Theorem lshplss 34268

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 2622	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		eqid 2622	. . . . 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 34266	. . . 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 222	. 2 ⊢ (𝜑 → (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊)))
10	9	simp1d 1073	1 ⊢ (𝜑 → 𝑈 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   ∪ cun 3572  {csn 4177  ‘cfv 5888  Basecbs 15857  LModclmod 18863  LSubSpclss 18932  LSpanclspn 18971  LSHypclsh 34262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-lshyp 34264

This theorem is referenced by:  lshpnel  34270  lshpnelb  34271  lshpne0  34273  lshpdisj  34274  lshpcmp  34275  lshpsmreu  34396  lshpkrlem1  34397  lshpkrlem5  34401  lshpkr  34404  dochshpncl  36673  dochshpsat  36743  lclkrlem2f  36801