P. 424 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42301-42400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dihjatcclem3 42301*	Lemma for dihjatcc 42303. (Contributed by NM, 28-Sep-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))    &   ⊢ 𝐶 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐺 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑃)    &   ⊢ 𝐷 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑄)    ⇒   ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉)
Theorem	dihjatcclem4 42302*	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))    &   ⊢ 𝐶 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐺 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑃)    &   ⊢ 𝐷 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑄)    &   ⊢ 𝑁 = (𝑎 ∈ 𝐸 ↦ (𝑑 ∈ 𝑇 ↦ ◡(𝑎‘𝑑)))    &   ⊢ 0 = (𝑑 ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝐽 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑑 ∈ 𝑇 ↦ ((𝑎‘𝑑) ∘ (𝑏‘𝑑))))    ⇒   ⊢ (𝜑 → (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))
Theorem	dihjatcc 42303	Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))    ⇒   ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))
Theorem	dihjat 42304	Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))
Theorem	dihprrnlem1N 42305	Lemma for dihprrn 42307, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    &   ⊢ (𝜑 → (◡𝐼‘(𝑁‘{𝑋})) ≤ 𝑊)    &   ⊢ (𝜑 → ¬ (◡𝐼‘(𝑁‘{𝑌})) ≤ 𝑊)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)
Theorem	dihprrnlem2 42306	Lemma for dihprrn 42307. (Contributed by NM, 29-Sep-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)
Theorem	dihprrn 42307	The span of a vector pair belongs to the range of isomorphism H i.e. is a closed subspace. (Contributed by NM, 29-Sep-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)
Theorem	djhlsmat 42308	The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 42307; should we directly use dihjat 42304? (Contributed by NM, 13-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = ((𝑁‘{𝑋}) ∨ (𝑁‘{𝑌})))
Theorem	dihjat1lem 42309	Subspace sum of a closed subspace and an atom. (pmapjat1 40734 analog.) TODO: merge into dihjat1 42310? (Contributed by NM, 18-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝑋 ∨ (𝑁‘{𝑇})) = (𝑋 ⊕ (𝑁‘{𝑇})))
Theorem	dihjat1 42310	Subspace sum of a closed subspace and an atom. (pmapjat1 40734 analog.) (Contributed by NM, 1-Oct-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ∨ (𝑁‘{𝑇})) = (𝑋 ⊕ (𝑁‘{𝑇})))
Theorem	dihsmsprn 42311	Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ⊕ (𝑁‘{𝑇})) ∈ ran 𝐼)
Theorem	dihjat2 42312	The subspace sum of a closed subspace and an atom is the same as their subspace join. (Contributed by NM, 1-Oct-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋 ∨ 𝑄) = (𝑋 ⊕ 𝑄))
Theorem	dihjat3 42313	Isomorphism H of lattice join with an atom. (Contributed by NM, 25-Apr-2015.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐼‘(𝑋 ∨ 𝑃)) = ((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))
Theorem	dihjat4 42314	Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋 ⊕ 𝑄) = (𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑄))))
Theorem	dihjat6 42315	Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (◡𝐼‘(𝑋 ⊕ 𝑄)) = ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑄)))
Theorem	dihsmsnrn 42316	The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ ran 𝐼)
Theorem	dihsmatrn 42317	The subspace sum of a closed subspace and an atom is closed. TODO: see if proof at https://math.stackexchange.com/a/1233211/50776 and Mon, 13 Apr 2015 20:44:07 -0400 email could be used instead of this and dihjat2 42312. (Contributed by NM, 15-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋 ⊕ 𝑄) ∈ ran 𝐼)
Theorem	dihjat5N 42318	Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋 ∨ 𝑃) = (◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃))))
Theorem	dvh4dimat 42319*	There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅))
Theorem	dvh3dimatN 42320*	There is an atom that is outside the subspace sum of 2 others. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ⊆ (𝑃 ⊕ 𝑄))
Theorem	dvh2dimatN 42321*	Given an atom, there exists another. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ 𝐴 𝑠 ≠ 𝑃)
Theorem	dvh1dimat 42322*	There exists an atom. (Contributed by NM, 25-Apr-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → ∃𝑠 𝑠 ∈ 𝐴)
Theorem	dvh1dim 42323*	There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 𝑧 ≠ 0 )
Theorem	dvh4dimlem 42324*	Lemma for dvh4dimN 42328. (Contributed by NM, 22-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    &   ⊢ (𝜑 → 𝑍 ≠ 0 )    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌, 𝑍}))
Theorem	dvhdimlem 42325*	Lemma for dvh2dim 42326 and dvh3dim 42327. TODO: make this obsolete and use dvh4dimlem 42324 directly? (Contributed by NM, 24-Apr-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))
Theorem	dvh2dim 42326*	There is a vector that is outside the span of another. (Contributed by NM, 25-Apr-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋}))
Theorem	dvh3dim 42327*	There is a vector that is outside the span of 2 others. (Contributed by NM, 24-Apr-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))
Theorem	dvh4dimN 42328*	There is a vector that is outside the span of 3 others. (Contributed by NM, 22-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌, 𝑍}))
Theorem	dvh3dim2 42329*	There is a vector that is outside of 2 spans with a common vector. (Contributed by NM, 13-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑍})))
Theorem	dvh3dim3N 42330*	There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 42329 everywhere. If this one is needed, make dvh3dim2 42329 into a lemma. (Contributed by NM, 21-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
Theorem	dochsnnz 42331	The orthocomplement of a singleton is nonzero. (Contributed by NM, 13-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ { 0 })
Theorem	dochsatshp 42332	The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 27-Jul-2014.) (Revised by Mario Carneiro, 1-Oct-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ 𝑌 = (LSHyp‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ 𝑌)
Theorem	dochsatshpb 42333	The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 29-Oct-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ 𝑌 = (LSHyp‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑄 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ( ⊥ ‘𝑄) ∈ 𝑌))
Theorem	dochsnshp 42334	The orthocomplement of a nonzero singleton is a hyperplane. (Contributed by NM, 3-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑌 = (LSHyp‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ 𝑌)
Theorem	dochshpsat 42335	A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ 𝑌 = (LSHyp‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑌)    ⇒   ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ↔ ( ⊥ ‘𝑋) ∈ 𝐴))
Theorem	dochkrsat 42336	The orthocomplement of a kernel is an atom iff it is nonzero. (Contributed by NM, 1-Nov-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ≠ { 0 } ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴))
Theorem	dochkrsat2 42337	The orthocomplement of a kernel is an atom iff the double orthocomplement is not the vector space. (Contributed by NM, 1-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴))
Theorem	dochsat0 42338	The orthocomplement of a kernel is either an atom or zero. (Contributed by NM, 29-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴 ∨ ( ⊥ ‘(𝐿‘𝐺)) = { 0 }))
Theorem	dochkrsm 42339	The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 42295 can be used to convert sum to join.) (Contributed by NM, 29-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘(𝐿‘𝐺))) ∈ ran 𝐼)
Theorem	dochexmidat 42340	Special case of excluded middle for the singleton of a vector. (Contributed by NM, 27-Oct-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (( ⊥ ‘{𝑋}) ⊕ (𝑁‘{𝑋})) = 𝑉)
Theorem	dochexmidlem1 42341	Lemma for dochexmid 42349. Holland's proof implicitly requires 𝑞 ≠ 𝑟, which we prove here. (Contributed by NM, 14-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑝 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑞 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑟 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑞 ⊆ ( ⊥ ‘𝑋))    &   ⊢ (𝜑 → 𝑟 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → 𝑞 ≠ 𝑟)
Theorem	dochexmidlem2 42342	Lemma for dochexmid 42349. (Contributed by NM, 14-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑝 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑞 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑟 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑞 ⊆ ( ⊥ ‘𝑋))    &   ⊢ (𝜑 → 𝑟 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑝 ⊆ (𝑟 ⊕ 𝑞))    ⇒   ⊢ (𝜑 → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
Theorem	dochexmidlem3 42343	Lemma for dochexmid 42349. Use atom exchange lsatexch1 39927 to swap 𝑝 and 𝑞. (Contributed by NM, 14-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑝 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑞 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑟 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑞 ⊆ ( ⊥ ‘𝑋))    &   ⊢ (𝜑 → 𝑟 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑞 ⊆ (𝑟 ⊕ 𝑝))    ⇒   ⊢ (𝜑 → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
Theorem	dochexmidlem4 42344	Lemma for dochexmid 42349. (Contributed by NM, 15-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑝 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑞 ∈ 𝐴)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑀 = (𝑋 ⊕ 𝑝)    &   ⊢ (𝜑 → 𝑋 ≠ { 0 })    &   ⊢ (𝜑 → 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀))    ⇒   ⊢ (𝜑 → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
Theorem	dochexmidlem5 42345	Lemma for dochexmid 42349. (Contributed by NM, 15-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑝 ∈ 𝐴)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑀 = (𝑋 ⊕ 𝑝)    &   ⊢ (𝜑 → 𝑋 ≠ { 0 })    &   ⊢ (𝜑 → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))    ⇒   ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑀) = { 0 })
Theorem	dochexmidlem6 42346	Lemma for dochexmid 42349. (Contributed by NM, 15-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑝 ∈ 𝐴)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑀 = (𝑋 ⊕ 𝑝)    &   ⊢ (𝜑 → 𝑋 ≠ { 0 })    &   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)    &   ⊢ (𝜑 → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))    ⇒   ⊢ (𝜑 → 𝑀 = 𝑋)
Theorem	dochexmidlem7 42347	Lemma for dochexmid 42349. Contradict dochexmidlem6 42346. (Contributed by NM, 15-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑝 ∈ 𝐴)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑀 = (𝑋 ⊕ 𝑝)    &   ⊢ (𝜑 → 𝑋 ≠ { 0 })    &   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)    &   ⊢ (𝜑 → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))    ⇒   ⊢ (𝜑 → 𝑀 ≠ 𝑋)
Theorem	dochexmidlem8 42348	Lemma for dochexmid 42349. The contradiction of dochexmidlem6 42346 and dochexmidlem7 42347 shows that there can be no atom 𝑝 that is not in 𝑋 + ( ⊥ ‘𝑋), which is therefore the whole atom space. (Contributed by NM, 15-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑋 ≠ { 0 })    &   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)    ⇒   ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) = 𝑉)
Theorem	dochexmid 42349	Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 42258. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables 𝑋, 𝑀, 𝑝, 𝑞, 𝑟 in place of Hollands' l, m, P, Q, L respectively. (pexmidALTN 40859 analog.) (Contributed by NM, 15-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)    ⇒   ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) = 𝑉)
Theorem	dochsnkrlem1 42350	Lemma for dochsnkr 42353. (Contributed by NM, 1-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)
Theorem	dochsnkrlem2 42351	Lemma for dochsnkr 42353. (Contributed by NM, 1-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    ⇒   ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴)
Theorem	dochsnkrlem3 42352	Lemma for dochsnkr 42353. (Contributed by NM, 2-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
Theorem	dochsnkr 42353	A (closed) kernel expressed in terms of a nonzero vector in its orthocomplement. TODO: consolidate lemmas unless they're needed for something else (in which case break out as theorems). (Contributed by NM, 2-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋}))
Theorem	dochsnkr2 42354*	Kernel of the explicit functional 𝐺 determined by a nonzero vector 𝑋. Compare the more general lshpkr 39998. (Contributed by NM, 27-Oct-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝐷)    &   ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋}))
Theorem	dochsnkr2cl 42355*	The 𝑋 determining functional 𝐺 belongs to the atom formed by the orthocomplement of the kernel. (Contributed by NM, 4-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝐷)    &   ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))
Theorem	dochflcl 42356*	Closure of the explicit functional 𝐺 determined by a nonzero vector 𝑋. Compare the more general lshpkrcl 39997. (Contributed by NM, 27-Oct-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝐷)    &   ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → 𝐺 ∈ 𝐹)
Theorem	dochfl1 42357*	The value of the explicit functional 𝐺 is 1 at the 𝑋 that determines it. (Contributed by NM, 27-Oct-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐷 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝐷)    &   ⊢ 1 = (1r‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))    ⇒   ⊢ (𝜑 → (𝐺‘𝑋) = 1 )
Theorem	dochfln0 42358	The value of a functional is nonzero at a nonzero vector in the orthocomplement of its kernel. (Contributed by NM, 2-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝑁 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐺‘𝑋) ≠ 𝑁)
Theorem	dochkr1 42359*	A nonzero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 39951. (Contributed by NM, 2-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })(𝐺‘𝑥) = 1 )
Theorem	dochkr1OLDN 42360*	A nonzero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 39951. (Contributed by NM, 2-Jan-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ( ⊥ ‘(𝐿‘𝐺))(𝐺‘𝑥) = 1 )
21.29.14  Construction of involution and inner product from a Hilbert lattice
Syntax	clpoN 42361	Extend class notation with all polarities of a left module or left vector space.
class LPol
Definition	df-lpolN 42362*	Define the set of all polarities of a left module or left vector space. A polarity is a kind of complementation operation on a subspace. The double polarity of a subspace is a closure operation. Based on Definition 3.2 of [Holland95] p. 214 for projective geometry polarities. For convenience, we open up the domain to include all vector subsets and not just subspaces, but any more restricted polarity can be converted to this one by taking the span of its argument. (Contributed by NM, 24-Nov-2014.)
⊢ LPol = (𝑤 ∈ V ↦ {𝑜 ∈ ((LSubSp‘𝑤) ↑m 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0g‘𝑤)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))})
Theorem	lpolsetN 42363*	The set of polarities of a left module or left vector space. (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ 𝑃 = (LPol‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → 𝑃 = {𝑜 ∈ (𝑆 ↑m 𝒫 𝑉) ∣ ((𝑜‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))})
Theorem	islpolN 42364*	The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ 𝑃 = (LPol‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)))))
Theorem	islpoldN 42365*	Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ 𝑃 = (LPol‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → ⊥ :𝒫 𝑉⟶𝑆)    &   ⊢ (𝜑 → ( ⊥ ‘𝑉) = { 0 })    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦)) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( ⊥ ‘𝑥) ∈ 𝐻)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥)    ⇒   ⊢ (𝜑 → ⊥ ∈ 𝑃)
Theorem	lpolfN 42366	Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑃 = (LPol‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → ⊥ ∈ 𝑃)    ⇒   ⊢ (𝜑 → ⊥ :𝒫 𝑉⟶𝑆)
Theorem	lpolvN 42367	The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑃 = (LPol‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → ⊥ ∈ 𝑃)    ⇒   ⊢ (𝜑 → ( ⊥ ‘𝑉) = { 0 })
Theorem	lpolconN 42368	Contraposition property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑃 = (LPol‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → ⊥ ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑉)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑌)    ⇒   ⊢ (𝜑 → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))
Theorem	lpolsatN 42369	The polarity of an atomic subspace is a hyperplane. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝐻 = (LSHyp‘𝑊)    &   ⊢ 𝑃 = (LPol‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → ⊥ ∈ 𝑃)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ 𝐻)
Theorem	lpolpolsatN 42370	Property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
⊢ 𝐴 = (LSAtoms‘𝑊)    &   ⊢ 𝑃 = (LPol‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑋)    &   ⊢ (𝜑 → ⊥ ∈ 𝑃)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄)
Theorem	dochpolN 42371	The subspace orthocomplement for the DVecH vector space is a polarity. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑃 = (LPol‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → ⊥ ∈ 𝑃)
Theorem	lcfl1lem 42372*	Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.)
⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    ⇒   ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
Theorem	lcfl1 42373*	Property of a functional with a closed kernel. (Contributed by NM, 31-Dec-2014.)
⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
Theorem	lcfl2 42374*	Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ∨ (𝐿‘𝐺) = 𝑉)))
Theorem	lcfl3 42375*	Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ (( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴 ∨ (𝐿‘𝐺) = 𝑉)))
Theorem	lcfl4N 42376*	Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑌 = (LSHyp‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ∨ (𝐿‘𝐺) = 𝑉)))
Theorem	lcfl5 42377*	Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ (𝐿‘𝐺) ∈ ran 𝐼))
Theorem	lcfl5a 42378	Property of a functional with a closed kernel. TODO: Make lcfl5 42377 etc. obsolete and rewrite without 𝐶 hypothesis? (Contributed by NM, 29-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ↔ (𝐿‘𝐺) ∈ ran 𝐼))
Theorem	lcfl6lem 42379*	Lemma for lcfl6 42381. A functional 𝐺 (whose kernel is closed by dochsnkr 42353) is completely determined by a vector 𝑋 in the orthocomplement in its kernel at which the functional value is 1. Note that the ∖ { 0 } in the 𝑋 hypothesis is redundant by the last hypothesis but allows easier use of other theorems. (Contributed by NM, 3-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 1 = (1r‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))    &   ⊢ (𝜑 → (𝐺‘𝑋) = 1 )    ⇒   ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))))
Theorem	lcfl7lem 42380*	Lemma for lcfl7N 42382. If two functionals 𝐺 and 𝐽 are equal, they are determined by the same vector. (Contributed by NM, 4-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))    &   ⊢ 𝐽 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑌})𝑣 = (𝑤 + (𝑘 · 𝑌))))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐺 = 𝐽)    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	lcfl6 42381*	Property of a functional with a closed kernel. Note that (𝐿‘𝐺) = 𝑉 means the functional is zero by lkr0f 39975. (Contributed by NM, 3-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ((𝐿‘𝐺) = 𝑉 ∨ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))
Theorem	lcfl7N 42382*	Property of a functional with a closed kernel. Every nonzero functional is determined by a unique nonzero vector. Note that (𝐿‘𝐺) = 𝑉 means the functional is zero by lkr0f 39975. (Contributed by NM, 4-Jan-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ((𝐿‘𝐺) = 𝑉 ∨ ∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))
Theorem	lcfl8 42383*	Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥})))
Theorem	lcfl8a 42384*	Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ↔ ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥})))
Theorem	lcfl8b 42385*	Property of a nonzero functional with a closed kernel. (Contributed by NM, 4-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑌 = (0g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 ∖ {𝑌}))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))
Theorem	lcfl9a 42386	Property implying that a functional has a closed kernel. (Contributed by NM, 16-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
Theorem	lclkrlem1 42387*	The set of functionals having closed kernels is closed under scalar product. (Contributed by NM, 28-Dec-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐶)    ⇒   ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐶)
Theorem	lclkrlem2a 42388	Lemma for lclkr 42414. Use lshpat 39937 to show that the intersection of a hyperplane with a noncomparable sum of atoms is an atom. (Contributed by NM, 16-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{𝐵}))    ⇒   ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)
Theorem	lclkrlem2b 42389	Lemma for lclkr 42414. (Contributed by NM, 17-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    ⇒   ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)
Theorem	lclkrlem2c 42390	Lemma for lclkr 42414. (Contributed by NM, 16-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ 𝐽 = (LSHyp‘𝑈)    ⇒   ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) ∈ 𝐽)
Theorem	lclkrlem2d 42391	Lemma for lclkr 42414. (Contributed by NM, 16-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) ∈ ran 𝐼)
Theorem	lclkrlem2e 42392	Lemma for lclkr 42414. The kernel of the sum is closed when the kernels of the summands are equal and closed. (Contributed by NM, 17-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐸) = (𝐿‘𝐺))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2f 42393	Lemma for lclkr 42414. Construct a closed hyperplane under the kernel of the sum. (Contributed by NM, 16-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺))    &   ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽)    ⇒   ⊢ (𝜑 → (((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) ⊆ (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2g 42394	Lemma for lclkr 42414. Comparable hyperplanes are equal, so the kernel of the sum is closed. (Contributed by NM, 16-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺))    &   ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽)    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2h 42395	Lemma for lclkr 42414. Eliminate the (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽 hypothesis. (Contributed by NM, 16-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2i 42396	Lemma for lclkr 42414. Eliminate the (𝐿‘𝐸) ≠ (𝐿‘𝐺) hypothesis. (Contributed by NM, 17-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2j 42397	Lemma for lclkr 42414. Kernel closure when 𝑌 is zero. (Contributed by NM, 18-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 = 0 )    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2k 42398	Lemma for lclkr 42414. Kernel closure when 𝑋 is zero. (Contributed by NM, 18-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ (𝜑 → 𝑋 = 0 )    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2l 42399	Lemma for lclkr 42414. Eliminate the 𝑋 ≠ 0, 𝑌 ≠ 0 hypotheses. (Contributed by NM, 18-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2m 42400	Lemma for lclkr 42414. Construct a vector 𝐵 that makes the sum of functionals zero. Combine with 𝐵 ∈ 𝑉 to shorten overall proof. (Contributed by NM, 17-Jan-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑈 ∈ LVec)    &   ⊢ 𝐵 = (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐵 ∈ 𝑉 ∧ ((𝐸 + 𝐺)‘𝐵) = 0 ))