P. 413 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 41201-41300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dihvalcq 41201	Value of isomorphism H for a lattice 𝐾 when ¬ 𝑋 ≤ 𝑊, given auxiliary atom 𝑄. TODO: Use dihvalcq2 41212 (with lhpmcvr3 39990 for (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 simplification) that changes 𝐶 and 𝐷 to 𝐼 and make this obsolete. Do to other theorems as well. (Contributed by NM, 6-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊)    &   ⊢ 𝐶 = ((DIsoC‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐼‘𝑋) = ((𝐶‘𝑄) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))
Theorem	dihvalb 41202	Value of isomorphism H for a lattice 𝐾 when 𝑋 ≤ 𝑊. (Contributed by NM, 4-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (𝐷‘𝑋))
Theorem	dihopelvalbN 41203*	Ordered pair member of the partial isomorphism H for argument under 𝑊. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋) ∧ 𝑆 = 𝑂)))
Theorem	dihvalcqat 41204	Value of isomorphism H for a lattice 𝐾 at an atom not under 𝑊. (Contributed by NM, 27-Mar-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = (𝐽‘𝑄))
Theorem	dih1dimb 41205*	Two expressions for a 1-dimensional subspace of vector space H (when 𝐹 is a nonzero vector i.e. non-identity translation). (Contributed by NM, 27-Apr-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐼‘(𝑅‘𝐹)) = (𝑁‘{⟨𝐹, 𝑂⟩}))
Theorem	dih1dimb2 41206*	Isomorphism H at an atom under 𝑊. (Contributed by NM, 27-Apr-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩})))
Theorem	dih1dimc 41207*	Isomorphism H at an atom not under 𝑊. (Contributed by NM, 27-Apr-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = (𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}))
Theorem	dib2dim 41208	Extend dia2dim 41042 to partial isomorphism B. (Contributed by NM, 22-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoB‘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	dih2dimb 41209	Extend dib2dim 41208 to isomorphism H. (Contributed by NM, 22-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘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	dih2dimbALTN 41210	Extend dia2dim 41042 to isomorphism H. (This version combines dib2dim 41208 and dih2dimb 41209 for shorter overall proof, but may be less easy to understand. TODO: decide which to use.) (Contributed by NM, 22-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘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	dihopelvalcqat 41211*	Ordered pair member of the partial isomorphism H for atom argument not under 𝑊. TODO: remove .t hypothesis. (Contributed by NM, 30-Mar-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)    &   ⊢ 𝐹 ∈ V    &   ⊢ 𝑆 ∈ V    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑄) ↔ (𝐹 = (𝑆‘𝐺) ∧ 𝑆 ∈ 𝐸)))
Theorem	dihvalcq2 41212	Value of isomorphism H for a lattice 𝐾 when ¬ 𝑋 ≤ 𝑊, given auxiliary atom 𝑄. (Contributed by NM, 26-Sep-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ 𝑋)) → (𝐼‘𝑋) = ((𝐼‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))))
Theorem	dihopelvalcpre 41213*	Member of value of isomorphism H for a lattice 𝐾 when ¬ 𝑋 ≤ 𝑊, given auxiliary atom 𝑄. TODO: refactor to be shorter and more understandable; add lemmas? (Contributed by NM, 13-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)    &   ⊢ 𝐹 ∈ V    &   ⊢ 𝑆 ∈ V    &   ⊢ 𝑍 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝑁 = ((DIsoB‘𝐾)‘𝑊)    &   ⊢ 𝐶 = ((DIsoC‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝑉 = (LSubSp‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑂 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑎‘ℎ) ∘ (𝑏‘ℎ))))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘(𝐹 ∘ ◡(𝑆‘𝐺))) ≤ 𝑋)))
Theorem	dihopelvalc 41214*	Member of value of isomorphism H for a lattice 𝐾 when ¬ 𝑋 ≤ 𝑊, given auxiliary atom 𝑄. (Contributed by NM, 13-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)    &   ⊢ 𝐹 ∈ V    &   ⊢ 𝑆 ∈ V    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘(𝐹 ∘ ◡(𝑆‘𝐺))) ≤ 𝑋)))
Theorem	dihlss 41215	The value of isomorphism H is a subspace. (Contributed by NM, 6-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝑆)
Theorem	dihss 41216	The value of isomorphism H is a set of vectors. (Contributed by NM, 14-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ⊆ 𝑉)
Theorem	dihssxp 41217	An isomorphism H value is included in the vector space (expressed as 𝑇 × 𝐸). (Contributed by NM, 26-Sep-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼‘𝑋) ⊆ (𝑇 × 𝐸))
Theorem	dihopcl 41218	Closure of an ordered pair (vector) member of a value of isomorphism H. (Contributed by NM, 26-Sep-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → ⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋))    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸))
Theorem	xihopellsmN 41219*	Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐴 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼‘𝑋) ⊕ (𝐼‘𝑌)) ↔ ∃𝑔∃𝑡∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) ∧ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢)))))
Theorem	dihopellsm 41220*	Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐴 = (𝑣 ∈ 𝐸, 𝑤 ∈ 𝐸 ↦ (𝑖 ∈ 𝑇 ↦ ((𝑣‘𝑖) ∘ (𝑤‘𝑖))))    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼‘𝑋) ⊕ (𝐼‘𝑌)) ↔ ∃𝑔∃𝑡∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) ∧ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢)))))
Theorem	dihord6apre 41221*	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑞)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → 𝑋 ≤ 𝑌)
Theorem	dihord3 41222	The isomorphism H for a lattice 𝐾 is order-preserving in the region under co-atom 𝑊. (Contributed by NM, 6-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌))
Theorem	dihord4 41223	The isomorphism H for a lattice 𝐾 is order-preserving in the region not under co-atom 𝑊. TODO: reformat (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) to eliminate adant*. (Contributed by NM, 6-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌))
Theorem	dihord5b 41224	Part of proof that isomorphism H is order-preserving. TODO: eliminate 3ad2ant1; combine with other way to have one lhpmcvr2 . (Contributed by NM, 7-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝐼‘𝑋) ⊆ (𝐼‘𝑌))
Theorem	dihord6b 41225	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝐼‘𝑋) ⊆ (𝐼‘𝑌))
Theorem	dihord6a 41226	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → 𝑋 ≤ 𝑌)
Theorem	dihord5apre 41227	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-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)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → 𝑋 ≤ 𝑌)
Theorem	dihord5a 41228	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → 𝑋 ≤ 𝑌)
Theorem	dihord 41229	The isomorphism H is order-preserving. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌))
Theorem	dih11 41230	The isomorphism H is one-to-one. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐼‘𝑋) = (𝐼‘𝑌) ↔ 𝑋 = 𝑌))
Theorem	dihf11lem 41231	Functionality of the isomorphism H. (Contributed by NM, 6-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:𝐵⟶𝑆)
Theorem	dihf11 41232	The isomorphism H for a lattice 𝐾 is a one-to-one function. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:𝐵–1-1→𝑆)
Theorem	dihfn 41233	Functionality and domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn 𝐵)
Theorem	dihdm 41234	Domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = 𝐵)
Theorem	dihcl 41235	Closure of isomorphism H. (Contributed by NM, 8-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ ran 𝐼)
Theorem	dihcnvcl 41236	Closure of isomorphism H converse. (Contributed by NM, 8-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (◡𝐼‘𝑋) ∈ 𝐵)
Theorem	dihcnvid1 41237	The converse isomorphism of an isomorphism. (Contributed by NM, 5-Aug-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (◡𝐼‘(𝐼‘𝑋)) = 𝑋)
Theorem	dihcnvid2 41238	The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝐼‘(◡𝐼‘𝑋)) = 𝑋)
Theorem	dihcnvord 41239	Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → ((◡𝐼‘𝑋) ≤ (◡𝐼‘𝑌) ↔ 𝑋 ⊆ 𝑌))
Theorem	dihcnv11 41240	The converse of isomorphism H is one-to-one. (Contributed by NM, 17-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → ((◡𝐼‘𝑋) = (◡𝐼‘𝑌) ↔ 𝑋 = 𝑌))
Theorem	dihsslss 41241	The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ran 𝐼 ⊆ 𝑆)
Theorem	dihrnlss 41242	The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ∈ 𝑆)
Theorem	dihrnss 41243	The isomorphism H maps to a set of vectors. (Contributed by NM, 14-Mar-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ⊆ 𝑉)
Theorem	dihvalrel 41244	The value of isomorphism H is a relation. (Contributed by NM, 9-Mar-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑋))
Theorem	dih0 41245	The value of isomorphism H at the lattice zero is the singleton of the zero vector i.e. the zero subspace. (Contributed by NM, 9-Mar-2014.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (0g‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {𝑂})
Theorem	dih0bN 41246	A lattice element is zero iff its isomorphism is the zero subspace. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑍 = (0g‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 = 0 ↔ (𝐼‘𝑋) = {𝑍}))
Theorem	dih0vbN 41247	A vector is zero iff its span is the isomorphism of lattice zero. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑍 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 = 𝑍 ↔ (𝑁‘{𝑋}) = (𝐼‘ 0 )))
Theorem	dih0cnv 41248	The isomorphism H converse value of the zero subspace is the lattice zero. (Contributed by NM, 19-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑍 = (0g‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (◡𝐼‘{𝑍}) = 0 )
Theorem	dih0rn 41249	The zero subspace belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 0 = (0g‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → { 0 } ∈ ran 𝐼)
Theorem	dih0sb 41250	A subspace is zero iff the converse of its isomorphism is lattice zero. (Contributed by NM, 17-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑍 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → (𝑋 = {𝑍} ↔ (◡𝐼‘𝑋) = 0 ))
Theorem	dih1 41251	The value of isomorphism H at the lattice unity is the set of all vectors. (Contributed by NM, 13-Mar-2014.)
⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 1 ) = 𝑉)
Theorem	dih1rn 41252	The full vector space belongs to the range of isomorphism H. (Contributed by NM, 19-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑉 ∈ ran 𝐼)
Theorem	dih1cnv 41253	The isomorphism H converse value of the full vector space is the lattice one. (Contributed by NM, 19-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (◡𝐼‘𝑉) = 1 )
Theorem	dihwN 41254*	Value of isomorphism H at the fiducial hyperplane 𝑊. (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → (𝐼‘𝑊) = (𝑇 × { 0 }))
Theorem	dihmeetlem1N 41255*	Isomorphism H of a conjunction. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑞)    &   ⊢ 0 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
Theorem	dihglblem5apreN 41256*	A conjunction property of isomorphism H. TODO: reduce antecedent size; general review for shorter proof. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑞)    &   ⊢ 0 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑊)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑊)))
Theorem	dihglblem5aN 41257	A conjunction property of isomorphism H. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘(𝑋 ∧ 𝑊)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑊)))
Theorem	dihglblem2aN 41258*	Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = {𝑢 ∈ 𝐵 ∣ ∃𝑣 ∈ 𝑆 𝑢 = (𝑣 ∧ 𝑊)}    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) → 𝑇 ≠ ∅)
Theorem	dihglblem2N 41259*	The GLB of a set of lattice elements 𝑆 is the same as that of the set 𝑇 with elements of 𝑆 cut down to be under 𝑊. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = {𝑢 ∈ 𝐵 ∣ ∃𝑣 ∈ 𝑆 𝑢 = (𝑣 ∧ 𝑊)}    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐺‘𝑆) = (𝐺‘𝑇))
Theorem	dihglblem3N 41260*	Isomorphism H of a lattice glb. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = {𝑢 ∈ 𝐵 ∣ ∃𝑣 ∈ 𝑆 𝑢 = (𝑣 ∧ 𝑊)}    &   ⊢ 𝐽 = ((DIsoB‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐼‘(𝐺‘𝑇)) = ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥))
Theorem	dihglblem3aN 41261*	Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = {𝑢 ∈ 𝐵 ∣ ∃𝑣 ∈ 𝑆 𝑢 = (𝑣 ∧ 𝑊)}    &   ⊢ 𝐽 = ((DIsoB‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥))
Theorem	dihglblem4 41262*	Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = {𝑢 ∈ 𝐵 ∣ ∃𝑣 ∈ 𝑆 𝑢 = (𝑣 ∧ 𝑊)}    &   ⊢ 𝐽 = ((DIsoB‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) ⊆ ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))
Theorem	dihglblem5 41263*	Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ⊆ 𝐵 ∧ 𝑇 ≠ ∅)) → ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥) ∈ 𝑆)
Theorem	dihmeetlem2N 41264	Isomorphism H of a conjunction. (Contributed by NM, 22-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑞)    &   ⊢ 0 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
Theorem	dihglbcpreN 41265*	Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane 𝑊. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ ¬ (𝐺‘𝑆) ≤ 𝑊) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))
Theorem	dihglbcN 41266*	Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane 𝑊. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ ¬ (𝐺‘𝑆) ≤ 𝑊) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))
Theorem	dihmeetcN 41267	Isomorphism H of a lattice meet when the meet is not under the fiducial hyperplane 𝑊. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
Theorem	dihmeetbN 41268	Isomorphism H of a lattice meet when one element is under the fiducial hyperplane 𝑊. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
Theorem	dihmeetbclemN 41269	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊) → (𝐼‘(𝑋 ∧ 𝑌)) = (((𝐼‘𝑋) ∩ (𝐼‘𝑌)) ∩ (𝐼‘𝑊)))
Theorem	dihmeetlem3N 41270	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → 𝑄 ≠ 𝑅)
Theorem	dihmeetlem4preN 41271*	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)    &   ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐼‘𝑄) ∩ (𝐼‘(𝑋 ∧ 𝑊))) = { 0 })
Theorem	dihmeetlem4N 41272	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 0 = (0g‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐼‘𝑄) ∩ (𝐼‘(𝑋 ∧ 𝑊))) = { 0 })
Theorem	dihmeetlem5 41273	Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → (𝑋 ∧ (𝑌 ∨ 𝑄)) = ((𝑋 ∧ 𝑌) ∨ 𝑄))
Theorem	dihmeetlem6 41274	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ 𝑋)) → ¬ (𝑋 ∧ (𝑌 ∨ 𝑄)) ≤ 𝑊)
Theorem	dihmeetlem7N 41275	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (((𝑋 ∧ 𝑌) ∨ 𝑝) ∧ 𝑌) = (𝑋 ∧ 𝑌))
Theorem	dihjatc1 41276	Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change ∨ order of (𝑋 ∧ 𝑌) ∨ 𝑄 here and down? (Contributed by NM, 6-Apr-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)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑄)) = ((𝐼‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑌))))
Theorem	dihjatc2N 41277	Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)
⊢ 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)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐼‘(𝑄 ∨ (𝑋 ∧ 𝑌))) = ((𝐼‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑌))))
Theorem	dihjatc3 41278	Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-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)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑄)) = ((𝐼‘(𝑋 ∧ 𝑌)) ⊕ (𝐼‘𝑄)))
Theorem	dihmeetlem8N 41279	Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change ∨ order of (𝑋 ∧ 𝑌) ∨ 𝑝 here and down? (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
⊢ 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)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑝 ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑝)) = ((𝐼‘𝑝) ⊕ (𝐼‘(𝑋 ∧ 𝑌))))
Theorem	dihmeetlem9N 41280	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
⊢ 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)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (((𝐼‘𝑝) ⊕ (𝐼‘(𝑋 ∧ 𝑌))) ∩ (𝐼‘𝑌)) = ((𝐼‘(𝑋 ∧ 𝑌)) ⊕ ((𝐼‘𝑝) ∩ (𝐼‘𝑌))))
Theorem	dihmeetlem10N 41281	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
⊢ 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)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑝)) = ((𝐼‘𝑋) ∩ (𝐼‘(𝑌 ∨ 𝑝))))
Theorem	dihmeetlem11N 41282	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
⊢ 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)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → ((𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑝)) ∩ (𝐼‘𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
Theorem	dihmeetlem12N 41283	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
⊢ 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)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → ((𝐼‘(𝑋 ∧ 𝑌)) ⊕ ((𝐼‘𝑝) ∩ (𝐼‘𝑌))) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
Theorem	dihmeetlem13N 41284*	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄)    &   ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) = { 0 })
Theorem	dihmeetlem14N 41285	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
⊢ 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)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → ((𝐼‘(𝑌 ∧ 𝑝)) ⊕ ((𝐼‘𝑟) ∩ (𝐼‘𝑝))) = ((𝐼‘𝑌) ∩ (𝐼‘𝑝)))
Theorem	dihmeetlem15N 41286	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
⊢ 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)    &   ⊢ 0 = (0g‘𝑈)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → ((𝐼‘𝑟) ∩ (𝐼‘𝑝)) = { 0 })
Theorem	dihmeetlem16N 41287	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
⊢ 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)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → (𝐼‘(𝑌 ∧ 𝑝)) = ((𝐼‘𝑌) ∩ (𝐼‘𝑝)))
Theorem	dihmeetlem17N 41288	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
⊢ 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)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → (𝑌 ∧ 𝑝) = 0 )
Theorem	dihmeetlem18N 41289	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
⊢ 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)    &   ⊢ 0 = (0g‘𝑈)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊))) → ((𝐼‘𝑌) ∩ (𝐼‘𝑝)) = { 0 })
Theorem	dihmeetlem19N 41290	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
⊢ 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)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊))) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
Theorem	dihmeetlem20N 41291	Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
⊢ 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)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
Theorem	dihmeetALTN 41292	Isomorphism H of a lattice meet. This version does not depend on the atomisticity of the constructed vector space. TODO: Delete? (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
Theorem	dih1dimatlem0 41293*	Lemma for dih1dimat 41295. (Contributed by NM, 11-Apr-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐶 = (Atoms‘𝐾)    &   ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝐹 = (Scalar‘𝑈)    &   ⊢ 𝐽 = (invr‘𝐹)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = (((𝐽‘𝑠)‘𝑓)‘𝑃))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ 𝑇 ∧ 𝑠 ∈ 𝐸) ∧ 𝑠 ≠ 𝑂) → ((𝑖 = (𝑝‘𝐺) ∧ 𝑝 ∈ 𝐸) ↔ ((𝑖 ∈ 𝑇 ∧ 𝑝 ∈ 𝐸) ∧ ∃𝑡 ∈ 𝐸 (𝑖 = (𝑡‘𝑓) ∧ 𝑝 = (𝑡 ∘ 𝑠)))))
Theorem	dih1dimatlem 41294*	Lemma for dih1dimat 41295. (Contributed by NM, 10-Apr-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐶 = (Atoms‘𝐾)    &   ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝐹 = (Scalar‘𝑈)    &   ⊢ 𝐽 = (invr‘𝐹)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = (((𝐽‘𝑠)‘𝑓)‘𝑃))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝐴) → 𝐷 ∈ ran 𝐼)
Theorem	dih1dimat 41295	Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ran 𝐼)
Theorem	dihlsprn 41296	The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran 𝐼)
Theorem	dihlspsnssN 41297	A subspace included in a 1-dim subspace belongs to the range of isomorphism H. (Contributed by NM, 26-Apr-2014.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) → (𝑇 ∈ 𝑆 ↔ 𝑇 ∈ ran 𝐼))
Theorem	dihlspsnat 41298	The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (◡𝐼‘(𝑁‘{𝑋})) ∈ 𝐴)
Theorem	dihatlat 41299	The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSAtoms‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) → (𝐼‘𝑄) ∈ 𝐿)
Theorem	dihat 41300	There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → (𝐼‘𝑃) ∈ 𝐴)