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Theorem List for Metamath Proof Explorer - 35401-35500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcdlemn8 35401* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑅)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑠𝐸𝑔𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → 𝑔 = (𝐺𝐹))

Theoremcdlemn9 35402* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑅)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑠𝐸𝑔𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → (𝑔𝑄) = 𝑅)

Theoremcdlemn10 35403 Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑆 (𝑄 𝑋))

Theoremcdlemn11a 35404* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &    = (LSSum‘𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → ⟨𝐺, ( I ↾ 𝑇)⟩ ∈ (𝐽𝑁))

Theoremcdlemn11b 35405* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &    = (LSSum‘𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → ⟨𝐺, ( I ↾ 𝑇)⟩ ∈ ((𝐽𝑄) (𝐼𝑋)))

Theoremcdlemn11c 35406* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &    = (LSSum‘𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → ∃𝑦 ∈ (𝐽𝑄)∃𝑧 ∈ (𝐼𝑋)⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧))

Theoremcdlemn11pre 35407* Part of proof of Lemma N of [Crawley] p. 121 line 37. TODO: combine cdlemn11a 35404, cdlemn11b 35405, cdlemn11c 35406, cdlemn11pre into one? (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &    = (LSSum‘𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → 𝑁 (𝑄 𝑋))

Theoremcdlemn11 35408 Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑅) ⊆ ((𝐽𝑄) (𝐼𝑋))) → 𝑅 (𝑄 𝑋))

Theoremcdlemn 35409 Lemma N of [Crawley] p. 121 line 27. (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑋𝐵𝑋 𝑊))) → (𝑅 (𝑄 𝑋) ↔ (𝐽𝑅) ⊆ ((𝐽𝑄) (𝐼𝑋))))

Theoremdihordlem6 35410* Part of proof of Lemma N of [Crawley] p. 122 line 35. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑅)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑠𝐸𝑔𝑇)) → (⟨(𝑠𝐺), 𝑠+𝑔, 𝑂⟩) = ⟨((𝑠𝐺) ∘ 𝑔), 𝑠⟩)

Theoremdihordlem7 35411* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑅)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑠𝐸𝑔𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠𝐺), 𝑠+𝑔, 𝑂⟩))) → (𝑓 = ((𝑠𝐺) ∘ 𝑔) ∧ 𝑂 = 𝑠))

Theoremdihordlem7b 35412* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑅)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑠𝐸𝑔𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠𝐺), 𝑠+𝑔, 𝑂⟩))) → (𝑓 = 𝑔𝑂 = 𝑠))

Theoremdihjustlem 35413 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑋𝐵) ∧ (𝑄 (𝑋 𝑊)) = (𝑅 (𝑋 𝑊))) → ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑅) (𝐼‘(𝑋 𝑊))))

Theoremdihjust 35414 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑋𝐵) ∧ (𝑄 (𝑋 𝑊)) = (𝑅 (𝑋 𝑊))) → ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) = ((𝐽𝑅) (𝐼‘(𝑋 𝑊))))

Theoremdihord1 35415 Part of proof after Lemma N of [Crawley] p. 122. Forward ordering property. TODO: change (𝑄 (𝑋 𝑊)) = 𝑋 to 𝑄 𝑋 using lhpmcvr3 34219, here and all theorems below. (Contributed by NM, 2-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))))

Theoremdihord2a 35416 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌 ∧ ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))))) → 𝑄 (𝑅 (𝑌 𝑊)))

Theoremdihord2b 35417 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊)))) → (𝐼‘(𝑋 𝑊)) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))))

Theoremdihord2cN 35418* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: needed? shorten other proof with it? (Contributed by NM, 3-Mar-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵 ∧ (𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊))) → ⟨𝑓, 𝑂⟩ ∈ (𝐼‘(𝑋 𝑊)))

Theoremdihord11b 35419* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑁) (𝐼‘(𝑌 𝑊)))) ∧ (𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊))) → ⟨𝑓, 𝑂⟩ ∈ ((𝐽𝑁) (𝐼‘(𝑌 𝑊))))

Theoremdihord10 35420* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊)) ∧ (𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ ((𝑠𝐸𝑔𝑇) ∧ (𝑅𝑔) (𝑌 𝑊) ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠𝐺), 𝑠+𝑔, 𝑂⟩))) → (𝑅𝑓) (𝑌 𝑊))

Theoremdihord11c 35421* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ (((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑁) (𝐼‘(𝑌 𝑊))) ∧ 𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊))) → ∃𝑦 ∈ (𝐽𝑁)∃𝑧 ∈ (𝐼‘(𝑌 𝑊))⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧))

Theoremdihord2pre 35422* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑁) (𝐼‘(𝑌 𝑊)))) → (𝑋 𝑊) (𝑌 𝑊))

Theoremdihord2pre2 35423* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 4-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑁 (𝑌 𝑊)) = 𝑌 ∧ ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑁) (𝐼‘(𝑌 𝑊))))) → (𝑄 (𝑋 𝑊)) (𝑁 (𝑌 𝑊)))

Theoremdihord2 35424 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: do we need ¬ 𝑋 𝑊 and ¬ 𝑌 𝑊? (Contributed by NM, 4-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑁 (𝑌 𝑊)) = 𝑌 ∧ ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑁) (𝐼‘(𝑌 𝑊))))) → 𝑋 𝑌)

Syntaxcdih 35425 Extend class notation with isomorphism H.
class DIsoH

Definitiondf-dih 35426* Define isomorphism H. (Contributed by NM, 28-Jan-2014.)
DIsoH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))))))

Theoremdihffval 35427* The isomorphism H for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (𝐾𝑉 → (DIsoH‘𝐾) = (𝑤𝐻 ↦ (𝑥𝐵 ↦ if(𝑥 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞𝐴 ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)))))))))

Theoremdihfval 35428* Isomorphism H for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-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)    &   𝑆 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)       ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))))

Theoremdihval 35429* Value of isomorphism H for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 3-Feb-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)    &   𝑆 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)       (((𝐾𝑉𝑊𝐻) ∧ 𝑋𝐵) → (𝐼𝑋) = if(𝑋 𝑊, (𝐷𝑋), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))))

Theoremdihvalc 35430* Value of isomorphism H for a lattice 𝐾 when ¬ 𝑋 𝑊. (Contributed by NM, 4-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)    &   𝑆 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → (𝐼𝑋) = (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))))

Theoremdihlsscpre 35431 Closure of isomorphism H for a lattice 𝐾 when ¬ 𝑋 𝑊. (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)    &   𝑆 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → (𝐼𝑋) ∈ 𝑆)

Theoremdihvalcqpre 35432 Value of isomorphism H for a lattice 𝐾 when ¬ 𝑋 𝑊, given auxiliary atom 𝑄. (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)    &   𝑆 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝐼𝑋) = ((𝐶𝑄) (𝐷‘(𝑋 𝑊))))

Theoremdihvalcq 35433 Value of isomorphism H for a lattice 𝐾 when ¬ 𝑋 𝑊, given auxiliary atom 𝑄. TODO: Use dihvalcq2 35444 (with lhpmcvr3 34219 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 ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝐼𝑋) = ((𝐶𝑄) (𝐷‘(𝑋 𝑊))))

Theoremdihvalb 35434 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)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = (𝐷𝑋))

TheoremdihopelvalbN 35435* 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 = (𝑔𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 𝑂)))

Theoremdihvalcqat 35436 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 ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = (𝐽𝑄))

Theoremdih1dimb 35437* 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 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑁 = (LSpan‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝐼‘(𝑅𝐹)) = (𝑁‘{⟨𝐹, 𝑂⟩}))

Theoremdih1dimb2 35438* 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 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑁 = (LSpan‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴𝑄 𝑊)) → ∃𝑓𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩})))

Theoremdih1dimc 35439* 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 = (𝑓𝑇 (𝑓𝑃) = 𝑄)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = (𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}))

Theoremdib2dim 35440 Extend dia2dim 35274 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 ∧ 𝑊𝐻))    &   (𝜑 → (𝑃𝐴𝑃 𝑊))    &   (𝜑 → (𝑄𝐴𝑄 𝑊))       (𝜑 → (𝐼‘(𝑃 𝑄)) ⊆ ((𝐼𝑃) (𝐼𝑄)))

Theoremdih2dimb 35441 Extend dib2dim 35440 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 ∧ 𝑊𝐻))    &   (𝜑 → (𝑃𝐴𝑃 𝑊))    &   (𝜑 → (𝑄𝐴𝑄 𝑊))       (𝜑 → (𝐼‘(𝑃 𝑄)) ⊆ ((𝐼𝑃) (𝐼𝑄)))

Theoremdih2dimbALTN 35442 Extend dia2dim 35274 to isomorphism H. (This version combines dib2dim 35440 and dih2dimb 35441 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 ∧ 𝑊𝐻))    &   (𝜑 → (𝑃𝐴𝑃 𝑊))    &   (𝜑 → (𝑄𝐴𝑄 𝑊))       (𝜑 → (𝐼‘(𝑃 𝑄)) ⊆ ((𝐼𝑃) (𝐼𝑄)))

Theoremdihopelvalcqat 35443* 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 = (𝑔𝑇 (𝑔𝑃) = 𝑄)    &   𝐹 ∈ V    &   𝑆 ∈ V       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑄) ↔ (𝐹 = (𝑆𝐺) ∧ 𝑆𝐸)))

Theoremdihvalcq2 35444 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 ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑄 𝑋)) → (𝐼𝑋) = ((𝐼𝑄) (𝐼‘(𝑋 𝑊))))

Theoremdihopelvalcpre 35445* 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 = (𝑔𝑇 (𝑔𝑃) = 𝑄)    &   𝐹 ∈ V    &   𝑆 ∈ V    &   𝑍 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑁 = ((DIsoB‘𝐾)‘𝑊)    &   𝐶 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝑉 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)    &   𝑂 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑇 ↦ ((𝑎) ∘ (𝑏))))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋)))

Theoremdihopelvalc 35446* 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 = (𝑔𝑇 (𝑔𝑃) = 𝑄)    &   𝐹 ∈ V    &   𝑆 ∈ V       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋)))

Theoremdihlss 35447 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 ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (𝐼𝑋) ∈ 𝑆)

Theoremdihss 35448 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 ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (𝐼𝑋) ⊆ 𝑉)

Theoremdihssxp 35449 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)       (𝜑 → (𝐼𝑋) ⊆ (𝑇 × 𝐸))

Theoremdihopcl 35450 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)    &   (𝜑 → ⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋))       (𝜑 → (𝐹𝑇𝑆𝐸))

TheoremxihopellsmN 35451* 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 = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐿 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼𝑋) (𝐼𝑌)) ↔ ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑋) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑌)) ∧ (𝐹 = (𝑔) ∧ 𝑆 = (𝑡𝐴𝑢)))))

Theoremdihopellsm 35452* 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 = (𝑣𝐸, 𝑤𝐸 ↦ (𝑖𝑇 ↦ ((𝑣𝑖) ∘ (𝑤𝑖))))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐿 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼𝑋) (𝐼𝑌)) ↔ ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑋) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑌)) ∧ (𝐹 = (𝑔) ∧ 𝑆 = (𝑡𝐴𝑢)))))

Theoremdihord6apre 35453* 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 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑞)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (𝐼𝑋) ⊆ (𝐼𝑌)) → 𝑋 𝑌)

Theoremdihord3 35454 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 ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → ((𝐼𝑋) ⊆ (𝐼𝑌) ↔ 𝑋 𝑌))

Theoremdihord4 35455 The isomorphism H for a lattice 𝐾 is order-preserving in the region not under co-atom 𝑊. TODO: reformat q e. 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 ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) → ((𝐼𝑋) ⊆ (𝐼𝑌) ↔ 𝑋 𝑌))

Theoremdihord5b 35456 Part of proof that isomorphism H is order-preserving. TODO: eliminate 3ad2ant1; combine w/ 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 ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (𝐼𝑋) ⊆ (𝐼𝑌))

Theoremdihord6b 35457 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 ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ 𝑋 𝑌) → (𝐼𝑋) ⊆ (𝐼𝑌))

Theoremdihord6a 35458 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 ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) ∧ (𝐼𝑋) ⊆ (𝐼𝑌)) → 𝑋 𝑌)

Theoremdihord5apre 35459 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 ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ (𝐼𝑋) ⊆ (𝐼𝑌)) → 𝑋 𝑌)

Theoremdihord5a 35460 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 ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵 ∧ ¬ 𝑌 𝑊)) ∧ (𝐼𝑋) ⊆ (𝐼𝑌)) → 𝑋 𝑌)

Theoremdihord 35461 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 ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) → ((𝐼𝑋) ⊆ (𝐼𝑌) ↔ 𝑋 𝑌))

Theoremdih11 35462 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 ∧ 𝑊𝐻) ∧ 𝑋𝐵𝑌𝐵) → ((𝐼𝑋) = (𝐼𝑌) ↔ 𝑋 = 𝑌))

Theoremdihf11lem 35463 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 ∧ 𝑊𝐻) → 𝐼:𝐵𝑆)

Theoremdihf11 35464 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 ∧ 𝑊𝐻) → 𝐼:𝐵1-1𝑆)

Theoremdihfn 35465 Functionality and domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐼 Fn 𝐵)

Theoremdihdm 35466 Domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → dom 𝐼 = 𝐵)

Theoremdihcl 35467 Closure of isomorphism H. (Contributed by NM, 8-Mar-2014.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (𝐼𝑋) ∈ ran 𝐼)

Theoremdihcnvcl 35468 Closure of isomorphism H converse. (Contributed by NM, 8-Mar-2014.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝐼𝑋) ∈ 𝐵)

Theoremdihcnvid1 35469 The converse isomorphism of an isomorphism. (Contributed by NM, 5-Aug-2014.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (𝐼‘(𝐼𝑋)) = 𝑋)

Theoremdihcnvid2 35470 The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝐼‘(𝐼𝑋)) = 𝑋)

Theoremdihcnvord 35471 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 ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → ((𝐼𝑋) (𝐼𝑌) ↔ 𝑋𝑌))

Theoremdihcnv11 35472 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 ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → ((𝐼𝑋) = (𝐼𝑌) ↔ 𝑋 = 𝑌))

Theoremdihsslss 35473 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 ∧ 𝑊𝐻) → ran 𝐼𝑆)

Theoremdihrnlss 35474 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 ∧ 𝑊𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋𝑆)

Theoremdihrnss 35475 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 ∧ 𝑊𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋𝑉)

Theoremdihvalrel 35476 The value of isomorphism H is a relation. (Contributed by NM, 9-Mar-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → Rel (𝐼𝑋))

Theoremdih0 35477 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 = (0g𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐼0 ) = {𝑂})

Theoremdih0bN 35478 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 = (0g𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)       (𝜑 → (𝑋 = 0 ↔ (𝐼𝑋) = {𝑍}))

Theoremdih0vbN 35479 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 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑋 = 𝑍 ↔ (𝑁‘{𝑋}) = (𝐼0 )))

Theoremdih0cnv 35480 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 = (0g𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐼‘{𝑍}) = 0 )

Theoremdih0rn 35481 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 = (0g𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → { 0 } ∈ ran 𝐼)

Theoremdih0sb 35482 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 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)       (𝜑 → (𝑋 = {𝑍} ↔ (𝐼𝑋) = 0 ))

Theoremdih1 35483 The value of isomorphism H at the lattice unit 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 ∧ 𝑊𝐻) → (𝐼1 ) = 𝑉)

Theoremdih1rn 35484 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 ∧ 𝑊𝐻) → 𝑉 ∈ ran 𝐼)

Theoremdih1cnv 35485 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 ∧ 𝑊𝐻) → (𝐼𝑉) = 1 )

TheoremdihwN 35486* 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 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝐼𝑊) = (𝑇 × { 0 }))

Theoremdihmeetlem1N 35487* 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 = (𝑇 (𝑃) = 𝑞)    &    0 = (𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋) ∩ (𝐼𝑌)))

Theoremdihglblem5apreN 35488* 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 = (𝑇 (𝑃) = 𝑞)    &    0 = (𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → (𝐼‘(𝑋 𝑊)) = ((𝐼𝑋) ∩ (𝐼𝑊)))

Theoremdihglblem5aN 35489 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 ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (𝐼‘(𝑋 𝑊)) = ((𝐼𝑋) ∩ (𝐼𝑊)))

Theoremdihglblem2aN 35490* 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 = {𝑢𝐵 ∣ ∃𝑣𝑆 𝑢 = (𝑣 𝑊)}       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐵𝑆 ≠ ∅)) → 𝑇 ≠ ∅)

Theoremdihglblem2N 35491* 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 = {𝑢𝐵 ∣ ∃𝑣𝑆 𝑢 = (𝑣 𝑊)}       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐵 ∧ (𝐺𝑆) 𝑊) → (𝐺𝑆) = (𝐺𝑇))

Theoremdihglblem3N 35492* 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 = {𝑢𝐵 ∣ ∃𝑣𝑆 𝑢 = (𝑣 𝑊)}    &   𝐽 = ((DIsoB‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐵𝑆 ≠ ∅) ∧ (𝐺𝑆) 𝑊) → (𝐼‘(𝐺𝑇)) = 𝑥𝑇 (𝐼𝑥))

Theoremdihglblem3aN 35493* 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 = {𝑢𝐵 ∣ ∃𝑣𝑆 𝑢 = (𝑣 𝑊)}    &   𝐽 = ((DIsoB‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐵𝑆 ≠ ∅) ∧ (𝐺𝑆) 𝑊) → (𝐼‘(𝐺𝑆)) = 𝑥𝑇 (𝐼𝑥))

Theoremdihglblem4 35494* 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 = {𝑢𝐵 ∣ ∃𝑣𝑆 𝑢 = (𝑣 𝑊)}    &   𝐽 = ((DIsoB‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐵𝑆 ≠ ∅)) → (𝐼‘(𝐺𝑆)) ⊆ 𝑥𝑆 (𝐼𝑥))

Theoremdihglblem5 35495* 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 ∧ 𝑊𝐻) ∧ (𝑇𝐵𝑇 ≠ ∅)) → 𝑥𝑇 (𝐼𝑥) ∈ 𝑆)

Theoremdihmeetlem2N 35496 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 = (𝑇 (𝑃) = 𝑞)    &    0 = (𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋) ∩ (𝐼𝑌)))

TheoremdihglbcpreN 35497* 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 = (𝑔𝑇 (𝑔𝑃) = 𝑞)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐵𝑆 ≠ ∅) ∧ ¬ (𝐺𝑆) 𝑊) → (𝐼‘(𝐺𝑆)) = 𝑥𝑆 (𝐼𝑥))

TheoremdihglbcN 35498* 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 ∧ 𝑊𝐻) ∧ (𝑆𝐵𝑆 ≠ ∅) ∧ ¬ (𝐺𝑆) 𝑊) → (𝐼‘(𝐺𝑆)) = 𝑥𝑆 (𝐼𝑥))

TheoremdihmeetcN 35499 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 ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵) ∧ ¬ (𝑋 𝑌) 𝑊) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋) ∩ (𝐼𝑌)))

TheoremdihmeetbN 35500 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 ∧ 𝑊𝐻) ∧ 𝑋𝐵 ∧ (𝑌𝐵𝑌 𝑊)) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋) ∩ (𝐼𝑌)))

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