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Theorem List for Metamath Proof Explorer - 39301-39400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdlemk50 39301* Part of proof of Lemma K of [Crawley] p. 118. Line 6, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. TODO: Combine into cdlemk52 39303? (Contributed by NM, 23-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 β‰  ( I β†Ύ 𝐡))) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐼 ∈ 𝑇 ∧ 𝐼 β‰  ( I β†Ύ 𝐡))) β†’ ((⦋𝐺 / π‘”β¦Œπ‘‹ ∘ ⦋𝐼 / π‘”β¦Œπ‘‹)β€˜π‘ƒ) ≀ (((⦋𝐺 / π‘”β¦Œπ‘‹β€˜π‘ƒ) ∨ (π‘…β€˜β¦‹πΌ / π‘”β¦Œπ‘‹)) ∧ ((⦋𝐼 / π‘”β¦Œπ‘‹β€˜π‘ƒ) ∨ (π‘…β€˜β¦‹πΊ / π‘”β¦Œπ‘‹))))
 
Theoremcdlemk51 39302* Part of proof of Lemma K of [Crawley] p. 118. Line 6, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. TODO: Combine into cdlemk52 39303? (Contributed by NM, 23-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 β‰  ( I β†Ύ 𝐡))) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐼 ∈ 𝑇 ∧ 𝐼 β‰  ( I β†Ύ 𝐡))) β†’ (((⦋𝐺 / π‘”β¦Œπ‘‹β€˜π‘ƒ) ∨ (π‘…β€˜β¦‹πΌ / π‘”β¦Œπ‘‹)) ∧ ((⦋𝐼 / π‘”β¦Œπ‘‹β€˜π‘ƒ) ∨ (π‘…β€˜β¦‹πΊ / π‘”β¦Œπ‘‹))) ≀ (((⦋𝐺 / π‘”β¦Œπ‘‹β€˜π‘ƒ) ∨ (π‘…β€˜πΌ)) ∧ ((⦋𝐼 / π‘”β¦Œπ‘‹β€˜π‘ƒ) ∨ (π‘…β€˜πΊ))))
 
Theoremcdlemk52 39303* Part of proof of Lemma K of [Crawley] p. 118. Line 6, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 23-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 β‰  ( I β†Ύ 𝐡))) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐼 ∈ 𝑇 ∧ 𝐼 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΌ))) β†’ ((⦋𝐺 / π‘”β¦Œπ‘‹ ∘ ⦋𝐼 / π‘”β¦Œπ‘‹)β€˜π‘ƒ) = (⦋(𝐺 ∘ 𝐼) / π‘”β¦Œπ‘‹β€˜π‘ƒ))
 
Theoremcdlemk53a 39304* Lemma for cdlemk53 39306. (Contributed by NM, 26-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡)) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 β‰  ( I β†Ύ 𝐡))) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐼 ∈ 𝑇 ∧ 𝐼 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΌ))) β†’ ⦋(𝐺 ∘ 𝐼) / π‘”β¦Œπ‘‹ = (⦋𝐺 / π‘”β¦Œπ‘‹ ∘ ⦋𝐼 / π‘”β¦Œπ‘‹))
 
Theoremcdlemk53b 39305* Lemma for cdlemk53 39306. (Contributed by NM, 26-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝑁 ∈ 𝑇) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ (𝐼 ∈ 𝑇 ∧ 𝐼 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΌ))) β†’ ⦋(𝐺 ∘ 𝐼) / π‘”β¦Œπ‘‹ = (⦋𝐺 / π‘”β¦Œπ‘‹ ∘ ⦋𝐼 / π‘”β¦Œπ‘‹))
 
Theoremcdlemk53 39306* Part of proof of Lemma K of [Crawley] p. 118. Line 7, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 26-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝑁 ∈ 𝑇) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ (𝐼 ∈ 𝑇 ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΌ))) β†’ ⦋(𝐺 ∘ 𝐼) / π‘”β¦Œπ‘‹ = (⦋𝐺 / π‘”β¦Œπ‘‹ ∘ ⦋𝐼 / π‘”β¦Œπ‘‹))
 
Theoremcdlemk54 39307* Part of proof of Lemma K of [Crawley] p. 118. Line 10, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 26-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝑁 ∈ 𝑇) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ ((𝐼 ∈ 𝑇 ∧ (π‘…β€˜πΊ) = (π‘…β€˜πΌ)) ∧ 𝑗 ∈ 𝑇 ∧ (𝑗 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘—) β‰  (π‘…β€˜πΊ) ∧ (π‘…β€˜π‘—) β‰  (π‘…β€˜(𝐺 ∘ 𝐼))))) β†’ (⦋(𝐺 ∘ 𝐼) / π‘”β¦Œπ‘‹ ∘ ⦋𝑗 / π‘”β¦Œπ‘‹) = ((⦋𝐺 / π‘”β¦Œπ‘‹ ∘ ⦋𝐼 / π‘”β¦Œπ‘‹) ∘ ⦋𝑗 / π‘”β¦Œπ‘‹))
 
Theoremcdlemk55a 39308* Lemma for cdlemk55 39310. (Contributed by NM, 26-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝑁 ∈ 𝑇) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ ((𝐼 ∈ 𝑇 ∧ (π‘…β€˜πΊ) = (π‘…β€˜πΌ)) ∧ 𝑗 ∈ 𝑇 ∧ (𝑗 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘—) β‰  (π‘…β€˜πΊ) ∧ (π‘…β€˜π‘—) β‰  (π‘…β€˜(𝐺 ∘ 𝐼))))) β†’ ⦋(𝐺 ∘ 𝐼) / π‘”β¦Œπ‘‹ = (⦋𝐺 / π‘”β¦Œπ‘‹ ∘ ⦋𝐼 / π‘”β¦Œπ‘‹))
 
Theoremcdlemk55b 39309* Lemma for cdlemk55 39310. (Contributed by NM, 26-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝑁 ∈ 𝑇) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ (𝐼 ∈ 𝑇 ∧ (π‘…β€˜πΊ) = (π‘…β€˜πΌ))) β†’ ⦋(𝐺 ∘ 𝐼) / π‘”β¦Œπ‘‹ = (⦋𝐺 / π‘”β¦Œπ‘‹ ∘ ⦋𝐼 / π‘”β¦Œπ‘‹))
 
Theoremcdlemk55 39310* Part of proof of Lemma K of [Crawley] p. 118. Line 11, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 26-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝑁 ∈ 𝑇) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ ⦋(𝐺 ∘ 𝐼) / π‘”β¦Œπ‘‹ = (⦋𝐺 / π‘”β¦Œπ‘‹ ∘ ⦋𝐼 / π‘”β¦Œπ‘‹))
 
TheoremcdlemkyyN 39311* Part of proof of Lemma K of [Crawley] p. 118. TODO: clean up (π‘π‘ŒπΊ) stuff. (Contributed by NM, 21-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‰ = (𝑑 ∈ 𝑇, 𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ (((π‘†β€˜π‘‘)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝑑))))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝑁 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ (𝑏 ∈ 𝑇 ∧ (𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΊ)))) β†’ (⦋𝐺 / π‘”β¦Œπ‘‹β€˜π‘ƒ) = ((𝑏𝑉𝐺)β€˜π‘ƒ))
 
Theoremcdlemk43N 39312* Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 31-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ 𝐹 β‰  𝑁) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ (𝑏 ∈ 𝑇 ∧ (𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΊ)))) β†’ ((π‘ˆβ€˜πΊ)β€˜π‘ƒ) = ⦋𝐺 / π‘”β¦Œπ‘Œ)
 
Theoremcdlemk35u 39313* Substitution version of cdlemk35 39261. (Contributed by NM, 31-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (π‘ˆβ€˜πΊ) ∈ 𝑇)
 
Theoremcdlemk55u1 39314* Lemma for cdlemk55u 39315. (Contributed by NM, 31-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((π‘…β€˜πΉ) = (π‘…β€˜π‘) ∧ 𝐹 β‰  𝑁) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (π‘ˆβ€˜(𝐺 ∘ 𝐼)) = ((π‘ˆβ€˜πΊ) ∘ (π‘ˆβ€˜πΌ)))
 
Theoremcdlemk55u 39315* Part of proof of Lemma K of [Crawley] p. 118. Line 11, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 31-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((π‘…β€˜πΉ) = (π‘…β€˜π‘) ∧ 𝐺 ∈ 𝑇 ∧ 𝐼 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (π‘ˆβ€˜(𝐺 ∘ 𝐼)) = ((π‘ˆβ€˜πΊ) ∘ (π‘ˆβ€˜πΌ)))
 
Theoremcdlemk39u1 39316* Lemma for cdlemk39u 39317. (Contributed by NM, 31-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((π‘…β€˜πΉ) = (π‘…β€˜π‘) ∧ 𝐹 β‰  𝑁 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (π‘…β€˜(π‘ˆβ€˜πΊ)) ≀ (π‘…β€˜πΊ))
 
Theoremcdlemk39u 39317* Part of proof of Lemma K of [Crawley] p. 118. Line 31, p. 119. Trace-preserving property of the value of tau, represented by (π‘ˆβ€˜πΊ). (Contributed by NM, 31-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((π‘…β€˜πΉ) = (π‘…β€˜π‘) ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (π‘…β€˜(π‘ˆβ€˜πΊ)) ≀ (π‘…β€˜πΊ))
 
Theoremcdlemk19u1 39318* cdlemk19 39218 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 31-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  𝑁 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ ((π‘ˆβ€˜πΉ)β€˜π‘ƒ) = (π‘β€˜π‘ƒ))
 
Theoremcdlemk19u 39319* Part of Lemma K of [Crawley] p. 118. Line 12, p. 120, "f (exponent) tau = k". We represent f, k, tau with 𝐹, 𝑁, π‘ˆ. (Contributed by NM, 31-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (π‘ˆβ€˜πΉ) = 𝑁)
 
Theoremcdlemk56 39320* Part of Lemma K of [Crawley] p. 118. Line 11, p. 120, "tau is in Delta" i.e. π‘ˆ is a trace-preserving endormorphism. (Contributed by NM, 31-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ π‘ˆ ∈ 𝐸)
 
Theoremcdlemk19w 39321* Use a fixed element to eliminate 𝑃 in cdlemk19u 39319. (Contributed by NM, 1-Aug-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    βŠ₯ = (ocβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = ( βŠ₯ β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) β†’ (π‘ˆβ€˜πΉ) = 𝑁)
 
Theoremcdlemk56w 39322* Use a fixed element to eliminate 𝑃 in cdlemk56 39320. (Contributed by NM, 1-Aug-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &    βŠ₯ = (ocβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = ( βŠ₯ β€˜π‘Š)    &   π‘ = ((𝑃 ∨ (π‘…β€˜π‘)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑏 ∘ ◑𝐹))))    &   π‘Œ = ((𝑃 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘ƒ) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) β†’ (π‘ˆ ∈ 𝐸 ∧ (π‘ˆβ€˜πΉ) = 𝑁))
 
Theoremcdlemk 39323* Lemma K of [Crawley] p. 118. Final result, lines 11 and 12 on p. 120: given two translations f and k with the same trace, there exists a trace-preserving endomorphism tau whose value at f is k. We use 𝐹, 𝑁, and 𝑒 to represent f, k, and tau. (Contributed by NM, 1-Aug-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) β†’ βˆƒπ‘’ ∈ 𝐸 (π‘’β€˜πΉ) = 𝑁)
 
Theoremtendoex 39324* Generalization of Lemma K of [Crawley] p. 118, cdlemk 39323. TODO: can this be used to shorten uses of cdlemk 39323? (Contributed by NM, 15-Oct-2013.)
≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (π‘…β€˜π‘) ≀ (π‘…β€˜πΉ)) β†’ βˆƒπ‘’ ∈ 𝐸 (π‘’β€˜πΉ) = 𝑁)
 
Theoremcdleml1N 39325 Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝑓 ∈ 𝑇) ∧ (𝑓 β‰  ( I β†Ύ 𝐡) ∧ (π‘ˆβ€˜π‘“) β‰  ( I β†Ύ 𝐡) ∧ (π‘‰β€˜π‘“) β‰  ( I β†Ύ 𝐡))) β†’ (π‘…β€˜(π‘ˆβ€˜π‘“)) = (π‘…β€˜(π‘‰β€˜π‘“)))
 
Theoremcdleml2N 39326* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝑓 ∈ 𝑇) ∧ (𝑓 β‰  ( I β†Ύ 𝐡) ∧ (π‘ˆβ€˜π‘“) β‰  ( I β†Ύ 𝐡) ∧ (π‘‰β€˜π‘“) β‰  ( I β†Ύ 𝐡))) β†’ βˆƒπ‘  ∈ 𝐸 (π‘ β€˜(π‘ˆβ€˜π‘“)) = (π‘‰β€˜π‘“))
 
Theoremcdleml3N 39327* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &    0 = (𝑔 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝑓 ∈ 𝑇) ∧ (𝑓 β‰  ( I β†Ύ 𝐡) ∧ π‘ˆ β‰  0 ∧ 𝑉 β‰  0 )) β†’ βˆƒπ‘  ∈ 𝐸 (𝑠 ∘ π‘ˆ) = 𝑉)
 
Theoremcdleml4N 39328* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &    0 = (𝑔 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (π‘ˆ β‰  0 ∧ 𝑉 β‰  0 )) β†’ βˆƒπ‘  ∈ 𝐸 (𝑠 ∘ π‘ˆ) = 𝑉)
 
Theoremcdleml5N 39329* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &    0 = (𝑔 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ π‘ˆ β‰  0 ) β†’ βˆƒπ‘  ∈ 𝐸 (𝑠 ∘ π‘ˆ) = 𝑉)
 
Theoremcdleml6 39330* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘„ = ((ocβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑄 ∨ (π‘…β€˜π‘)) ∧ ((β„Žβ€˜π‘„) ∨ (π‘…β€˜(𝑏 ∘ β—‘(π‘ β€˜β„Ž)))))    &   π‘Œ = ((𝑄 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜(π‘ β€˜β„Ž)) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘„) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if((π‘ β€˜β„Ž) = β„Ž, 𝑔, 𝑋))    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &    0 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ β„Ž ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 β‰  0 )) β†’ (π‘ˆ ∈ 𝐸 ∧ (π‘ˆβ€˜(π‘ β€˜β„Ž)) = β„Ž))
 
Theoremcdleml7 39331* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘„ = ((ocβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑄 ∨ (π‘…β€˜π‘)) ∧ ((β„Žβ€˜π‘„) ∨ (π‘…β€˜(𝑏 ∘ β—‘(π‘ β€˜β„Ž)))))    &   π‘Œ = ((𝑄 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜(π‘ β€˜β„Ž)) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘„) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if((π‘ β€˜β„Ž) = β„Ž, 𝑔, 𝑋))    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &    0 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ β„Ž ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 β‰  0 )) β†’ ((π‘ˆ ∘ 𝑠)β€˜β„Ž) = (( I β†Ύ 𝑇)β€˜β„Ž))
 
Theoremcdleml8 39332* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘„ = ((ocβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑄 ∨ (π‘…β€˜π‘)) ∧ ((β„Žβ€˜π‘„) ∨ (π‘…β€˜(𝑏 ∘ β—‘(π‘ β€˜β„Ž)))))    &   π‘Œ = ((𝑄 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜(π‘ β€˜β„Ž)) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘„) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if((π‘ β€˜β„Ž) = β„Ž, 𝑔, 𝑋))    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &    0 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (β„Ž ∈ 𝑇 ∧ β„Ž β‰  ( I β†Ύ 𝐡)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 β‰  0 )) β†’ (π‘ˆ ∘ 𝑠) = ( I β†Ύ 𝑇))
 
Theoremcdleml9 39333* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘„ = ((ocβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑄 ∨ (π‘…β€˜π‘)) ∧ ((β„Žβ€˜π‘„) ∨ (π‘…β€˜(𝑏 ∘ β—‘(π‘ β€˜β„Ž)))))    &   π‘Œ = ((𝑄 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜(π‘ β€˜β„Ž)) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘„) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if((π‘ β€˜β„Ž) = β„Ž, 𝑔, 𝑋))    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &    0 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (β„Ž ∈ 𝑇 ∧ β„Ž β‰  ( I β†Ύ 𝐡)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 β‰  0 )) β†’ π‘ˆ β‰  0 )
 
Theoremdva1dim 39334* Two expressions for the 1-dimensional subspaces of partial vector space A. Remark in [Crawley] p. 120 line 21, but using a non-identity translation (nonzero vector) 𝐹 whose trace is 𝑃 rather than 𝑃 itself; 𝐹 exists by cdlemf 38912. 𝐸 is the division ring base by erngdv 39342, and π‘ β€˜πΉ is the scalar product by dvavsca 39366. 𝐹 must be a non-identity translation for the expression to be a 1-dimensional subspace, although the theorem doesn't require it. (Contributed by NM, 14-Oct-2013.)
≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ {𝑔 ∣ βˆƒπ‘  ∈ 𝐸 𝑔 = (π‘ β€˜πΉ)} = {𝑔 ∈ 𝑇 ∣ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ)})
 
Theoremdvhb1dimN 39335* Two expressions for the 1-dimensional subspaces of vector space H, in the isomorphism B case where the 2nd vector component is zero. (Contributed by NM, 23-Feb-2014.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &    0 = (β„Ž ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ {𝑔 ∈ (𝑇 Γ— 𝐸) ∣ βˆƒπ‘  ∈ 𝐸 𝑔 = ⟨(π‘ β€˜πΉ), 0 ⟩} = {𝑔 ∈ (𝑇 Γ— 𝐸) ∣ ((π‘…β€˜(1st β€˜π‘”)) ≀ (π‘…β€˜πΉ) ∧ (2nd β€˜π‘”) = 0 )})
 
Theoremerng1lem 39336 Value of the endomorphism division ring unity. (Contributed by NM, 12-Oct-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Ring)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (1rβ€˜π·) = ( I β†Ύ 𝑇))
 
Theoremerngdvlem1 39337* Lemma for eringring 39341. (Contributed by NM, 4-Aug-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &   π΅ = (Baseβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (π‘Ž ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘Žβ€˜π‘“) ∘ (π‘β€˜π‘“))))    &    0 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    &   πΌ = (π‘Ž ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘Žβ€˜π‘“)))    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Grp)
 
Theoremerngdvlem2N 39338* Lemma for eringring 39341. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &   π΅ = (Baseβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (π‘Ž ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘Žβ€˜π‘“) ∘ (π‘β€˜π‘“))))    &    0 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    &   πΌ = (π‘Ž ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘Žβ€˜π‘“)))    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Abel)
 
Theoremerngdvlem3 39339* Lemma for eringring 39341. (Contributed by NM, 6-Aug-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &   π΅ = (Baseβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (π‘Ž ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘Žβ€˜π‘“) ∘ (π‘β€˜π‘“))))    &    0 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    &   πΌ = (π‘Ž ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘Žβ€˜π‘“)))    &    + = (π‘Ž ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (π‘Ž ∘ 𝑏))    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Ring)
 
Theoremerngdvlem4 39340* Lemma for erngdv 39342. (Contributed by NM, 11-Aug-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &   π΅ = (Baseβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (π‘Ž ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘Žβ€˜π‘“) ∘ (π‘β€˜π‘“))))    &    0 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    &   πΌ = (π‘Ž ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘Žβ€˜π‘“)))    &    + = (π‘Ž ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (π‘Ž ∘ 𝑏))    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘„ = ((ocβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑄 ∨ (π‘…β€˜π‘)) ∧ ((β„Žβ€˜π‘„) ∨ (π‘…β€˜(𝑏 ∘ β—‘(π‘ β€˜β„Ž)))))    &   π‘Œ = ((𝑄 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜(π‘ β€˜β„Ž)) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘„) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if((π‘ β€˜β„Ž) = β„Ž, 𝑔, 𝑋))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (β„Ž ∈ 𝑇 ∧ β„Ž β‰  ( I β†Ύ 𝐡))) β†’ 𝐷 ∈ DivRing)
 
Theoremeringring 39341 An endomorphism ring is a ring. TODO: fix comment. (Contributed by NM, 4-Aug-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Ring)
 
Theoremerngdv 39342 An endomorphism ring is a division ring. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ DivRing)
 
Theoremerng0g 39343* The division ring zero of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    &    0 = (0gβ€˜π·)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 = 𝑂)
 
Theoremerng1r 39344 The division ring unity of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &    1 = (1rβ€˜π·)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 1 = ( I β†Ύ 𝑇))
 
Theoremerngdvlem1-rN 39345* Lemma for eringring 39341. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &   π΅ = (Baseβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (π‘Ž ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘Žβ€˜π‘“) ∘ (π‘β€˜π‘“))))    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    &   πΌ = (π‘Ž ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘Žβ€˜π‘“)))    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Grp)
 
Theoremerngdvlem2-rN 39346* Lemma for eringring 39341. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &   π΅ = (Baseβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (π‘Ž ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘Žβ€˜π‘“) ∘ (π‘β€˜π‘“))))    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    &   πΌ = (π‘Ž ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘Žβ€˜π‘“)))    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Abel)
 
Theoremerngdvlem3-rN 39347* Lemma for eringring 39341. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &   π΅ = (Baseβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (π‘Ž ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘Žβ€˜π‘“) ∘ (π‘β€˜π‘“))))    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    &   πΌ = (π‘Ž ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘Žβ€˜π‘“)))    &   π‘€ = (π‘Ž ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑏 ∘ π‘Ž))    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Ring)
 
Theoremerngdvlem4-rN 39348* Lemma for erngdv 39342. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &   π΅ = (Baseβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (π‘Ž ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘Žβ€˜π‘“) ∘ (π‘β€˜π‘“))))    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    &   πΌ = (π‘Ž ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘Žβ€˜π‘“)))    &   π‘€ = (π‘Ž ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑏 ∘ π‘Ž))    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘„ = ((ocβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑄 ∨ (π‘…β€˜π‘)) ∧ ((β„Žβ€˜π‘„) ∨ (π‘…β€˜(𝑏 ∘ β—‘(π‘ β€˜β„Ž)))))    &   π‘Œ = ((𝑄 ∨ (π‘…β€˜π‘”)) ∧ (𝑍 ∨ (π‘…β€˜(𝑔 ∘ ◑𝑏))))    &   π‘‹ = (℩𝑧 ∈ 𝑇 βˆ€π‘ ∈ 𝑇 ((𝑏 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜(π‘ β€˜β„Ž)) ∧ (π‘…β€˜π‘) β‰  (π‘…β€˜π‘”)) β†’ (π‘§β€˜π‘„) = π‘Œ))    &   π‘ˆ = (𝑔 ∈ 𝑇 ↦ if((π‘ β€˜β„Ž) = β„Ž, 𝑔, 𝑋))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (β„Ž ∈ 𝑇 ∧ β„Ž β‰  ( I β†Ύ 𝐡))) β†’ 𝐷 ∈ DivRing)
 
Theoremerngring-rN 39349 An endomorphism ring is a ring. TODO: fix comment. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ Ring)
 
Theoremerngdv-rN 39350 An endomorphism ring is a division ring. TODO: fix comment. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 ∈ DivRing)
 
Syntaxcdveca 39351 Extend class notation with constructed vector space A.
class DVecA
 
Definitiondf-dveca 39352* Define constructed partial vector space A. (Contributed by NM, 8-Oct-2013.)
DVecA = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ ({⟨(Baseβ€˜ndx), ((LTrnβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜π‘˜)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (π‘ β€˜π‘“))⟩})))
 
Theoremdvafset 39353* The constructed partial vector space A for a lattice 𝐾. (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHypβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ (DVecAβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ ({⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘€)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (π‘ β€˜π‘“))⟩})))
 
Theoremdvaset 39354* The constructed partial vector space A for a lattice 𝐾. (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ = ({⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalarβ€˜ndx), 𝐷⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (π‘ β€˜π‘“))⟩}))
 
Theoremdvasca 39355 The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom π‘Š). (Contributed by NM, 22-Jun-2014.)
𝐻 = (LHypβ€˜πΎ)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘ˆ)    β‡’   ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ 𝐹 = 𝐷)
 
Theoremdvabase 39356 The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom π‘Š). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHypβ€˜πΎ)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘ˆ)    &   πΆ = (Baseβ€˜πΉ)    β‡’   ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ 𝐢 = 𝐸)
 
Theoremdvafplusg 39357* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘ˆ)    &    + = (+gβ€˜πΉ)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ + = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“)))))
 
Theoremdvaplusg 39358* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘ˆ)    &    + = (+gβ€˜πΉ)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸)) β†’ (𝑅 + 𝑆) = (𝑓 ∈ 𝑇 ↦ ((π‘…β€˜π‘“) ∘ (π‘†β€˜π‘“))))
 
Theoremdvaplusgv 39359 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘ˆ)    &    + = (+gβ€˜πΉ)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇)) β†’ ((𝑅 + 𝑆)β€˜πΊ) = ((π‘…β€˜πΊ) ∘ (π‘†β€˜πΊ)))
 
Theoremdvafmulr 39360* Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘ˆ)    &    Β· = (.rβ€˜πΉ)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ Β· = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑠 ∘ 𝑑)))
 
Theoremdvamulr 39361 Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘ˆ)    &    Β· = (.rβ€˜πΉ)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸)) β†’ (𝑅 Β· 𝑆) = (𝑅 ∘ 𝑆))
 
Theoremdvavbase 39362 The vectors (vector base set) of the constructed partial vector space A are all translations (for a fiducial co-atom π‘Š). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    β‡’   ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ 𝑉 = 𝑇)
 
Theoremdvafvadd 39363* The vector sum operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π‘ˆ)    β‡’   ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ + = (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔)))
 
Theoremdvavadd 39364 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π‘ˆ)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) β†’ (𝐹 + 𝐺) = (𝐹 ∘ 𝐺))
 
Theoremdvafvsca 39365* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ Β· = (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (π‘ β€˜π‘“)))
 
Theoremdvavsca 39366 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) β†’ (𝑅 Β· 𝐹) = (π‘…β€˜πΉ))
 
Theoremtendospcl 39367 Closure of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ π‘ˆ ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ (π‘ˆβ€˜πΉ) ∈ 𝑇)
 
Theoremtendospass 39368 Associative law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) β†’ ((π‘ˆ ∘ 𝑉)β€˜πΉ) = (π‘ˆβ€˜(π‘‰β€˜πΉ)))
 
Theoremtendospdi1 39369 Forward distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) β†’ (π‘ˆβ€˜(𝐹 ∘ 𝐺)) = ((π‘ˆβ€˜πΉ) ∘ (π‘ˆβ€˜πΊ)))
 
Theoremtendocnv 39370 Converse of a trace-preserving endomorphism value. (Contributed by NM, 7-Apr-2014.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ β—‘(π‘†β€˜πΉ) = (π‘†β€˜β—‘πΉ))
 
Theoremtendospdi2 39371* Reverse distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    β‡’   ((π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ ((π‘ˆπ‘ƒπ‘‰)β€˜πΉ) = ((π‘ˆβ€˜πΉ) ∘ (π‘‰β€˜πΉ)))
 
TheoremtendospcanN 39372* Cancellation law for trace-preserving endomorphism values (used as scalar product). (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 ∈ 𝐸 ∧ 𝑆 β‰  𝑂) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) β†’ ((π‘†β€˜πΉ) = (π‘†β€˜πΊ) ↔ 𝐹 = 𝐺))
 
Theoremdvaabl 39373 The constructed partial vector space A for a lattice 𝐾 is an abelian group. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ Abel)
 
Theoremdvalveclem 39374 Lemma for dvalvec 39375. (Contributed by NM, 11-Oct-2013.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π‘ˆ)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜πΎ)    &    ⨣ = (+gβ€˜π·)    &    Γ— = (.rβ€˜π·)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ LVec)
 
Theoremdvalvec 39375 The constructed partial vector space A for a lattice 𝐾 is a left vector space. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ ∈ LVec)
 
Theoremdva0g 39376 The zero vector of partial vector space A. (Contributed by NM, 9-Sep-2014.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &    0 = (0gβ€˜π‘ˆ)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 = ( I β†Ύ 𝐡))
 
Syntaxcdia 39377 Extend class notation with partial isomorphism A.
class DIsoA
 
Definitiondf-disoa 39378* Define partial isomorphism A. (Contributed by NM, 15-Oct-2013.)
DIsoA = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ (π‘₯ ∈ {𝑦 ∈ (Baseβ€˜π‘˜) ∣ 𝑦(leβ€˜π‘˜)𝑀} ↦ {𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ∣ (((trLβ€˜π‘˜)β€˜π‘€)β€˜π‘“)(leβ€˜π‘˜)π‘₯})))
 
Theoremdiaffval 39379* The partial isomorphism A for a lattice 𝐾. (Contributed by NM, 15-Oct-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ (DIsoAβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ 𝑀} ↦ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ∣ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) ≀ π‘₯})))
 
Theoremdiafval 39380* The partial isomorphism A for a lattice 𝐾. (Contributed by NM, 15-Oct-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 = (π‘₯ ∈ {𝑦 ∈ 𝐡 ∣ 𝑦 ≀ π‘Š} ↦ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ π‘₯}))
 
Theoremdiaval 39381* The partial isomorphism A for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ 𝑋})
 
Theoremdiaelval 39382 Member of the partial isomorphism A for a lattice 𝐾. (Contributed by NM, 3-Dec-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (𝐹 ∈ (πΌβ€˜π‘‹) ↔ (𝐹 ∈ 𝑇 ∧ (π‘…β€˜πΉ) ≀ 𝑋)))
 
Theoremdiafn 39383* Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 Fn {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š})
 
Theoremdiadm 39384* Domain of the partial isomorphism A. (Contributed by NM, 3-Dec-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ dom 𝐼 = {π‘₯ ∈ 𝐡 ∣ π‘₯ ≀ π‘Š})
 
Theoremdiaeldm 39385 Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)))
 
TheoremdiadmclN 39386 A member of domain of the partial isomorphism A is a lattice element. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ 𝑋 ∈ 𝐡)
 
TheoremdiadmleN 39387 A member of domain of the partial isomorphism A is under the fiducial hyperplane. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ 𝑋 ≀ π‘Š)
 
Theoremdian0 39388 The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) β‰  βˆ…)
 
Theoremdia0eldmN 39389 The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
0 = (0.β€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 ∈ dom 𝐼)
 
Theoremdia1eldmN 39390 The fiducial hyperplane (the largest allowed lattice element) belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘Š ∈ dom 𝐼)
 
Theoremdiass 39391 The value of the partial isomorphism A is a set of translations, i.e., a set of vectors. (Contributed by NM, 26-Nov-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) βŠ† 𝑇)
 
Theoremdiael 39392 A member of the value of the partial isomorphism A is a translation, i.e., a vector. (Contributed by NM, 17-Jan-2014.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ 𝐹 ∈ (πΌβ€˜π‘‹)) β†’ 𝐹 ∈ 𝑇)
 
Theoremdiatrl 39393 Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ 𝐹 ∈ (πΌβ€˜π‘‹)) β†’ (π‘…β€˜πΉ) ≀ 𝑋)
 
TheoremdiaelrnN 39394 Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ ran 𝐼) β†’ 𝑆 βŠ† 𝑇)
 
Theoremdialss 39395 The value of partial isomorphism A is a subspace of partial vector space A. Part of Lemma M of [Crawley] p. 120 line 26. (Contributed by NM, 17-Jan-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecAβ€˜πΎ)β€˜π‘Š)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) ∈ 𝑆)
 
Theoremdiaord 39396 The partial isomorphism A for a lattice 𝐾 is order-preserving in the region under co-atom π‘Š. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 26-Nov-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ ((πΌβ€˜π‘‹) βŠ† (πΌβ€˜π‘Œ) ↔ 𝑋 ≀ π‘Œ))
 
Theoremdia11N 39397 The partial isomorphism A for a lattice 𝐾 is one-to-one in the region under co-atom π‘Š. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 25-Nov-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ ((πΌβ€˜π‘‹) = (πΌβ€˜π‘Œ) ↔ 𝑋 = π‘Œ))
 
Theoremdiaf11N 39398 The partial isomorphism A for a lattice 𝐾 is a one-to-one function. Part of Lemma M of [Crawley] p. 120 line 27. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐼:dom 𝐼–1-1-ontoβ†’ran 𝐼)
 
TheoremdiaclN 39399 Closure of partial isomorphism A for a lattice 𝐾. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ (πΌβ€˜π‘‹) ∈ ran 𝐼)
 
TheoremdiacnvclN 39400 Closure of partial isomorphism A converse. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   πΌ = ((DIsoAβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) β†’ (β—‘πΌβ€˜π‘‹) ∈ dom 𝐼)
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-46966
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