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Theorem List for Metamath Proof Explorer - 39101-39200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxctendo 39101 Extend class notation with translation group endomorphisms.
class TEndo
 
Syntaxcedring 39102 Extend class notation with division ring on trace-preserving endomorphisms.
class EDRing
 
Syntaxcedring-rN 39103 Extend class notation with division ring on trace-preserving endomorphisms, with multiplication reversed. TODO: remove EDRingR theorems if not used.
class EDRingR
 
Definitiondf-tendo 39104* Define trace-preserving endomorphisms on the set of translations. (Contributed by NM, 8-Jun-2013.)
TEndo = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {𝑓 ∣ (𝑓:((LTrnβ€˜π‘˜)β€˜π‘€)⟢((LTrnβ€˜π‘˜)β€˜π‘€) ∧ βˆ€π‘₯ ∈ ((LTrnβ€˜π‘˜)β€˜π‘€)βˆ€π‘¦ ∈ ((LTrnβ€˜π‘˜)β€˜π‘€)(π‘“β€˜(π‘₯ ∘ 𝑦)) = ((π‘“β€˜π‘₯) ∘ (π‘“β€˜π‘¦)) ∧ βˆ€π‘₯ ∈ ((LTrnβ€˜π‘˜)β€˜π‘€)(((trLβ€˜π‘˜)β€˜π‘€)β€˜(π‘“β€˜π‘₯))(leβ€˜π‘˜)(((trLβ€˜π‘˜)β€˜π‘€)β€˜π‘₯))}))
 
Definitiondf-edring-rN 39105* Define division ring on trace-preserving endomorphisms. Definition of E of [Crawley] p. 117, 4th line from bottom. (Contributed by NM, 8-Jun-2013.)
EDRingR = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {⟨(Baseβ€˜ndx), ((TEndoβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€) ↦ (𝑑 ∘ 𝑠))⟩}))
 
Definitiondf-edring 39106* Define division ring on trace-preserving endomorphisms. The multiplication operation is reversed composition, per the definition of E of [Crawley] p. 117, 4th line from bottom. (Contributed by NM, 8-Jun-2013.)
EDRing = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {⟨(Baseβ€˜ndx), ((TEndoβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€) ↦ (𝑠 ∘ 𝑑))⟩}))
 
Theoremtendofset 39107* The set of all trace-preserving endomorphisms on the set of translations for a lattice 𝐾. (Contributed by NM, 8-Jun-2013.)
≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ (TEndoβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {𝑠 ∣ (𝑠:((LTrnβ€˜πΎ)β€˜π‘€)⟢((LTrnβ€˜πΎ)β€˜π‘€) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(((trLβ€˜πΎ)β€˜π‘€)β€˜(π‘ β€˜π‘“)) ≀ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“))}))
 
Theoremtendoset 39108* The set of trace-preserving endomorphisms on the set of translations for a fiducial co-atom π‘Š. (Contributed by NM, 8-Jun-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 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐸 = {𝑠 ∣ (𝑠:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))})
 
Theoremistendo 39109* The predicate "is a trace-preserving endomorphism". Similar to definition of trace-preserving endomorphism in [Crawley] p. 117, penultimate line. (Contributed by NM, 8-Jun-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 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑆 ∈ 𝐸 ↔ (𝑆:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“))))
 
Theoremtendotp 39110 Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-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 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜(π‘†β€˜πΉ)) ≀ (π‘…β€˜πΉ))
 
Theoremistendod 39111* Deduce the predicate "is a trace-preserving endomorphism". (Contributed by NM, 9-Jun-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)    &   (πœ‘ β†’ (𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑆:π‘‡βŸΆπ‘‡)    &   ((πœ‘ ∧ 𝑓 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇) β†’ (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)))    &   ((πœ‘ ∧ 𝑓 ∈ 𝑇) β†’ (π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“))    β‡’   (πœ‘ β†’ 𝑆 ∈ 𝐸)
 
Theoremtendof 39112 Functionality of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-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) ∧ 𝑆 ∈ 𝐸) β†’ 𝑆:π‘‡βŸΆπ‘‡)
 
Theoremtendoeq1 39113* Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013.)
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) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ βˆ€π‘“ ∈ 𝑇 (π‘ˆβ€˜π‘“) = (π‘‰β€˜π‘“)) β†’ π‘ˆ = 𝑉)
 
Theoremtendovalco 39114 Value of composition of translations in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-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 ∧ 𝑆 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) β†’ (π‘†β€˜(𝐹 ∘ 𝐺)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜πΊ)))
 
Theoremtendocoval 39115 Value of composition of endomorphisms in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-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)))
 
Theoremtendocl 39116 Closure of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-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) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ (π‘†β€˜πΉ) ∈ 𝑇)
 
Theoremtendoco2 39117 Distribution of compositions in preparation for endomorphism sum definition. (Contributed by NM, 10-Jun-2013.)
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) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) β†’ ((π‘ˆβ€˜(𝐹 ∘ 𝐺)) ∘ (π‘‰β€˜(𝐹 ∘ 𝐺))) = (((π‘ˆβ€˜πΉ) ∘ (π‘‰β€˜πΉ)) ∘ ((π‘ˆβ€˜πΊ) ∘ (π‘‰β€˜πΊ))))
 
Theoremtendoidcl 39118 The identity is a trace-preserving endomorphism. (Contributed by NM, 30-Jul-2013.)
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) β†’ ( I β†Ύ 𝑇) ∈ 𝐸)
 
Theoremtendo1mul 39119 Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)
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) ∧ π‘ˆ ∈ 𝐸) β†’ (( I β†Ύ 𝑇) ∘ π‘ˆ) = π‘ˆ)
 
Theoremtendo1mulr 39120 Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)
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) ∧ π‘ˆ ∈ 𝐸) β†’ (π‘ˆ ∘ ( I β†Ύ 𝑇)) = π‘ˆ)
 
Theoremtendococl 39121 The composition of two trace-preserving endomorphisms (multiplication in the endormorphism ring) is a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) β†’ (𝑆 ∘ 𝑇) ∈ 𝐸)
 
Theoremtendoid 39122 The identity value of a trace-preserving endomorphism. (Contributed by NM, 21-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) β†’ (π‘†β€˜( I β†Ύ 𝐡)) = ( I β†Ύ 𝐡))
 
Theoremtendoeq2 39123* Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 39173, we show that we only need to consider a single non-identity translation. (Contributed by NM, 21-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ βˆ€π‘“ ∈ 𝑇 (𝑓 β‰  ( I β†Ύ 𝐡) β†’ (π‘ˆβ€˜π‘“) = (π‘‰β€˜π‘“))) β†’ π‘ˆ = 𝑉)
 
Theoremtendoplcbv 39124* Define sum operation for trace-preserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.)
𝑃 = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    β‡’   π‘ƒ = (𝑒 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((π‘’β€˜π‘”) ∘ (π‘£β€˜π‘”))))
 
Theoremtendopl 39125* Value of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
𝑃 = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   ((π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) β†’ (π‘ˆπ‘ƒπ‘‰) = (𝑔 ∈ 𝑇 ↦ ((π‘ˆβ€˜π‘”) ∘ (π‘‰β€˜π‘”))))
 
Theoremtendopl2 39126* Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
𝑃 = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   ((π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ ((π‘ˆπ‘ƒπ‘‰)β€˜πΉ) = ((π‘ˆβ€˜πΉ) ∘ (π‘‰β€˜πΉ)))
 
Theoremtendoplcl2 39127* Value of result of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝐹 ∈ 𝑇) β†’ ((π‘ˆπ‘ƒπ‘‰)β€˜πΉ) ∈ 𝑇)
 
Theoremtendoplco2 39128* Value of result of endomorphism sum operation on a translation composition. (Contributed by NM, 10-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) β†’ ((π‘ˆπ‘ƒπ‘‰)β€˜(𝐹 ∘ 𝐺)) = (((π‘ˆπ‘ƒπ‘‰)β€˜πΉ) ∘ ((π‘ˆπ‘ƒπ‘‰)β€˜πΊ)))
 
Theoremtendopltp 39129* Trace-preserving property of endomorphism sum operation 𝑃, based on Theorems trlco 39076. Part of remark in [Crawley] p. 118, 2nd line, "it is clear from the second part of G (our trlco 39076) that Delta is a subring of E." (In our development, we will bypass their E and go directly to their Delta, whose base set is our (TEndoβ€˜πΎ)β€˜π‘Š.) (Contributed by NM, 9-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    &    ≀ = (leβ€˜πΎ)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜((π‘ˆπ‘ƒπ‘‰)β€˜πΉ)) ≀ (π‘…β€˜πΉ))
 
Theoremtendoplcl 39130* Endomorphism sum is a trace-preserving endomorphism. (Contributed by NM, 10-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) β†’ (π‘ˆπ‘ƒπ‘‰) ∈ 𝐸)
 
Theoremtendoplcom 39131* The endomorphism sum operation is commutative. (Contributed by NM, 11-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) β†’ (π‘ˆπ‘ƒπ‘‰) = (π‘‰π‘ƒπ‘ˆ))
 
Theoremtendoplass 39132* The endomorphism sum operation is associative. (Contributed by NM, 11-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 ∈ 𝐸 ∧ π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) β†’ ((π‘†π‘ƒπ‘ˆ)𝑃𝑉) = (𝑆𝑃(π‘ˆπ‘ƒπ‘‰)))
 
Theoremtendodi1 39133* Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 ∈ 𝐸 ∧ π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) β†’ (𝑆 ∘ (π‘ˆπ‘ƒπ‘‰)) = ((𝑆 ∘ π‘ˆ)𝑃(𝑆 ∘ 𝑉)))
 
Theoremtendodi2 39134* Endomorphism composition distributes over sum. (Contributed by NM, 13-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑆 ∈ 𝐸 ∧ π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) β†’ ((π‘†π‘ƒπ‘ˆ) ∘ 𝑉) = ((𝑆 ∘ 𝑉)𝑃(π‘ˆ ∘ 𝑉)))
 
Theoremtendo0cbv 39135* Define additive identity for trace-preserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 11-Jun-2013.)
𝑂 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   π‘‚ = (𝑔 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
 
Theoremtendo02 39136* Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)
𝑂 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    &   π΅ = (Baseβ€˜πΎ)    β‡’   (𝐹 ∈ 𝑇 β†’ (π‘‚β€˜πΉ) = ( I β†Ύ 𝐡))
 
Theoremtendo0co2 39137* The additive identity trace-preserving endormorphism preserves composition of translations. TODO: why isn't this a special case of tendospdi1 39369? (Contributed by NM, 11-Jun-2013.)
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) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ (π‘‚β€˜(𝐹 ∘ 𝐺)) = ((π‘‚β€˜πΉ) ∘ (π‘‚β€˜πΊ)))
 
Theoremtendo0tp 39138* Trace-preserving property of endomorphism additive identity. (Contributed by NM, 11-Jun-2013.)
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))    &    ≀ = (leβ€˜πΎ)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜(π‘‚β€˜πΉ)) ≀ (π‘…β€˜πΉ))
 
Theoremtendo0cl 39139* The additive identity is a trace-preserving endormorphism. (Contributed by NM, 12-Jun-2013.)
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) β†’ 𝑂 ∈ 𝐸)
 
Theoremtendo0pl 39140* Property of the additive identity endormorphism. (Contributed by NM, 12-Jun-2013.)
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))    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) β†’ (𝑂𝑃𝑆) = 𝑆)
 
Theoremtendo0plr 39141* Property of the additive identity endormorphism. (Contributed by NM, 21-Feb-2014.)
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))    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) β†’ (𝑆𝑃𝑂) = 𝑆)
 
Theoremtendoicbv 39142* Define inverse function for trace-preserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 12-Jun-2013.)
𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘ β€˜π‘“)))    β‡’   πΌ = (𝑒 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ β—‘(π‘’β€˜π‘”)))
 
Theoremtendoi 39143* Value of inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘ β€˜π‘“)))    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (𝑆 ∈ 𝐸 β†’ (πΌβ€˜π‘†) = (𝑔 ∈ 𝑇 ↦ β—‘(π‘†β€˜π‘”)))
 
Theoremtendoi2 39144* Value of additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘ β€˜π‘“)))    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ ((πΌβ€˜π‘†)β€˜πΉ) = β—‘(π‘†β€˜πΉ))
 
Theoremtendoicl 39145* Closure of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   πΌ = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘ β€˜π‘“)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) β†’ (πΌβ€˜π‘†) ∈ 𝐸)
 
Theoremtendoipl 39146* Property of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   πΌ = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘ β€˜π‘“)))    &   π΅ = (Baseβ€˜πΎ)    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) β†’ ((πΌβ€˜π‘†)𝑃𝑆) = 𝑂)
 
Theoremtendoipl2 39147* Property of the additive inverse endomorphism. (Contributed by NM, 29-Sep-2014.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   πΌ = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘ β€˜π‘“)))    &   π΅ = (Baseβ€˜πΎ)    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) β†’ (𝑆𝑃(πΌβ€˜π‘†)) = 𝑂)
 
Theoremerngfset 39148* The division rings on trace-preserving endomorphisms for a lattice 𝐾. (Contributed by NM, 8-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ (EDRingβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑠 ∘ 𝑑))⟩}))
 
Theoremerngset 39149* The division ring on trace-preserving endomorphisms for a fiducial co-atom π‘Š. (Contributed by NM, 5-Jun-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)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑠 ∘ 𝑑))⟩})
 
Theoremerngbase 39150 The base set of the division ring on trace-preserving endomorphisms is the set of all trace-preserving endomorphisms (for a fiducial co-atom π‘Š). TODO: the .t hypothesis isn't used. (Also look at others.) (Contributed by NM, 9-Jun-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)    &   πΆ = (Baseβ€˜π·)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐢 = 𝐸)
 
Theoremerngfplus 39151* Ring addition operation. (Contributed by NM, 9-Jun-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)    &    + = (+gβ€˜π·)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ + = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“)))))
 
Theoremerngplus 39152* Ring addition operation. (Contributed by NM, 10-Jun-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)    &    + = (+gβ€˜π·)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) β†’ (π‘ˆ + 𝑉) = (𝑓 ∈ 𝑇 ↦ ((π‘ˆβ€˜π‘“) ∘ (π‘‰β€˜π‘“))))
 
Theoremerngplus2 39153 Ring addition operation. (Contributed by NM, 10-Jun-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)    &    + = (+gβ€˜π·)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) β†’ ((π‘ˆ + 𝑉)β€˜πΉ) = ((π‘ˆβ€˜πΉ) ∘ (π‘‰β€˜πΉ)))
 
Theoremerngfmul 39154* Ring multiplication operation. (Contributed by NM, 9-Jun-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)    &    Β· = (.rβ€˜π·)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ Β· = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑠 ∘ 𝑑)))
 
Theoremerngmul 39155 Ring addition operation. (Contributed by NM, 10-Jun-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)    &    Β· = (.rβ€˜π·)    β‡’   (((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) β†’ (π‘ˆ Β· 𝑉) = (π‘ˆ ∘ 𝑉))
 
Theoremerngfset-rN 39156* The division rings on trace-preserving endomorphisms for a lattice 𝐾. (Contributed by NM, 8-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ (EDRingRβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑑 ∘ 𝑠))⟩}))
 
Theoremerngset-rN 39157* The division ring on trace-preserving endomorphisms for a fiducial co-atom π‘Š. (Contributed by NM, 5-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑑 ∘ 𝑠))⟩})
 
Theoremerngbase-rN 39158 The base set of the division ring on trace-preserving endomorphisms is the set of all trace-preserving endomorphisms (for a fiducial co-atom π‘Š). (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &   πΆ = (Baseβ€˜π·)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐢 = 𝐸)
 
Theoremerngfplus-rN 39159* Ring addition operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π·)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ + = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“)))))
 
Theoremerngplus-rN 39160* Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π·)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) β†’ (π‘ˆ + 𝑉) = (𝑓 ∈ 𝑇 ↦ ((π‘ˆβ€˜π‘“) ∘ (π‘‰β€˜π‘“))))
 
Theoremerngplus2-rN 39161 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π·)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) β†’ ((π‘ˆ + 𝑉)β€˜πΉ) = ((π‘ˆβ€˜πΉ) ∘ (π‘‰β€˜πΉ)))
 
Theoremerngfmul-rN 39162* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &    Β· = (.rβ€˜π·)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ Β· = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑑 ∘ 𝑠)))
 
Theoremerngmul-rN 39163 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &    Β· = (.rβ€˜π·)    β‡’   (((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) β†’ (π‘ˆ Β· 𝑉) = (𝑉 ∘ π‘ˆ))
 
Theoremcdlemh1 39164 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-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Ί)) ∧ (𝑄 ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≀ (𝑃 ∨ (π‘…β€˜πΉ)) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ))) β†’ (𝑆 ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))) = (𝑄 ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))
 
Theoremcdlemh2 39165 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 16-Jun-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Ί)) ∧ (𝑄 ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))    &    0 = (0.β€˜πΎ)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ))) β†’ (𝑆 ∧ π‘Š) = 0 )
 
Theoremcdlemh 39166 Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-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Ί)) ∧ (𝑄 ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ (𝑃 ∨ (π‘…β€˜πΉ))) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ))) β†’ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š))
 
Theoremcdlemi1 39167 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-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Έ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ ((π‘ˆβ€˜πΊ)β€˜π‘ƒ) ≀ (𝑃 ∨ (π‘…β€˜πΊ)))
 
Theoremcdlemi2 39168 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-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Έ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ ((π‘ˆβ€˜πΊ)β€˜π‘ƒ) ≀ (((π‘ˆβ€˜πΉ)β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))
 
Theoremcdlemi 39169 Lemma I of [Crawley] p. 118. (Contributed by NM, 19-Jun-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Έ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘† = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ (((π‘ˆβ€˜πΉ)β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (π‘ˆ ∈ 𝐸 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ))) β†’ ((π‘ˆβ€˜πΊ)β€˜π‘ƒ) = 𝑆)
 
Theoremcdlemj1 39170 Part of proof of Lemma J of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)
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)    &    ≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (π‘ˆβ€˜πΉ) = (π‘‰β€˜πΉ)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡) ∧ β„Ž ∈ 𝑇)) ∧ (β„Ž β‰  ( I β†Ύ 𝐡) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 β‰  ( I β†Ύ 𝐡)) ∧ ((π‘…β€˜πΉ) β‰  (π‘…β€˜π‘”) ∧ (π‘…β€˜π‘”) β‰  (π‘…β€˜β„Ž) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š))) β†’ ((π‘ˆβ€˜β„Ž)β€˜π‘) = ((π‘‰β€˜β„Ž)β€˜π‘))
 
Theoremcdlemj2 39171 Part of proof of Lemma J of [Crawley] p. 118. Eliminate 𝑝. (Contributed by NM, 20-Jun-2013.)
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 ∧ (π‘ˆβ€˜πΉ) = (π‘‰β€˜πΉ)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡) ∧ β„Ž ∈ 𝑇)) ∧ (β„Ž β‰  ( I β†Ύ 𝐡) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 β‰  ( I β†Ύ 𝐡)) ∧ ((π‘…β€˜πΉ) β‰  (π‘…β€˜π‘”) ∧ (π‘…β€˜π‘”) β‰  (π‘…β€˜β„Ž))) β†’ (π‘ˆβ€˜β„Ž) = (π‘‰β€˜β„Ž))
 
Theoremcdlemj3 39172 Part of proof of Lemma J of [Crawley] p. 118. Eliminate 𝑔. (Contributed by NM, 20-Jun-2013.)
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 ∧ (π‘ˆβ€˜πΉ) = (π‘‰β€˜πΉ)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡) ∧ β„Ž ∈ 𝑇)) ∧ β„Ž β‰  ( I β†Ύ 𝐡)) β†’ (π‘ˆβ€˜β„Ž) = (π‘‰β€˜β„Ž))
 
Theoremtendocan 39173 Cancellation law: if the values of two trace-preserving endormorphisms are equal, so are the endormorphisms. Lemma J of [Crawley] p. 118. (Contributed by NM, 21-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (π‘ˆβ€˜πΉ) = (π‘‰β€˜πΉ)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡))) β†’ π‘ˆ = 𝑉)
 
Theoremtendoid0 39174* A trace-preserving endomorphism is the additive identity iff at least one of its values (at a non-identity translation) is the identity translation. (Contributed by NM, 1-Aug-2013.)
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) ∧ π‘ˆ ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡))) β†’ ((π‘ˆβ€˜πΉ) = ( I β†Ύ 𝐡) ↔ π‘ˆ = 𝑂))
 
Theoremtendo0mul 39175* Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 1-Aug-2013.)
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) ∧ π‘ˆ ∈ 𝐸) β†’ (𝑂 ∘ π‘ˆ) = 𝑂)
 
Theoremtendo0mulr 39176* Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 13-Feb-2014.)
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) ∧ π‘ˆ ∈ 𝐸) β†’ (π‘ˆ ∘ 𝑂) = 𝑂)
 
Theoremtendo1ne0 39177* The identity (unity) is not equal to the zero trace-preserving endomorphism. (Contributed by NM, 8-Aug-2013.)
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) β†’ ( I β†Ύ 𝑇) β‰  𝑂)
 
Theoremtendoconid 39178* The composition (product) of trace-preserving endormorphisms is nonzero when each argument is nonzero. (Contributed by NM, 8-Aug-2013.)
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) ∧ (π‘ˆ ∈ 𝐸 ∧ π‘ˆ β‰  𝑂) ∧ (𝑉 ∈ 𝐸 ∧ 𝑉 β‰  𝑂)) β†’ (π‘ˆ ∘ 𝑉) β‰  𝑂)
 
Theoremtendotr 39179* The trace of the value of a nonzero trace-preserving endomorphism equals the trace of the argument. (Contributed by NM, 11-Aug-2013.)
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)    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ π‘ˆ β‰  𝑂) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜(π‘ˆβ€˜πΉ)) = (π‘…β€˜πΉ))
 
Theoremcdlemk1 39180 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜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†’ (𝑃 ∨ (π‘β€˜π‘ƒ)) = ((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜πΉ)))
 
Theoremcdlemk2 39181 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ ((πΊβ€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))) = ((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))
 
Theoremcdlemk3 39182 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹)))) = (πΉβ€˜π‘ƒ))
 
Theoremcdlemk4 39183 Part of proof of Lemma K of [Crawley] p. 118, last line. We use 𝑋 for their h, since 𝐻 is already used. (Contributed by NM, 24-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (πΉβ€˜π‘ƒ) ≀ ((π‘‹β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹))))
 
Theoremcdlemk5a 39184 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ ((π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹)))) ≀ ((π‘‹β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹))))
 
Theoremcdlemk5 39185 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 25-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ))) β†’ ((𝑃 ∨ (π‘β€˜π‘ƒ)) ∧ ((πΊβ€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹)))) ≀ ((π‘‹β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹))))
 
Theoremcdlemk6 39186 Part of proof of Lemma K of [Crawley] p. 118. Apply dalaw 38235. (Contributed by NM, 25-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ ((π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜πΉ)))) β†’ ((𝑃 ∨ (πΊβ€˜π‘ƒ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹)))) ≀ ((((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐹)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹)))) ∨ (((π‘‹β€˜π‘ƒ) ∨ 𝑃) ∧ ((π‘…β€˜(𝑋 ∘ ◑𝐹)) ∨ (π‘β€˜π‘ƒ)))))
 
Theoremcdlemk8 39187 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 26-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ ((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) = ((πΊβ€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐺))))
 
Theoremcdlemk9 39188 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 29-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ π‘Š) = (π‘…β€˜(𝑋 ∘ ◑𝐺)))
 
Theoremcdlemk9bN 39189 Part of proof of Lemma K of [Crawley] p. 118. TODO: is this needed? If so, shorten with cdlemk9 39188 if that one is also needed. (Contributed by NM, 28-Jun-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ π‘Š) = (π‘…β€˜(𝐺 ∘ ◑𝑋)))
 
Theoremcdlemki 39190* Part of proof of Lemma K of [Crawley] p. 118. TODO: Eliminate and put into cdlemksel 39194. (Contributed by NM, 25-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   πΌ = (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹)))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ))) β†’ 𝐼 ∈ 𝑇)
 
Theoremcdlemkvcl 39191 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 27-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘‰ = (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐹)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) β†’ 𝑉 ∈ 𝐡)
 
Theoremcdlemk10 39192 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 29-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘‰ = (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐹)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ 𝑉 ≀ (π‘…β€˜(𝑋 ∘ ◑𝐺)))
 
Theoremcdlemksv 39193* Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function. (Contributed by NM, 26-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    β‡’   (𝐺 ∈ 𝑇 β†’ (π‘†β€˜πΊ) = (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))))
 
Theoremcdlemksel 39194* Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma(p) function to be a translation. TODO: combine cdlemki 39190? (Contributed by NM, 26-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ))) β†’ (π‘†β€˜πΊ) ∈ 𝑇)
 
Theoremcdlemksat 39195* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 27-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ))) β†’ ((π‘†β€˜πΊ)β€˜π‘ƒ) ∈ 𝐴)
 
Theoremcdlemksv2 39196* Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function 𝑆 at the fixed 𝑃 parameter. (Contributed by NM, 26-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ))) β†’ ((π‘†β€˜πΊ)β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹)))))
 
Theoremcdlemk7 39197* Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119. (Contributed by NM, 27-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‰ = (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐹)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ 𝑋 β‰  ( I β†Ύ 𝐡)) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜πΉ))) β†’ ((π‘†β€˜πΊ)β€˜π‘ƒ) ≀ (((π‘†β€˜π‘‹)β€˜π‘ƒ) ∨ 𝑉))
 
Theoremcdlemk11 39198* Part of proof of Lemma K of [Crawley] p. 118. Eq. 3, line 8, p. 119. (Contributed by NM, 29-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‰ = (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐹)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ 𝑋 β‰  ( I β†Ύ 𝐡)) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜πΉ))) β†’ ((π‘†β€˜πΊ)β€˜π‘ƒ) ≀ (((π‘†β€˜π‘‹)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐺))))
 
Theoremcdlemk12 39199* Part of proof of Lemma K of [Crawley] p. 118. Eq. 4, line 10, p. 119. (Contributed by NM, 30-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ 𝑋 β‰  ( I β†Ύ 𝐡)) ∧ ((π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜πΉ)) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜π‘‹))) β†’ ((π‘†β€˜πΊ)β€˜π‘ƒ) = ((𝑃 ∨ (πΊβ€˜π‘ƒ)) ∧ (((π‘†β€˜π‘‹)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐺)))))
 
Theoremcdlemkoatnle 39200* Utility lemma. (Contributed by NM, 2-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‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΉ))) β†’ ((π‘‚β€˜π‘ƒ) ∈ 𝐴 ∧ Β¬ (π‘‚β€˜π‘ƒ) ≀ π‘Š))
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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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