Theorem tendofset

Description: The set of all trace-preserving endomorphisms on the set of translations for a lattice K . (Contributed by NM, 8-Jun-2013)

	Ref	Expression
Hypotheses	tendoset.l	⊢ ≤ = ( le ‘ 𝐾 )
	tendoset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	tendofset	⊢ ( 𝐾 ∈ 𝑉 → ( TEndo ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } ) )

Proof

Step	Hyp	Ref	Expression
1		tendoset.l	⊢ ≤ = ( le ‘ 𝐾 )
2		tendoset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3		elex	⊢ ( 𝐾 ∈ 𝑉 → 𝐾 ∈ V )
4		fveq2	⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = ( LHyp ‘ 𝐾 ) )
5	4 2	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = 𝐻 )
6		fveq2	⊢ ( 𝑘 = 𝐾 → ( LTrn ‘ 𝑘 ) = ( LTrn ‘ 𝐾 ) )
7	6	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) )
8	7 7	feq23d	⊢ ( 𝑘 = 𝐾 → ( 𝑠 : ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↔ 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ) )
9	7	raleqdv	⊢ ( 𝑘 = 𝐾 → ( ∀ 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ↔ ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ) )
10	7 9	raleqbidv	⊢ ( 𝑘 = 𝐾 → ( ∀ 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ↔ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ) )
11		fveq2	⊢ ( 𝑘 = 𝐾 → ( trL ‘ 𝑘 ) = ( trL ‘ 𝐾 ) )
12	11	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) )
13	12	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) = ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) )
14		fveq2	⊢ ( 𝑘 = 𝐾 → ( le ‘ 𝑘 ) = ( le ‘ 𝐾 ) )
15	14 1	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( le ‘ 𝑘 ) = ≤ )
16	12	fveq1d	⊢ ( 𝑘 = 𝐾 → ( ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑓 ) = ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) )
17	13 15 16	breq123d	⊢ ( 𝑘 = 𝐾 → ( ( ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ( le ‘ 𝑘 ) ( ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑓 ) ↔ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) )
18	7 17	raleqbidv	⊢ ( 𝑘 = 𝐾 → ( ∀ 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ( le ‘ 𝑘 ) ( ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑓 ) ↔ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) )
19	8 10 18	3anbi123d	⊢ ( 𝑘 = 𝐾 → ( ( 𝑠 : ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ( le ‘ 𝑘 ) ( ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑓 ) ) ↔ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) ) )
20	19	abbidv	⊢ ( 𝑘 = 𝐾 → { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ( le ‘ 𝑘 ) ( ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } = { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } )
21	5 20	mpteq12dv	⊢ ( 𝑘 = 𝐾 → ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ( le ‘ 𝑘 ) ( ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } ) )
22		df-tendo	⊢ TEndo = ( 𝑘 ∈ V ↦ ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ( le ‘ 𝑘 ) ( ( ( trL ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } ) )
23	21 22 2	mptfvmpt	⊢ ( 𝐾 ∈ V → ( TEndo ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } ) )
24	3 23	syl	⊢ ( 𝐾 ∈ 𝑉 → ( TEndo ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑠 ∣ ( 𝑠 : ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ⟶ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∀ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑠 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑠 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( 𝑠 ‘ 𝑓 ) ) ≤ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ) } ) )