Theorem tendofset 41389

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

Hypotheses

Ref	Expression
tendoset.l	⊢ ≤ = (le‘𝐾)
tendoset.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
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))}))

Distinct variable groups:   𝑤,𝐻   𝑤,𝑠,𝑓,𝑔,𝐾

Allowed substitution hints:   𝐻(𝑓,𝑔,𝑠)   ≤ (𝑤,𝑓,𝑔,𝑠)   𝑉(𝑤,𝑓,𝑔,𝑠)

Proof of Theorem tendofset

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3478	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6871	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		tendoset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2818	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6871	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
6	5	fveq1d 6873	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
7	6, 6	feq23d 6690	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑠:((LTrn‘𝑘)‘𝑤)⟶((LTrn‘𝑘)‘𝑤) ↔ 𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤)))
8	6	raleqdv 3323	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (∀𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ↔ ∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔))))
9	6, 8	raleqbidv 3339	. . . . . 6 ⊢ (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))))
10		fveq2 6871	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (trL‘𝑘) = (trL‘𝐾))
11	10	fveq1d 6873	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((trL‘𝑘)‘𝑤) = ((trL‘𝐾)‘𝑤))
12	11	fveq1d 6873	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((trL‘𝑘)‘𝑤)‘(𝑠‘𝑓)) = (((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)))
13		fveq2 6871	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
14		tendoset.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
15	13, 14	eqtr4di 2818	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
16	11	fveq1d 6873	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((trL‘𝑘)‘𝑤)‘𝑓) = (((trL‘𝐾)‘𝑤)‘𝑓))
17	12, 15, 16	breq123d 5118	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((((trL‘𝑘)‘𝑤)‘(𝑠‘𝑓))(le‘𝑘)(((trL‘𝑘)‘𝑤)‘𝑓) ↔ (((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)) ≤ (((trL‘𝐾)‘𝑤)‘𝑓)))
18	6, 17	raleqbidv 3339	. . . . . 6 ⊢ (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	7, 9, 18	3anbi123d 1460	. . . . 5 ⊢ (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 2831	. . . 4 ⊢ (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	4, 20	mpteq12dv 5191	. . 3 ⊢ (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 41386	. . 3 ⊢ 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, 3	mptfvmpt 7216	. 2 ⊢ (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	1, 23	syl 18	1 ⊢ (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))}))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  {cab 2743  ∀wral 3079  Vcvv 3457   class class class wbr 5104   ↦ cmpt 5185   ∘ ccom 5655  ⟶wf 6521  ‘cfv 6525  lecple 17305  LHypclh 40615  LTrncltrn 40732  trLctrl 40789  TEndoctendo 41383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-tendo 41386

This theorem is referenced by:  tendoset  41390