Theorem tendofset 38395

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 3416	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6674	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		tendoset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2791	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6674	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
6	5	fveq1d 6676	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
7	6, 6	feq23d 6499	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑠:((LTrn‘𝑘)‘𝑤)⟶((LTrn‘𝑘)‘𝑤) ↔ 𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤)))
8	6	raleqdv 3316	. . . . . . 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 3304	. . . . . 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 6674	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (trL‘𝑘) = (trL‘𝐾))
11	10	fveq1d 6676	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((trL‘𝑘)‘𝑤) = ((trL‘𝐾)‘𝑤))
12	11	fveq1d 6676	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((trL‘𝑘)‘𝑤)‘(𝑠‘𝑓)) = (((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)))
13		fveq2 6674	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
14		tendoset.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
15	13, 14	eqtr4di 2791	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
16	11	fveq1d 6676	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((trL‘𝑘)‘𝑤)‘𝑓) = (((trL‘𝐾)‘𝑤)‘𝑓))
17	12, 15, 16	breq123d 5044	. . . . . . 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 3304	. . . . . 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 1437	. . . . 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 2802	. . . 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 5115	. . 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 38392	. . 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 7001	. 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 17	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 1088   = wceq 1542   ∈ wcel 2114  {cab 2716  ∀wral 3053  Vcvv 3398   class class class wbr 5030   ↦ cmpt 5110   ∘ ccom 5529  ⟶wf 6335  ‘cfv 6339  lecple 16675  LHypclh 37621  LTrncltrn 37738  trLctrl 37795  TEndoctendo 38389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-tendo 38392

This theorem is referenced by:  tendoset  38396