Theorem diaffval 38781

Description: The partial isomorphism A for a lattice 𝐾. (Contributed by NM, 15-Oct-2013.)

Hypotheses

Ref	Expression
diaval.b	⊢ 𝐵 = (Base‘𝐾)
diaval.l	⊢ ≤ = (le‘𝐾)
diaval.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
diaffval	⊢ (𝐾 ∈ 𝑉 → (DIsoA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥})))

Distinct variable groups:   𝑥,𝑤,𝑦, ≤   𝑤,𝐵,𝑥,𝑦   𝑤,𝐻   𝑤,𝑓,𝑥,𝑦,𝐾

Allowed substitution hints:   𝐵(𝑓)   𝐻(𝑥,𝑦,𝑓)   ≤ (𝑓)   𝑉(𝑥,𝑦,𝑤,𝑓)

Proof of Theorem diaffval

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3426	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6717	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		diaval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2796	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6717	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
6		diaval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
7	5, 6	eqtr4di 2796	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
8		fveq2 6717	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9		diaval.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
10	8, 9	eqtr4di 2796	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
11	10	breqd 5064	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑤 ↔ 𝑦 ≤ 𝑤))
12	7, 11	rabeqbidv 3396	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑦 ∈ (Base‘𝑘) ∣ 𝑦(le‘𝑘)𝑤} = {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤})
13		fveq2 6717	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
14	13	fveq1d 6719	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
15		fveq2 6717	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (trL‘𝑘) = (trL‘𝐾))
16	15	fveq1d 6719	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((trL‘𝑘)‘𝑤) = ((trL‘𝐾)‘𝑤))
17	16	fveq1d 6719	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (((trL‘𝑘)‘𝑤)‘𝑓) = (((trL‘𝐾)‘𝑤)‘𝑓))
18		eqidd 2738	. . . . . . 7 ⊢ (𝑘 = 𝐾 → 𝑥 = 𝑥)
19	17, 10, 18	breq123d 5067	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥 ↔ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥))
20	14, 19	rabeqbidv 3396	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ∣ (((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥} = {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥})
21	12, 20	mpteq12dv 5140	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ {𝑦 ∈ (Base‘𝑘) ∣ 𝑦(le‘𝑘)𝑤} ↦ {𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ∣ (((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥}) = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥}))
22	4, 21	mpteq12dv 5140	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ {𝑦 ∈ (Base‘𝑘) ∣ 𝑦(le‘𝑘)𝑤} ↦ {𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ∣ (((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥})) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥})))
23		df-disoa 38780	. . 3 ⊢ DIsoA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ {𝑦 ∈ (Base‘𝑘) ∣ 𝑦(le‘𝑘)𝑤} ↦ {𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ∣ (((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥})))
24	22, 23, 3	mptfvmpt 7044	. 2 ⊢ (𝐾 ∈ V → (DIsoA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥})))
25	1, 24	syl 17	1 ⊢ (𝐾 ∈ 𝑉 → (DIsoA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥})))

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110  {crab 3065  Vcvv 3408   class class class wbr 5053   ↦ cmpt 5135  ‘cfv 6380  Basecbs 16760  lecple 16809  LHypclh 37735  LTrncltrn 37852  trLctrl 37909  DIsoAcdia 38779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-disoa 38780

This theorem is referenced by:  diafval  38782