Theorem diaffval 41012

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 3459	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6826	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		diaval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2782	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6826	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
6		diaval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
7	5, 6	eqtr4di 2782	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
8		fveq2 6826	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9		diaval.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
10	8, 9	eqtr4di 2782	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
11	10	breqd 5106	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑤 ↔ 𝑦 ≤ 𝑤))
12	7, 11	rabeqbidv 3415	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑦 ∈ (Base‘𝑘) ∣ 𝑦(le‘𝑘)𝑤} = {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤})
13		fveq2 6826	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
14	13	fveq1d 6828	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
15		fveq2 6826	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (trL‘𝑘) = (trL‘𝐾))
16	15	fveq1d 6828	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((trL‘𝑘)‘𝑤) = ((trL‘𝐾)‘𝑤))
17	16	fveq1d 6828	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (((trL‘𝑘)‘𝑤)‘𝑓) = (((trL‘𝐾)‘𝑤)‘𝑓))
18		eqidd 2730	. . . . . . 7 ⊢ (𝑘 = 𝐾 → 𝑥 = 𝑥)
19	17, 10, 18	breq123d 5109	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥 ↔ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥))
20	14, 19	rabeqbidv 3415	. . . . 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 5182	. . . 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 5182	. . 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 41011	. . 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 7168	. 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 1540   ∈ wcel 2109  {crab 3396  Vcvv 3438   class class class wbr 5095   ↦ cmpt 5176  ‘cfv 6486  Basecbs 17138  lecple 17186  LHypclh 39966  LTrncltrn 40083  trLctrl 40140  DIsoAcdia 41010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-disoa 41011

This theorem is referenced by:  diafval  41013