Theorem diafval 41013

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)
diaval.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
diaval.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
diaval.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)

Assertion

Ref	Expression
diafval	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}))

Distinct variable groups:   𝑥,𝑦, ≤   𝑥,𝐵,𝑦   𝑥,𝑓,𝑦,𝐾   𝑥,𝑅   𝑇,𝑓,𝑥   𝑓,𝑊,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑓)   𝑅(𝑦,𝑓)   𝑇(𝑦)   𝐻(𝑥,𝑦,𝑓)   𝐼(𝑥,𝑦,𝑓)   ≤ (𝑓)   𝑉(𝑥,𝑦,𝑓)

Proof of Theorem diafval

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		diaval.i	. . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
2		diaval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
3		diaval.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		diaval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5	2, 3, 4	diaffval 41012	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (DIsoA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥})))
6	5	fveq1d 6828	. . 3 ⊢ (𝐾 ∈ 𝑉 → ((DIsoA‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥}))‘𝑊))
7	1, 6	eqtrid 2776	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐼 = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥}))‘𝑊))
8		breq2 5099	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑦 ≤ 𝑤 ↔ 𝑦 ≤ 𝑊))
9	8	rabbidv 3404	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} = {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊})
10		fveq2 6826	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
11		diaval.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
12	10, 11	eqtr4di 2782	. . . . 5 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
13		fveq2 6826	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = ((trL‘𝐾)‘𝑊))
14		diaval.r	. . . . . . . 8 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
15	13, 14	eqtr4di 2782	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = 𝑅)
16	15	fveq1d 6828	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((trL‘𝐾)‘𝑤)‘𝑓) = (𝑅‘𝑓))
17	16	breq1d 5105	. . . . 5 ⊢ (𝑤 = 𝑊 → ((((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥 ↔ (𝑅‘𝑓) ≤ 𝑥))
18	12, 17	rabeqbidv 3415	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥} = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})
19	9, 18	mpteq12dv 5182	. . 3 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥}) = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}))
20		eqid 2729	. . 3 ⊢ (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥})) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥}))
21	2	fvexi 6840	. . . 4 ⊢ 𝐵 ∈ V
22	21	mptrabex 7165	. . 3 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}) ∈ V
23	19, 20, 22	fvmpt 6934	. 2 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥}))‘𝑊) = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}))
24	7, 23	sylan9eq 2784	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3396   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-pw 4555  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:  diaval  41014  diafn  41016