Theorem diafval 41863

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 41862	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (DIsoA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥})))
6	5	fveq1d 6887	. . 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 2812	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐼 = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥}))‘𝑊))
8		breq2 5115	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑦 ≤ 𝑤 ↔ 𝑦 ≤ 𝑊))
9	8	rabbidv 3425	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} = {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊})
10		fveq2 6885	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
11		diaval.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
12	10, 11	eqtr4di 2818	. . . . 5 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
13		fveq2 6885	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = ((trL‘𝐾)‘𝑊))
14		diaval.r	. . . . . . . 8 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
15	13, 14	eqtr4di 2818	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = 𝑅)
16	15	fveq1d 6887	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((trL‘𝐾)‘𝑤)‘𝑓) = (𝑅‘𝑓))
17	16	breq1d 5121	. . . . 5 ⊢ (𝑤 = 𝑊 → ((((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥 ↔ (𝑅‘𝑓) ≤ 𝑥))
18	12, 17	rabeqbidv 3436	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥} = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})
19	9, 18	mpteq12dv 5200	. . 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 2765	. . 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 6899	. . . 4 ⊢ 𝐵 ∈ V
22	21	mptrabex 7230	. . 3 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}) ∈ V
23	19, 20, 22	fvmpt 6993	. 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 2820	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {crab 3418   class class class wbr 5111   ↦ cmpt 5194  ‘cfv 6540  Basecbs 17291  lecple 17339  LHypclh 40816  LTrncltrn 40933  trLctrl 40990  DIsoAcdia 41860

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-disoa 41861

This theorem is used by:  diaval  41864  diafn  41866