Theorem dicffval 41157

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

Hypotheses

Ref	Expression
dicval.l	⊢ ≤ = (le‘𝐾)
dicval.a	⊢ 𝐴 = (Atoms‘𝐾)
dicval.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
dicffval	⊢ (𝐾 ∈ 𝑉 → (DIsoC‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))})))

Distinct variable groups:   𝐴,𝑟   𝑤,𝐻   𝑓,𝑔,𝑞,𝑟,𝑠,𝑤,𝐾

Allowed substitution hints:   𝐴(𝑤,𝑓,𝑔,𝑠,𝑞)   𝐻(𝑓,𝑔,𝑠,𝑟,𝑞)   ≤ (𝑤,𝑓,𝑔,𝑠,𝑟,𝑞)   𝑉(𝑤,𝑓,𝑔,𝑠,𝑟,𝑞)

Proof of Theorem dicffval

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3499	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6907	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		dicval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2793	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6907	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
6		dicval.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
7	5, 6	eqtr4di 2793	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
8		fveq2 6907	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9		dicval.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
10	8, 9	eqtr4di 2793	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
11	10	breqd 5159	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑟(le‘𝑘)𝑤 ↔ 𝑟 ≤ 𝑤))
12	11	notbid 318	. . . . . 6 ⊢ (𝑘 = 𝐾 → (¬ 𝑟(le‘𝑘)𝑤 ↔ ¬ 𝑟 ≤ 𝑤))
13	7, 12	rabeqbidv 3452	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑟 ∈ (Atoms‘𝑘) ∣ ¬ 𝑟(le‘𝑘)𝑤} = {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤})
14		fveq2 6907	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
15	14	fveq1d 6909	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
16		fveq2 6907	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17	16	fveq1d 6909	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑤) = ((oc‘𝐾)‘𝑤))
18	17	fveqeq2d 6915	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞 ↔ (𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞))
19	15, 18	riotaeqbidv 7391	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (℩𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞) = (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞))
20	19	fveq2d 6911	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑠‘(℩𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)))
21	20	eqeq2d 2746	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ↔ 𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞))))
22		fveq2 6907	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (TEndo‘𝑘) = (TEndo‘𝐾))
23	22	fveq1d 6909	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((TEndo‘𝑘)‘𝑤) = ((TEndo‘𝐾)‘𝑤))
24	23	eleq2d 2825	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑠 ∈ ((TEndo‘𝑘)‘𝑤) ↔ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤)))
25	21, 24	anbi12d 632	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝑘)‘𝑤)) ↔ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))))
26	25	opabbidv 5214	. . . . 5 ⊢ (𝑘 = 𝐾 → {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝑘)‘𝑤))} = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))})
27	13, 26	mpteq12dv 5239	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑞 ∈ {𝑟 ∈ (Atoms‘𝑘) ∣ ¬ 𝑟(le‘𝑘)𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝑘)‘𝑤))}) = (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))}))
28	4, 27	mpteq12dv 5239	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑞 ∈ {𝑟 ∈ (Atoms‘𝑘) ∣ ¬ 𝑟(le‘𝑘)𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝑘)‘𝑤))})) = (𝑤 ∈ 𝐻 ↦ (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))})))
29		df-dic 41156	. . 3 ⊢ DIsoC = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑞 ∈ {𝑟 ∈ (Atoms‘𝑘) ∣ ¬ 𝑟(le‘𝑘)𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝑘)‘𝑤))})))
30	28, 29, 3	mptfvmpt 7248	. 2 ⊢ (𝐾 ∈ V → (DIsoC‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))})))
31	1, 30	syl 17	1 ⊢ (𝐾 ∈ 𝑉 → (DIsoC‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {crab 3433  Vcvv 3478   class class class wbr 5148  {copab 5210   ↦ cmpt 5231  ‘cfv 6563  ℩crio 7387  lecple 17305  occoc 17306  Atomscatm 39245  LHypclh 39967  LTrncltrn 40084  TEndoctendo 40735  DIsoCcdic 41155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-dic 41156

This theorem is referenced by:  dicfval  41158