Theorem dicfval 41213

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)
dicval.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
dicval.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dicval.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dicval.i	⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)

Assertion

Ref	Expression
dicfval	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)}))

Distinct variable groups:   𝐴,𝑟   𝑓,𝑔,𝑞,𝑟,𝑠,𝐾   ≤ ,𝑞   𝐴,𝑞   𝑇,𝑔   𝑓,𝑊,𝑔,𝑞,𝑟,𝑠

Allowed substitution hints:   𝐴(𝑓,𝑔,𝑠)   𝑃(𝑓,𝑔,𝑠,𝑟,𝑞)   𝑇(𝑓,𝑠,𝑟,𝑞)   𝐸(𝑓,𝑔,𝑠,𝑟,𝑞)   𝐻(𝑓,𝑔,𝑠,𝑟,𝑞)   𝐼(𝑓,𝑔,𝑠,𝑟,𝑞)   ≤ (𝑓,𝑔,𝑠,𝑟)   𝑉(𝑓,𝑔,𝑠,𝑟,𝑞)

Proof of Theorem dicfval

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dicval.i	. . 3 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
2		dicval.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		dicval.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
4		dicval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5	2, 3, 4	dicffval 41212	. . . 4 ⊢ (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))})))
6	5	fveq1d 6824	. . 3 ⊢ (𝐾 ∈ 𝑉 → ((DIsoC‘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))}))‘𝑊))
7	1, 6	eqtrid 2778	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐼 = ((𝑤 ∈ 𝐻 ↦ (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))}))‘𝑊))
8		breq2 5095	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑟 ≤ 𝑤 ↔ 𝑟 ≤ 𝑊))
9	8	notbid 318	. . . . 5 ⊢ (𝑤 = 𝑊 → (¬ 𝑟 ≤ 𝑤 ↔ ¬ 𝑟 ≤ 𝑊))
10	9	rabbidv 3402	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤} = {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊})
11		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
12		dicval.t	. . . . . . . . . 10 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
13	11, 12	eqtr4di 2784	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
14		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ((oc‘𝐾)‘𝑤) = ((oc‘𝐾)‘𝑊))
15		dicval.p	. . . . . . . . . . 11 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
16	14, 15	eqtr4di 2784	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((oc‘𝐾)‘𝑤) = 𝑃)
17	16	fveqeq2d 6830	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞 ↔ (𝑔‘𝑃) = 𝑞))
18	13, 17	riotaeqbidv 7306	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞) = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞))
19	18	fveq2d 6826	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)))
20	19	eqeq2d 2742	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ↔ 𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞))))
21		fveq2 6822	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = ((TEndo‘𝐾)‘𝑊))
22		dicval.e	. . . . . . . 8 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
23	21, 22	eqtr4di 2784	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = 𝐸)
24	23	eleq2d 2817	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤) ↔ 𝑠 ∈ 𝐸))
25	20, 24	anbi12d 632	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤)) ↔ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)))
26	25	opabbidv 5157	. . . 4 ⊢ (𝑤 = 𝑊 → {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))} = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)})
27	10, 26	mpteq12dv 5178	. . 3 ⊢ (𝑤 = 𝑊 → (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 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 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)}))
28		eqid 2731	. . 3 ⊢ (𝑤 ∈ 𝐻 ↦ (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 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	3	fvexi 6836	. . . 4 ⊢ 𝐴 ∈ V
30	29	mptrabex 7159	. . 3 ⊢ (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)}) ∈ V
31	27, 28, 30	fvmpt 6929	. 2 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))}))‘𝑊) = (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)}))
32	7, 31	sylan9eq 2786	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395   class class class wbr 5091  {copab 5153   ↦ cmpt 5172  ‘cfv 6481  ℩crio 7302  lecple 17165  occoc 17166  Atomscatm 39301  LHypclh 40022  LTrncltrn 40139  TEndoctendo 40790  DIsoCcdic 41210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-dic 41211

This theorem is referenced by:  dicval  41214  dicfnN  41221