Theorem dicfval 42435

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 42434	. . . 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 6913	. . 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 2810	. 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 5110	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑟 ≤ 𝑤 ↔ 𝑟 ≤ 𝑊))
9	8	notbid 321	. . . . 5 ⊢ (𝑤 = 𝑊 → (¬ 𝑟 ≤ 𝑤 ↔ ¬ 𝑟 ≤ 𝑊))
10	9	rabbidv 3423	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤} = {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊})
11		fveq2 6911	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
12		dicval.t	. . . . . . . . . 10 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
13	11, 12	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
14		fveq2 6911	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ((oc‘𝐾)‘𝑤) = ((oc‘𝐾)‘𝑊))
15		dicval.p	. . . . . . . . . . 11 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
16	14, 15	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((oc‘𝐾)‘𝑤) = 𝑃)
17	16	fveqeq2d 6919	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞 ↔ (𝑔‘𝑃) = 𝑞))
18	13, 17	riotaeqbidv 7406	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞) = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞))
19	18	fveq2d 6915	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)))
20	19	eqeq2d 2774	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ↔ 𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞))))
21		fveq2 6911	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = ((TEndo‘𝐾)‘𝑊))
22		dicval.e	. . . . . . . 8 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
23	21, 22	eqtr4di 2816	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = 𝐸)
24	23	eleq2d 2850	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤) ↔ 𝑠 ∈ 𝐸))
25	20, 24	anbi12d 644	. . . . 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 5175	. . . 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 5196	. . 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 2763	. . 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 6925	. . . 4 ⊢ 𝐴 ∈ V
30	29	mptrabex 7257	. . 3 ⊢ (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)}) ∈ V
31	27, 28, 30	fvmpt 7019	. 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 2818	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416   class class class wbr 5106  {copab 5171   ↦ cmpt 5190  ‘cfv 6561  ℩crio 7402  lecple 17467  occoc 17468  Atomscatm 40523  LHypclh 41244  LTrncltrn 41361  TEndoctendo 42012  DIsoCcdic 42432

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pr 5396

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-dic 42433

This theorem is used by:  dicval  42436  dicfnN  42443