Theorem dicfval 40886

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 40885	. . . 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 6894	. . 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 5149	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑟 ≤ 𝑤 ↔ 𝑟 ≤ 𝑊))
9	8	notbid 317	. . . . 5 ⊢ (𝑤 = 𝑊 → (¬ 𝑟 ≤ 𝑤 ↔ ¬ 𝑟 ≤ 𝑊))
10	9	rabbidv 3428	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤} = {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊})
11		fveq2 6892	. . . . . . . . . 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 6892	. . . . . . . . . . 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 6900	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞 ↔ (𝑔‘𝑃) = 𝑞))
18	13, 17	riotaeqbidv 7374	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞) = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞))
19	18	fveq2d 6896	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)))
20	19	eqeq2d 2737	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ↔ 𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞))))
21		fveq2 6892	. . . . . . . 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 2812	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤) ↔ 𝑠 ∈ 𝐸))
25	20, 24	anbi12d 630	. . . . 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 5211	. . . 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 5236	. . 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 2726	. . 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 6906	. . . 4 ⊢ 𝐴 ∈ V
30	29	mptrabex 7233	. . 3 ⊢ (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)}) ∈ V
31	27, 28, 30	fvmpt 7000	. 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 394   = wceq 1534   ∈ wcel 2099  {crab 3420   class class class wbr 5145  {copab 5207   ↦ cmpt 5228  ‘cfv 6545  ℩crio 7370  lecple 17267  occoc 17268  Atomscatm 38973  LHypclh 39695  LTrncltrn 39812  TEndoctendo 40463  DIsoCcdic 40883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3466  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4325  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-iun 4997  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fo 6551  df-f1o 6552  df-fv 6553  df-riota 7371  df-dic 40884

This theorem is referenced by:  dicval  40887  dicfnN  40894