Theorem dicfval 41193

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 41192	. . . 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 6819	. . 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 2777	. 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 5093	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑟 ≤ 𝑤 ↔ 𝑟 ≤ 𝑊))
9	8	notbid 318	. . . . 5 ⊢ (𝑤 = 𝑊 → (¬ 𝑟 ≤ 𝑤 ↔ ¬ 𝑟 ≤ 𝑊))
10	9	rabbidv 3400	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤} = {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊})
11		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
12		dicval.t	. . . . . . . . . 10 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
13	11, 12	eqtr4di 2783	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
14		fveq2 6817	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ((oc‘𝐾)‘𝑤) = ((oc‘𝐾)‘𝑊))
15		dicval.p	. . . . . . . . . . 11 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
16	14, 15	eqtr4di 2783	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((oc‘𝐾)‘𝑤) = 𝑃)
17	16	fveqeq2d 6825	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞 ↔ (𝑔‘𝑃) = 𝑞))
18	13, 17	riotaeqbidv 7301	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞) = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞))
19	18	fveq2d 6821	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)))
20	19	eqeq2d 2741	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ↔ 𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞))))
21		fveq2 6817	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = ((TEndo‘𝐾)‘𝑊))
22		dicval.e	. . . . . . . 8 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
23	21, 22	eqtr4di 2783	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = 𝐸)
24	23	eleq2d 2815	. . . . . 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 5155	. . . 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 5176	. . 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 2730	. . 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 6831	. . . 4 ⊢ 𝐴 ∈ V
30	29	mptrabex 7154	. . 3 ⊢ (𝑞 ∈ {𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞)) ∧ 𝑠 ∈ 𝐸)}) ∈ V
31	27, 28, 30	fvmpt 6924	. 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 2785	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 2110  {crab 3393   class class class wbr 5089  {copab 5151   ↦ cmpt 5170  ‘cfv 6477  ℩crio 7297  lecple 17160  occoc 17161  Atomscatm 39281  LHypclh 40002  LTrncltrn 40119  TEndoctendo 40770  DIsoCcdic 41190

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-dic 41191

This theorem is referenced by:  dicval  41194  dicfnN  41201