Theorem dihffval 41187

Description: The isomorphism H for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)

Hypotheses

Ref	Expression
dihval.b	⊢ 𝐵 = (Base‘𝐾)
dihval.l	⊢ ≤ = (le‘𝐾)
dihval.j	⊢ ∨ = (join‘𝐾)
dihval.m	⊢ ∧ = (meet‘𝐾)
dihval.a	⊢ 𝐴 = (Atoms‘𝐾)
dihval.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
dihffval	⊢ (𝐾 ∈ 𝑉 → (DIsoH‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))))))))

Distinct variable groups:   𝐴,𝑞   𝑤,𝐻   𝑢,𝑞,𝑤,𝑥,𝐾

Allowed substitution hints:   𝐴(𝑥,𝑤,𝑢)   𝐵(𝑥,𝑤,𝑢,𝑞)   𝐻(𝑥,𝑢,𝑞)   ∨ (𝑥,𝑤,𝑢,𝑞)   ≤ (𝑥,𝑤,𝑢,𝑞)   ∧ (𝑥,𝑤,𝑢,𝑞)   𝑉(𝑥,𝑤,𝑢,𝑞)

Proof of Theorem dihffval

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3509	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6920	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		dihval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2798	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6920	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
6		dihval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
7	5, 6	eqtr4di 2798	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
8		fveq2 6920	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9		dihval.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
10	8, 9	eqtr4di 2798	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
11	10	breqd 5177	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤 ↔ 𝑥 ≤ 𝑤))
12		fveq2 6920	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (DIsoB‘𝑘) = (DIsoB‘𝐾))
13	12	fveq1d 6922	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((DIsoB‘𝑘)‘𝑤) = ((DIsoB‘𝐾)‘𝑤))
14	13	fveq1d 6922	. . . . . 6 ⊢ (𝑘 = 𝐾 → (((DIsoB‘𝑘)‘𝑤)‘𝑥) = (((DIsoB‘𝐾)‘𝑤)‘𝑥))
15		fveq2 6920	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
16	15	fveq1d 6922	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
17	16	fveq2d 6924	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LSubSp‘((DVecH‘𝑘)‘𝑤)) = (LSubSp‘((DVecH‘𝐾)‘𝑤)))
18		fveq2 6920	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
19		dihval.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
20	18, 19	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
21	10	breqd 5177	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑞(le‘𝑘)𝑤 ↔ 𝑞 ≤ 𝑤))
22	21	notbid 318	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (¬ 𝑞(le‘𝑘)𝑤 ↔ ¬ 𝑞 ≤ 𝑤))
23		fveq2 6920	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
24		dihval.j	. . . . . . . . . . . . 13 ⊢ ∨ = (join‘𝐾)
25	23, 24	eqtr4di 2798	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
26		eqidd 2741	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → 𝑞 = 𝑞)
27		fveq2 6920	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
28		dihval.m	. . . . . . . . . . . . . 14 ⊢ ∧ = (meet‘𝐾)
29	27, 28	eqtr4di 2798	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = ∧ )
30	29	oveqd 7465	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (𝑥(meet‘𝑘)𝑤) = (𝑥 ∧ 𝑤))
31	25, 26, 30	oveq123d 7469	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = (𝑞 ∨ (𝑥 ∧ 𝑤)))
32	31	eqeq1d 2742	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥 ↔ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥))
33	22, 32	anbi12d 631	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) ↔ (¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥)))
34	16	fveq2d 6924	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (LSSum‘((DVecH‘𝑘)‘𝑤)) = (LSSum‘((DVecH‘𝐾)‘𝑤)))
35		fveq2 6920	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (DIsoC‘𝑘) = (DIsoC‘𝐾))
36	35	fveq1d 6922	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → ((DIsoC‘𝑘)‘𝑤) = ((DIsoC‘𝐾)‘𝑤))
37	36	fveq1d 6922	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (((DIsoC‘𝑘)‘𝑤)‘𝑞) = (((DIsoC‘𝐾)‘𝑤)‘𝑞))
38	13, 30	fveq12d 6927	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)) = (((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))
39	34, 37, 38	oveq123d 7469	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))) = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))))
40	39	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))) ↔ 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))))
41	33, 40	imbi12d 344	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))) ↔ ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))))))
42	20, 41	raleqbidv 3354	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))) ↔ ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))))))
43	17, 42	riotaeqbidv 7407	. . . . . 6 ⊢ (𝑘 = 𝐾 → (℩𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))))) = (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))))))
44	11, 14, 43	ifbieq12d 4576	. . . . 5 ⊢ (𝑘 = 𝐾 → if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))) = if(𝑥 ≤ 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))))))
45	7, 44	mpteq12dv 5257	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))))))) = (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))))))))
46	4, 45	mpteq12dv 5257	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))))) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))))))))
47		df-dih 41186	. . 3 ⊢ DIsoH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))))))
48	46, 47, 3	mptfvmpt 7265	. 2 ⊢ (𝐾 ∈ V → (DIsoH‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))))))))
49	1, 48	syl 17	1 ⊢ (𝐾 ∈ 𝑉 → (DIsoH‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))))))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488  ifcif 4548   class class class wbr 5166   ↦ cmpt 5249  ‘cfv 6573  ℩crio 7403  (class class class)co 7448  Basecbs 17258  lecple 17318  joincjn 18381  meetcmee 18382  LSSumclsm 19676  LSubSpclss 20952  Atomscatm 39219  LHypclh 39941  DVecHcdvh 41035  DIsoBcdib 41095  DIsoCcdic 41129  DIsoHcdih 41185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-dih 41186

This theorem is referenced by:  dihfval  41188