Theorem dihffval 38360

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 3513	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6665	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		dihval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	syl6eqr 2874	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6665	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
6		dihval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
7	5, 6	syl6eqr 2874	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
8		fveq2 6665	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9		dihval.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
10	8, 9	syl6eqr 2874	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
11	10	breqd 5070	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤 ↔ 𝑥 ≤ 𝑤))
12		fveq2 6665	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (DIsoB‘𝑘) = (DIsoB‘𝐾))
13	12	fveq1d 6667	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((DIsoB‘𝑘)‘𝑤) = ((DIsoB‘𝐾)‘𝑤))
14	13	fveq1d 6667	. . . . . 6 ⊢ (𝑘 = 𝐾 → (((DIsoB‘𝑘)‘𝑤)‘𝑥) = (((DIsoB‘𝐾)‘𝑤)‘𝑥))
15		fveq2 6665	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
16	15	fveq1d 6667	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
17	16	fveq2d 6669	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LSubSp‘((DVecH‘𝑘)‘𝑤)) = (LSubSp‘((DVecH‘𝐾)‘𝑤)))
18		fveq2 6665	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
19		dihval.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
20	18, 19	syl6eqr 2874	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
21	10	breqd 5070	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑞(le‘𝑘)𝑤 ↔ 𝑞 ≤ 𝑤))
22	21	notbid 320	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (¬ 𝑞(le‘𝑘)𝑤 ↔ ¬ 𝑞 ≤ 𝑤))
23		fveq2 6665	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
24		dihval.j	. . . . . . . . . . . . 13 ⊢ ∨ = (join‘𝐾)
25	23, 24	syl6eqr 2874	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
26		eqidd 2822	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → 𝑞 = 𝑞)
27		fveq2 6665	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
28		dihval.m	. . . . . . . . . . . . . 14 ⊢ ∧ = (meet‘𝐾)
29	27, 28	syl6eqr 2874	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = ∧ )
30	29	oveqd 7167	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (𝑥(meet‘𝑘)𝑤) = (𝑥 ∧ 𝑤))
31	25, 26, 30	oveq123d 7171	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = (𝑞 ∨ (𝑥 ∧ 𝑤)))
32	31	eqeq1d 2823	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥 ↔ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥))
33	22, 32	anbi12d 632	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) ↔ (¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥)))
34	16	fveq2d 6669	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (LSSum‘((DVecH‘𝑘)‘𝑤)) = (LSSum‘((DVecH‘𝐾)‘𝑤)))
35		fveq2 6665	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (DIsoC‘𝑘) = (DIsoC‘𝐾))
36	35	fveq1d 6667	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → ((DIsoC‘𝑘)‘𝑤) = ((DIsoC‘𝐾)‘𝑤))
37	36	fveq1d 6667	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (((DIsoC‘𝑘)‘𝑤)‘𝑞) = (((DIsoC‘𝐾)‘𝑤)‘𝑞))
38	13, 30	fveq12d 6672	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)) = (((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))
39	34, 37, 38	oveq123d 7171	. . . . . . . . . 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 2832	. . . . . . . . 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 347	. . . . . . . 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 3402	. . . . . . 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 7111	. . . . . 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 4494	. . . . 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 5144	. . . 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 5144	. . 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 38359	. . 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 6984	. 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 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3495  ifcif 4467   class class class wbr 5059   ↦ cmpt 5139  ‘cfv 6350  ℩crio 7107  (class class class)co 7150  Basecbs 16477  lecple 16566  joincjn 17548  meetcmee 17549  LSSumclsm 18753  LSubSpclss 19697  Atomscatm 36393  LHypclh 37114  DVecHcdvh 38208  DIsoBcdib 38268  DIsoCcdic 38302  DIsoHcdih 38358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-dih 38359

This theorem is referenced by:  dihfval  38361