Theorem dihffval 41402

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 3458	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6831	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		dihval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2786	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6831	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
6		dihval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
7	5, 6	eqtr4di 2786	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
8		fveq2 6831	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9		dihval.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
10	8, 9	eqtr4di 2786	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
11	10	breqd 5106	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤 ↔ 𝑥 ≤ 𝑤))
12		fveq2 6831	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (DIsoB‘𝑘) = (DIsoB‘𝐾))
13	12	fveq1d 6833	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((DIsoB‘𝑘)‘𝑤) = ((DIsoB‘𝐾)‘𝑤))
14	13	fveq1d 6833	. . . . . 6 ⊢ (𝑘 = 𝐾 → (((DIsoB‘𝑘)‘𝑤)‘𝑥) = (((DIsoB‘𝐾)‘𝑤)‘𝑥))
15		fveq2 6831	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
16	15	fveq1d 6833	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
17	16	fveq2d 6835	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LSubSp‘((DVecH‘𝑘)‘𝑤)) = (LSubSp‘((DVecH‘𝐾)‘𝑤)))
18		fveq2 6831	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
19		dihval.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
20	18, 19	eqtr4di 2786	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
21	10	breqd 5106	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑞(le‘𝑘)𝑤 ↔ 𝑞 ≤ 𝑤))
22	21	notbid 318	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (¬ 𝑞(le‘𝑘)𝑤 ↔ ¬ 𝑞 ≤ 𝑤))
23		fveq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
24		dihval.j	. . . . . . . . . . . . 13 ⊢ ∨ = (join‘𝐾)
25	23, 24	eqtr4di 2786	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
26		eqidd 2734	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → 𝑞 = 𝑞)
27		fveq2 6831	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
28		dihval.m	. . . . . . . . . . . . . 14 ⊢ ∧ = (meet‘𝐾)
29	27, 28	eqtr4di 2786	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = ∧ )
30	29	oveqd 7372	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (𝑥(meet‘𝑘)𝑤) = (𝑥 ∧ 𝑤))
31	25, 26, 30	oveq123d 7376	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = (𝑞 ∨ (𝑥 ∧ 𝑤)))
32	31	eqeq1d 2735	. . . . . . . . . 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 6835	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (LSSum‘((DVecH‘𝑘)‘𝑤)) = (LSSum‘((DVecH‘𝐾)‘𝑤)))
35		fveq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (DIsoC‘𝑘) = (DIsoC‘𝐾))
36	35	fveq1d 6833	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → ((DIsoC‘𝑘)‘𝑤) = ((DIsoC‘𝐾)‘𝑤))
37	36	fveq1d 6833	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (((DIsoC‘𝑘)‘𝑤)‘𝑞) = (((DIsoC‘𝐾)‘𝑤)‘𝑞))
38	13, 30	fveq12d 6838	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)) = (((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))
39	34, 37, 38	oveq123d 7376	. . . . . . . . . 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 2744	. . . . . . . . 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 3313	. . . . . . 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 7315	. . . . . 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 4505	. . . . 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 5182	. . . 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 5182	. . 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 41401	. . 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 7171	. 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 1541   ∈ wcel 2113  ∀wral 3048  Vcvv 3437  ifcif 4476   class class class wbr 5095   ↦ cmpt 5176  ‘cfv 6489  ℩crio 7311  (class class class)co 7355  Basecbs 17127  lecple 17175  joincjn 18225  meetcmee 18226  LSSumclsm 19554  LSubSpclss 20873  Atomscatm 39435  LHypclh 40156  DVecHcdvh 41250  DIsoBcdib 41310  DIsoCcdic 41344  DIsoHcdih 41400

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-dih 41401

This theorem is referenced by:  dihfval  41403