Theorem dihfval 42554

Description: 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)
dihval.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihval.d	⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊)
dihval.c	⊢ 𝐶 = ((DIsoC‘𝐾)‘𝑊)
dihval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihval.s	⊢ 𝑆 = (LSubSp‘𝑈)
dihval.p	⊢ ⊕ = (LSSum‘𝑈)

Assertion

Ref	Expression
dihfval	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑊, (𝐷‘𝑥), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))))))))

Distinct variable groups:   𝐴,𝑞   𝑢,𝑞,𝑥,𝐾   𝑥,𝐵   𝑢,𝑆   𝑊,𝑞,𝑢,𝑥

Allowed substitution hints:   𝐴(𝑥, 𝑢)   𝐵(𝑢, 𝑞)   𝐶(𝑥, 𝑢, 𝑞)   𝐷(𝑥, 𝑢, 𝑞)   ⊕ (𝑥, 𝑢, 𝑞)   𝑆(𝑥, 𝑞)   𝑈(𝑥, 𝑢, 𝑞)   𝐻(𝑥, 𝑢, 𝑞)   𝐼(𝑥, 𝑢, 𝑞)   ∨ (𝑥, 𝑢, 𝑞)   ≤ (𝑥, 𝑢, 𝑞)   ∧ (𝑥, 𝑢, 𝑞)   𝑉(𝑥, 𝑢, 𝑞)

Proof of Theorem dihfval

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dihval.i	. . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
2		dihval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
3		dihval.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		dihval.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		dihval.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
6		dihval.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
7		dihval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
8	2, 3, 4, 5, 6, 7	dihffval 42553	. . . 4 ⊢ (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)))))))))
9	8	fveq1d 6918	. . 3 ⊢ (𝐾 ∈ 𝑉 → ((DIsoH‘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))))))))‘𝑊))
10	1, 9	eqtrid 2810	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐼 = ((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))))))))‘𝑊))
11		breq2 5112	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑊))
12		fveq2 6916	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((DIsoB‘𝐾)‘𝑤) = ((DIsoB‘𝐾)‘𝑊))
13		dihval.d	. . . . . . 7 ⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊)
14	12, 13	eqtr4di 2816	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((DIsoB‘𝐾)‘𝑤) = 𝐷)
15	14	fveq1d 6918	. . . . 5 ⊢ (𝑤 = 𝑊 → (((DIsoB‘𝐾)‘𝑤)‘𝑥) = (𝐷‘𝑥))
16		fveq2 6916	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
17		dihval.u	. . . . . . . . 9 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
18	16, 17	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
19	18	fveq2d 6920	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = (LSubSp‘𝑈))
20		dihval.s	. . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑈)
21	19, 20	eqtr4di 2816	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = 𝑆)
22		breq2 5112	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑞 ≤ 𝑤 ↔ 𝑞 ≤ 𝑊))
23	22	notbid 321	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (¬ 𝑞 ≤ 𝑤 ↔ ¬ 𝑞 ≤ 𝑊))
24		oveq2 7459	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (𝑥 ∧ 𝑤) = (𝑥 ∧ 𝑊))
25	24	oveq2d 7467	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑞 ∨ (𝑥 ∧ 𝑤)) = (𝑞 ∨ (𝑥 ∧ 𝑊)))
26	25	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥 ↔ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥))
27	23, 26	anbi12d 644	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) ↔ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥)))
28	18	fveq2d 6920	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (LSSum‘((DVecH‘𝐾)‘𝑤)) = (LSSum‘𝑈))
29		dihval.p	. . . . . . . . . . 11 ⊢ ⊕ = (LSSum‘𝑈)
30	28, 29	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (LSSum‘((DVecH‘𝐾)‘𝑤)) = ⊕ )
31		fveq2 6916	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ((DIsoC‘𝐾)‘𝑤) = ((DIsoC‘𝐾)‘𝑊))
32		dihval.c	. . . . . . . . . . . 12 ⊢ 𝐶 = ((DIsoC‘𝐾)‘𝑊)
33	31, 32	eqtr4di 2816	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ((DIsoC‘𝐾)‘𝑤) = 𝐶)
34	33	fveq1d 6918	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (((DIsoC‘𝐾)‘𝑤)‘𝑞) = (𝐶‘𝑞))
35	14, 24	fveq12d 6923	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)) = (𝐷‘(𝑥 ∧ 𝑊)))
36	30, 34, 35	oveq123d 7472	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))) = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))))
37	36	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))) ↔ 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊)))))
38	27, 37	imbi12d 347	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))) ↔ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))))))
39	38	ralbidv 3189	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))) ↔ ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))))))
40	21, 39	riotaeqbidv 7411	. . . . 5 ⊢ (𝑤 = 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))))) = (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))))))
41	11, 15, 40	ifbieq12d 4514	. . . 4 ⊢ (𝑤 = 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)))))) = if(𝑥 ≤ 𝑊, (𝐷‘𝑥), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊)))))))
42	41	mpteq2dv 5205	. . 3 ⊢ (𝑤 = 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))))))) = (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑊, (𝐷‘𝑥), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))))))))
43		eqid 2763	. . 3 ⊢ (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)))))))) = (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))))))))
44	42, 43, 2	mptfvmpt 7265	. 2 ⊢ (𝑊 ∈ 𝐻 → ((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))))))))‘𝑊) = (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑊, (𝐷‘𝑥), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))))))))
45	10, 44	sylan9eq 2818	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑊, (𝐷‘𝑥), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))))))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ifcif 4485   class class class wbr 5108   ↦ cmpt 5192  ‘cfv 6566  ℩crio 7407  (class class class)co 7451  Basecbs 17431  lecple 17479  joincjn 18535  meetcmee 18536  LSSumclsm 19899  LSubSpclss 21257  Atomscatm 40586  LHypclh 41307  DVecHcdvh 42401  DIsoBcdib 42461  DIsoCcdic 42495  DIsoHcdih 42551

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-dih 42552

This theorem is used by:  dihval  42555  dihf11lem  42589