Theorem dihfval 41816

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 41815	. . . 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 6864	. . 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 2808	. 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 5101	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑊))
12		fveq2 6862	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((DIsoB‘𝐾)‘𝑤) = ((DIsoB‘𝐾)‘𝑊))
13		dihval.d	. . . . . . 7 ⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊)
14	12, 13	eqtr4di 2814	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((DIsoB‘𝐾)‘𝑤) = 𝐷)
15	14	fveq1d 6864	. . . . 5 ⊢ (𝑤 = 𝑊 → (((DIsoB‘𝐾)‘𝑤)‘𝑥) = (𝐷‘𝑥))
16		fveq2 6862	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
17		dihval.u	. . . . . . . . 9 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
18	16, 17	eqtr4di 2814	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
19	18	fveq2d 6866	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = (LSubSp‘𝑈))
20		dihval.s	. . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑈)
21	19, 20	eqtr4di 2814	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = 𝑆)
22		breq2 5101	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑞 ≤ 𝑤 ↔ 𝑞 ≤ 𝑊))
23	22	notbid 320	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (¬ 𝑞 ≤ 𝑤 ↔ ¬ 𝑞 ≤ 𝑊))
24		oveq2 7399	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (𝑥 ∧ 𝑤) = (𝑥 ∧ 𝑊))
25	24	oveq2d 7407	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑞 ∨ (𝑥 ∧ 𝑤)) = (𝑞 ∨ (𝑥 ∧ 𝑊)))
26	25	eqeq1d 2763	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥 ↔ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥))
27	23, 26	anbi12d 641	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) ↔ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥)))
28	18	fveq2d 6866	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (LSSum‘((DVecH‘𝐾)‘𝑤)) = (LSSum‘𝑈))
29		dihval.p	. . . . . . . . . . 11 ⊢ ⊕ = (LSSum‘𝑈)
30	28, 29	eqtr4di 2814	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (LSSum‘((DVecH‘𝐾)‘𝑤)) = ⊕ )
31		fveq2 6862	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ((DIsoC‘𝐾)‘𝑤) = ((DIsoC‘𝐾)‘𝑊))
32		dihval.c	. . . . . . . . . . . 12 ⊢ 𝐶 = ((DIsoC‘𝐾)‘𝑊)
33	31, 32	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ((DIsoC‘𝐾)‘𝑤) = 𝐶)
34	33	fveq1d 6864	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (((DIsoC‘𝐾)‘𝑤)‘𝑞) = (𝐶‘𝑞))
35	14, 24	fveq12d 6869	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)) = (𝐷‘(𝑥 ∧ 𝑊)))
36	30, 34, 35	oveq123d 7412	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))) = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))))
37	36	eqeq2d 2772	. . . . . . . 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 346	. . . . . . 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 3184	. . . . . 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 7351	. . . . 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 4506	. . . 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 5191	. . 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 2761	. . 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 7207	. 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 2816	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))))))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ifcif 4477   class class class wbr 5097   ↦ cmpt 5178  ‘cfv 6516  ℩crio 7347  (class class class)co 7391  Basecbs 17236  lecple 17284  joincjn 18334  meetcmee 18335  LSSumclsm 19665  LSubSpclss 20986  Atomscatm 39848  LHypclh 40569  DVecHcdvh 41663  DIsoBcdib 41723  DIsoCcdic 41757  DIsoHcdih 41813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-dih 41814

This theorem is referenced by:  dihval  41817  dihf11lem  41851