Theorem dihval 41891

Description: Value of isomorphism H for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 3-Feb-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
dihval	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) = 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,𝐾   𝑢,𝑆   𝑊,𝑞,𝑢   𝑋,𝑞,𝑢

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

Proof of Theorem dihval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dihval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		dihval.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		dihval.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		dihval.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
5		dihval.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6		dihval.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
7		dihval.i	. . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
8		dihval.d	. . . 4 ⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊)
9		dihval.c	. . . 4 ⊢ 𝐶 = ((DIsoC‘𝐾)‘𝑊)
10		dihval.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
11		dihval.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑈)
12		dihval.p	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	dihfval 41890	. . 3 ⊢ ((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))))))))
14	13	fveq1d 6881	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐼‘𝑋) = ((𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑊, (𝐷‘𝑥), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊)))))))‘𝑋))
15		breq1 5113	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
16		fveq2 6879	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐷‘𝑥) = (𝐷‘𝑋))
17		oveq1 7415	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ 𝑊) = (𝑋 ∧ 𝑊))
18	17	oveq2d 7424	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑞 ∨ (𝑥 ∧ 𝑊)) = (𝑞 ∨ (𝑋 ∧ 𝑊)))
19		id 23	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
20	18, 19	eqeq12d 2785	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥 ↔ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
21	20	anbi2d 641	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) ↔ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)))
22		fvoveq1 7431	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝐷‘(𝑥 ∧ 𝑊)) = (𝐷‘(𝑋 ∧ 𝑊)))
23	22	oveq2d 7424	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))) = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))
24	23	eqeq2d 2780	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))) ↔ 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))
25	21, 24	imbi12d 347	. . . . . 6 ⊢ (𝑥 = 𝑋 → (((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊)))) ↔ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))))
26	25	ralbidv 3194	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊)))) ↔ ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))))
27	26	riotabidv 7367	. . . 4 ⊢ (𝑥 = 𝑋 → (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))))) = (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))))
28	15, 16, 27	ifbieq12d 4518	. . 3 ⊢ (𝑥 = 𝑋 → if(𝑥 ≤ 𝑊, (𝐷‘𝑥), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊)))))) = if(𝑋 ≤ 𝑊, (𝐷‘𝑋), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))))
29		eqid 2769	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑊, (𝐷‘𝑥), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(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)))))))
30		fvex 6892	. . . 4 ⊢ (𝐷‘𝑋) ∈ V
31		riotaex 7369	. . . 4 ⊢ (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))) ∈ V
32	30, 31	ifex 4540	. . 3 ⊢ if(𝑋 ≤ 𝑊, (𝐷‘𝑋), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))) ∈ V
33	28, 29, 32	fvmpt 6987	. 2 ⊢ (𝑋 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑊, (𝐷‘𝑥), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊)))))))‘𝑋) = if(𝑋 ≤ 𝑊, (𝐷‘𝑋), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))))
34	14, 33	sylan9eq 2824	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) = 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 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ifcif 4489   class class class wbr 5110   ↦ cmpt 5193  ‘cfv 6533  ℩crio 7364  (class class class)co 7408  Basecbs 17265  lecple 17313  joincjn 18363  meetcmee 18364  LSSumclsm 19700  LSubSpclss 21026  Atomscatm 39922  LHypclh 40643  DVecHcdvh 41737  DIsoBcdib 41797  DIsoCcdic 41831  DIsoHcdih 41887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-dih 41888

This theorem is referenced by:  dihvalc  41892  dihvalb  41896