Theorem dihval 40832

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 40831	. . 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 6898	. 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 5152	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
16		fveq2 6896	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐷‘𝑥) = (𝐷‘𝑋))
17		oveq1 7426	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ 𝑊) = (𝑋 ∧ 𝑊))
18	17	oveq2d 7435	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑞 ∨ (𝑥 ∧ 𝑊)) = (𝑞 ∨ (𝑋 ∧ 𝑊)))
19		id 22	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
20	18, 19	eqeq12d 2741	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥 ↔ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
21	20	anbi2d 628	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) ↔ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)))
22		fvoveq1 7442	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝐷‘(𝑥 ∧ 𝑊)) = (𝐷‘(𝑋 ∧ 𝑊)))
23	22	oveq2d 7435	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))) = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))
24	23	eqeq2d 2736	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))) ↔ 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))
25	21, 24	imbi12d 343	. . . . . 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 3167	. . . . 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 7377	. . . 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 4558	. . 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 2725	. . 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 6909	. . . 4 ⊢ (𝐷‘𝑋) ∈ V
31		riotaex 7379	. . . 4 ⊢ (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))) ∈ V
32	30, 31	ifex 4580	. . 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 7004	. 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 2785	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 394   = wceq 1533   ∈ wcel 2098  ∀wral 3050  ifcif 4530   class class class wbr 5149   ↦ cmpt 5232  ‘cfv 6549  ℩crio 7374  (class class class)co 7419  Basecbs 17183  lecple 17243  joincjn 18306  meetcmee 18307  LSSumclsm 19601  LSubSpclss 20827  Atomscatm 38862  LHypclh 39584  DVecHcdvh 40678  DIsoBcdib 40738  DIsoCcdic 40772  DIsoHcdih 40828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-dih 40829

This theorem is referenced by:  dihvalc  40833  dihvalb  40837