Theorem dihval 42383

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 42382	. . 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 6888	. 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 5105	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
16		fveq2 6886	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐷‘𝑥) = (𝐷‘𝑋))
17		oveq1 7428	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ 𝑊) = (𝑋 ∧ 𝑊))
18	17	oveq2d 7437	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑞 ∨ (𝑥 ∧ 𝑊)) = (𝑞 ∨ (𝑋 ∧ 𝑊)))
19		id 23	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
20	18, 19	eqeq12d 2776	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥 ↔ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
21	20	anbi2d 642	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) ↔ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)))
22		fvoveq1 7444	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝐷‘(𝑥 ∧ 𝑊)) = (𝐷‘(𝑋 ∧ 𝑊)))
23	22	oveq2d 7437	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))) = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))
24	23	eqeq2d 2771	. . . . . . 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 3185	. . . . 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 7380	. . . 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 4510	. . 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 2760	. . 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 6899	. . . 4 ⊢ (𝐷‘𝑋) ∈ V
31		riotaex 7382	. . . 4 ⊢ (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))) ∈ V
32	30, 31	ifex 4532	. . 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 6994	. 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 2815	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)))))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ifcif 4481   class class class wbr 5102   ↦ cmpt 5185  ‘cfv 6539  ℩crio 7377  (class class class)co 7421  Basecbs 17393  lecple 17441  joincjn 18491  meetcmee 18492  LSSumclsm 19854  LSubSpclss 21212  Atomscatm 40414  LHypclh 41135  DVecHcdvh 42229  DIsoBcdib 42289  DIsoCcdic 42323  DIsoHcdih 42379

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-dih 42380

This theorem is used by:  dihvalc  42384  dihvalb  42388