Theorem isoml 34525

Description: The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)

Hypotheses

Ref	Expression
isoml.b	⊢ 𝐵 = (Base‘𝐾)
isoml.l	⊢ ≤ = (le‘𝐾)
isoml.j	⊢ ∨ = (join‘𝐾)
isoml.m	⊢ ∧ = (meet‘𝐾)
isoml.o	⊢ ⊥ = (oc‘𝐾)

Assertion

Ref	Expression
isoml	⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦

Allowed substitution hints:   ∨ (𝑥,𝑦)   ≤ (𝑥,𝑦)   ∧ (𝑥,𝑦)   ⊥ (𝑥,𝑦)

Proof of Theorem isoml

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2		isoml.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	syl6eqr 2674	. . 3 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4		fveq2 6191	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5		isoml.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
6	4, 5	syl6eqr 2674	. . . . . 6 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
7	6	breqd 4664	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦 ↔ 𝑥 ≤ 𝑦))
8		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9		isoml.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
10	8, 9	syl6eqr 2674	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
11		eqidd 2623	. . . . . . 7 ⊢ (𝑘 = 𝐾 → 𝑥 = 𝑥)
12		fveq2 6191	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
13		isoml.m	. . . . . . . . 9 ⊢ ∧ = (meet‘𝐾)
14	12, 13	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = ∧ )
15		eqidd 2623	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → 𝑦 = 𝑦)
16		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17		isoml.o	. . . . . . . . . 10 ⊢ ⊥ = (oc‘𝐾)
18	16, 17	syl6eqr 2674	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = ⊥ )
19	18	fveq1d 6193	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑥) = ( ⊥ ‘𝑥))
20	14, 15, 19	oveq123d 6671	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)) = (𝑦 ∧ ( ⊥ ‘𝑥)))
21	10, 11, 20	oveq123d 6671	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))
22	21	eqeq2d 2632	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) ↔ 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))))
23	7, 22	imbi12d 334	. . . 4 ⊢ (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
24	3, 23	raleqbidv 3152	. . 3 ⊢ (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
25	3, 24	raleqbidv 3152	. 2 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
26		df-oml 34466	. 2 ⊢ OML = {𝑘 ∈ OL ∣ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))))}
27	25, 26	elrab2 3366	1 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  lecple 15948  occoc 15949  joincjn 16944  meetcmee 16945  OLcol 34461  OMLcoml 34462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-oml 34466

This theorem is referenced by:  isomliN  34526  omlol  34527  omllaw  34530