Theorem omllaw 39874

Description: The orthomodular law. (Contributed by NM, 18-Sep-2011.)

Hypotheses

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

Assertion

Ref	Expression
omllaw	⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))))

Proof of Theorem omllaw

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omllaw.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		omllaw.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		omllaw.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
4		omllaw.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
5		omllaw.o	. . . . 5 ⊢ ⊥ = (oc‘𝐾)
6	1, 2, 3, 4, 5	isoml 39869	. . . 4 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
7	6	simprbi 502	. . 3 ⊢ (𝐾 ∈ OML → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))))
8		breq1 5107	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
9		id 23	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
10		fveq2 6871	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
11	10	oveq2d 7416	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑦 ∧ ( ⊥ ‘𝑥)) = (𝑦 ∧ ( ⊥ ‘𝑋)))
12	9, 11	oveq12d 7418	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))) = (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋))))
13	12	eqeq2d 2776	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))) ↔ 𝑦 = (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋)))))
14	8, 13	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))) ↔ (𝑋 ≤ 𝑦 → 𝑦 = (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋))))))
15		breq2 5108	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
16		id 23	. . . . . 6 ⊢ (𝑦 = 𝑌 → 𝑦 = 𝑌)
17		oveq1 7407	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∧ ( ⊥ ‘𝑋)) = (𝑌 ∧ ( ⊥ ‘𝑋)))
18	17	oveq2d 7416	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋))) = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))
19	16, 18	eqeq12d 2781	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 = (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋))) ↔ 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))))
20	15, 19	imbi12d 347	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 → 𝑦 = (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋)))) ↔ (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))))
21	14, 20	rspc2v 3595	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))))
22	7, 21	syl5com 32	. 2 ⊢ (𝐾 ∈ OML → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))))
23	22	3impib 1132	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079   class class class wbr 5104  ‘cfv 6525  (class class class)co 7400  Basecbs 17257  lecple 17305  occoc 17306  joincjn 18355  meetcmee 18356  OLcol 39805  OMLcoml 39806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-iota 6481  df-fv 6533  df-ov 7403  df-oml 39810

This theorem is referenced by:  omllaw2N  39875  omllaw3  39876  omllaw4  39877