Theorem omllaw 39499

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 39494	. . . 4 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
7	6	simprbi 496	. . 3 ⊢ (𝐾 ∈ OML → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))))
8		breq1 5101	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
9		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
10		fveq2 6834	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
11	10	oveq2d 7374	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑦 ∧ ( ⊥ ‘𝑥)) = (𝑦 ∧ ( ⊥ ‘𝑋)))
12	9, 11	oveq12d 7376	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))) = (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋))))
13	12	eqeq2d 2747	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))) ↔ 𝑦 = (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋)))))
14	8, 13	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))) ↔ (𝑋 ≤ 𝑦 → 𝑦 = (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋))))))
15		breq2 5102	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
16		id 22	. . . . . 6 ⊢ (𝑦 = 𝑌 → 𝑦 = 𝑌)
17		oveq1 7365	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∧ ( ⊥ ‘𝑋)) = (𝑌 ∧ ( ⊥ ‘𝑋)))
18	17	oveq2d 7374	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋))) = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))
19	16, 18	eqeq12d 2752	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 = (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋))) ↔ 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))))
20	15, 19	imbi12d 344	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 → 𝑦 = (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋)))) ↔ (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))))
21	14, 20	rspc2v 3587	. . 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 31	. 2 ⊢ (𝐾 ∈ OML → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))))
23	22	3impib 1116	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  lecple 17184  occoc 17185  joincjn 18234  meetcmee 18235  OLcol 39430  OMLcoml 39431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-oml 39435

This theorem is referenced by:  omllaw2N  39500  omllaw3  39501  omllaw4  39502