Theorem omllaw 39613

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 39608	. . . 4 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
7	6	simprbi 497	. . 3 ⊢ (𝐾 ∈ OML → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))))
8		breq1 5103	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
9		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
10		fveq2 6842	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
11	10	oveq2d 7384	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑦 ∧ ( ⊥ ‘𝑥)) = (𝑦 ∧ ( ⊥ ‘𝑋)))
12	9, 11	oveq12d 7386	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))) = (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋))))
13	12	eqeq2d 2748	. . . . 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 5104	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
16		id 22	. . . . . 6 ⊢ (𝑦 = 𝑌 → 𝑦 = 𝑌)
17		oveq1 7375	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∧ ( ⊥ ‘𝑋)) = (𝑌 ∧ ( ⊥ ‘𝑋)))
18	17	oveq2d 7384	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋))) = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))
19	16, 18	eqeq12d 2753	. . . . 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 3589	. . 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 1117	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  lecple 17196  occoc 17197  joincjn 18246  meetcmee 18247  OLcol 39544  OMLcoml 39545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-oml 39549

This theorem is referenced by:  omllaw2N  39614  omllaw3  39615  omllaw4  39616