Theorem omllaw 36373

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 36368	. . . 4 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
7	6	simprbi 499	. . 3 ⊢ (𝐾 ∈ OML → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))))
8		breq1 5061	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
9		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
10		fveq2 6664	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
11	10	oveq2d 7166	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑦 ∧ ( ⊥ ‘𝑥)) = (𝑦 ∧ ( ⊥ ‘𝑋)))
12	9, 11	oveq12d 7168	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))) = (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋))))
13	12	eqeq2d 2832	. . . . 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 5062	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
16		id 22	. . . . . 6 ⊢ (𝑦 = 𝑌 → 𝑦 = 𝑌)
17		oveq1 7157	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∧ ( ⊥ ‘𝑋)) = (𝑌 ∧ ( ⊥ ‘𝑋)))
18	17	oveq2d 7166	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 ∨ (𝑦 ∧ ( ⊥ ‘𝑋))) = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))
19	16, 18	eqeq12d 2837	. . . . 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 3632	. . 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 1112	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138   class class class wbr 5058  ‘cfv 6349  (class class class)co 7150  Basecbs 16477  lecple 16566  occoc 16567  joincjn 17548  meetcmee 17549  OLcol 36304  OMLcoml 36305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-iota 6308  df-fv 6357  df-ov 7153  df-oml 36309

This theorem is referenced by:  omllaw2N  36374  omllaw3  36375  omllaw4  36376