Theorem oposlem 39205

Description: Lemma for orthoposet properties. (Contributed by NM, 20-Oct-2011.)

Hypotheses

Ref	Expression
oposlem.b	⊢ 𝐵 = (Base‘𝐾)
oposlem.l	⊢ ≤ = (le‘𝐾)
oposlem.o	⊢ ⊥ = (oc‘𝐾)
oposlem.j	⊢ ∨ = (join‘𝐾)
oposlem.m	⊢ ∧ = (meet‘𝐾)
oposlem.f	⊢ 0 = (0.‘𝐾)
oposlem.u	⊢ 1 = (1.‘𝐾)

Assertion

Ref	Expression
oposlem	⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋))) ∧ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ))

Proof of Theorem oposlem

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

Step	Hyp	Ref	Expression
1		oposlem.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2736	. . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2736	. . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾)
4		oposlem.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
5		oposlem.o	. . . . 5 ⊢ ⊥ = (oc‘𝐾)
6		oposlem.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
7		oposlem.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
8		oposlem.f	. . . . 5 ⊢ 0 = (0.‘𝐾)
9		oposlem.u	. . . . 5 ⊢ 1 = (1.‘𝐾)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	isopos 39203	. . . 4 ⊢ (𝐾 ∈ OP ↔ ((𝐾 ∈ Poset ∧ 𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ∧ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ∧ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 )))
11	10	simprbi 496	. . 3 ⊢ (𝐾 ∈ OP → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ∧ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ∧ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 ))
12		fveq2 6881	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
13	12	eleq1d 2820	. . . . . 6 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘𝑥) ∈ 𝐵 ↔ ( ⊥ ‘𝑋) ∈ 𝐵))
14		2fveq3 6886	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑋)))
15		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
16	14, 15	eqeq12d 2752	. . . . . 6 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
17		breq1 5127	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
18	12	breq2d 5136	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥) ↔ ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋)))
19	17, 18	imbi12d 344	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥)) ↔ (𝑋 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋))))
20	13, 16, 19	3anbi123d 1438	. . . . 5 ⊢ (𝑥 = 𝑋 → ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋)))))
21	15, 12	oveq12d 7428	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ ( ⊥ ‘𝑥)) = (𝑋 ∨ ( ⊥ ‘𝑋)))
22	21	eqeq1d 2738	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ↔ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ))
23	15, 12	oveq12d 7428	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ ( ⊥ ‘𝑥)) = (𝑋 ∧ ( ⊥ ‘𝑋)))
24	23	eqeq1d 2738	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ∧ ( ⊥ ‘𝑥)) = 0 ↔ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ))
25	20, 22, 24	3anbi123d 1438	. . . 4 ⊢ (𝑥 = 𝑋 → (((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ∧ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ∧ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 ) ↔ ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋))) ∧ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 )))
26		breq2 5128	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
27		fveq2 6881	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘𝑦) = ( ⊥ ‘𝑌))
28	27	breq1d 5134	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋) ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))
29	26, 28	imbi12d 344	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋)) ↔ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋))))
30	29	3anbi3d 1444	. . . . 5 ⊢ (𝑦 = 𝑌 → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋))) ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))))
31	30	3anbi1d 1442	. . . 4 ⊢ (𝑦 = 𝑌 → (((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋))) ∧ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ) ↔ ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋))) ∧ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 )))
32	25, 31	rspc2v 3617	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ∧ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ∧ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 ) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋))) ∧ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 )))
33	11, 32	mpan9 506	. 2 ⊢ ((𝐾 ∈ OP ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋))) ∧ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ))
34	33	3impb 1114	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋))) ∧ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052   class class class wbr 5124  dom cdm 5659  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  lecple 17283  occoc 17284  Posetcpo 18324  lubclub 18326  glbcglb 18327  joincjn 18328  meetcmee 18329  0.cp0 18438  1.cp1 18439  OPcops 39195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-nul 5281

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-dm 5669  df-iota 6489  df-fv 6544  df-ov 7413  df-oposet 39199

This theorem is referenced by:  opoccl  39217  opococ  39218  oplecon3  39222  opexmid  39230  opnoncon  39231