Theorem oposlem 39165

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 2729	. . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2729	. . . . 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 39163	. . . 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 6822	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
13	12	eleq1d 2813	. . . . . 6 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘𝑥) ∈ 𝐵 ↔ ( ⊥ ‘𝑋) ∈ 𝐵))
14		2fveq3 6827	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑋)))
15		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
16	14, 15	eqeq12d 2745	. . . . . 6 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
17		breq1 5095	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
18	12	breq2d 5104	. . . . . . 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 7367	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ ( ⊥ ‘𝑥)) = (𝑋 ∨ ( ⊥ ‘𝑋)))
22	21	eqeq1d 2731	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ↔ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ))
23	15, 12	oveq12d 7367	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ ( ⊥ ‘𝑥)) = (𝑋 ∧ ( ⊥ ‘𝑋)))
24	23	eqeq1d 2731	. . . . 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 5096	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
27		fveq2 6822	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘𝑦) = ( ⊥ ‘𝑌))
28	27	breq1d 5102	. . . . . . 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 3588	. . 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 3044   class class class wbr 5092  dom cdm 5619  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  lecple 17168  occoc 17169  Posetcpo 18213  lubclub 18215  glbcglb 18216  joincjn 18217  meetcmee 18218  0.cp0 18327  1.cp1 18328  OPcops 39155

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 2701  ax-nul 5245

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 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-dm 5629  df-iota 6438  df-fv 6490  df-ov 7352  df-oposet 39159

This theorem is referenced by:  opoccl  39177  opococ  39178  oplecon3  39182  opexmid  39190  opnoncon  39191