Theorem oposlem 37123

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 2738	. . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2738	. . . . 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 37121	. . . 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 6756	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
13	12	eleq1d 2823	. . . . . 6 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘𝑥) ∈ 𝐵 ↔ ( ⊥ ‘𝑋) ∈ 𝐵))
14		2fveq3 6761	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑋)))
15		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
16	14, 15	eqeq12d 2754	. . . . . 6 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
17		breq1 5073	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
18	12	breq2d 5082	. . . . . . 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 1434	. . . . 5 ⊢ (𝑥 = 𝑋 → ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋)))))
21	15, 12	oveq12d 7273	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ ( ⊥ ‘𝑥)) = (𝑋 ∨ ( ⊥ ‘𝑋)))
22	21	eqeq1d 2740	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ↔ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ))
23	15, 12	oveq12d 7273	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ ( ⊥ ‘𝑥)) = (𝑋 ∧ ( ⊥ ‘𝑋)))
24	23	eqeq1d 2740	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ∧ ( ⊥ ‘𝑥)) = 0 ↔ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ))
25	20, 22, 24	3anbi123d 1434	. . . 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 5074	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
27		fveq2 6756	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘𝑦) = ( ⊥ ‘𝑌))
28	27	breq1d 5080	. . . . . . 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 1440	. . . . 5 ⊢ (𝑦 = 𝑌 → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋))) ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))))
31	30	3anbi1d 1438	. . . 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 3562	. . 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 1113	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 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   class class class wbr 5070  dom cdm 5580  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  lecple 16895  occoc 16896  Posetcpo 17940  lubclub 17942  glbcglb 17943  joincjn 17944  meetcmee 17945  0.cp0 18056  1.cp1 18057  OPcops 37113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-dm 5590  df-iota 6376  df-fv 6426  df-ov 7258  df-oposet 37117

This theorem is referenced by:  opoccl  37135  opococ  37136  oplecon3  37140  opexmid  37148  opnoncon  37149