Theorem oposlem 40376

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 2760	. . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2760	. . . . 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 40374	. . . 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 503	. . 3 ⊢ (𝐾 ∈ OP → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ∧ (𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ∧ (𝑥 ∧ ( ⊥ ‘𝑥)) = 0 ))
12		fveq2 6890	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
13	12	eleq1d 2845	. . . . . 6 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘𝑥) ∈ 𝐵 ↔ ( ⊥ ‘𝑋) ∈ 𝐵))
14		2fveq3 6895	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑋)))
15		id 23	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
16	14, 15	eqeq12d 2776	. . . . . 6 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
17		breq1 5105	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
18	12	breq2d 5114	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥) ↔ ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋)))
19	17, 18	imbi12d 347	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥)) ↔ (𝑋 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋))))
20	13, 16, 19	3anbi123d 1464	. . . . 5 ⊢ (𝑥 = 𝑋 → ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋)))))
21	15, 12	oveq12d 7442	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ ( ⊥ ‘𝑥)) = (𝑋 ∨ ( ⊥ ‘𝑋)))
22	21	eqeq1d 2762	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ↔ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ))
23	15, 12	oveq12d 7442	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ ( ⊥ ‘𝑥)) = (𝑋 ∧ ( ⊥ ‘𝑋)))
24	23	eqeq1d 2762	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ∧ ( ⊥ ‘𝑥)) = 0 ↔ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ))
25	20, 22, 24	3anbi123d 1464	. . . 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 5106	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
27		fveq2 6890	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘𝑦) = ( ⊥ ‘𝑌))
28	27	breq1d 5112	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋) ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))
29	26, 28	imbi12d 347	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋)) ↔ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋))))
30	29	3anbi3d 1470	. . . . 5 ⊢ (𝑦 = 𝑌 → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋))) ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))))
31	30	3anbi1d 1468	. . . 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 3586	. . 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 516	. 2 ⊢ ((𝐾 ∈ OP ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋))) ∧ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ))
34	33	3impb 1132	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋))) ∧ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ∧ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   class class class wbr 5102  dom cdm 5651  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  lecple 17444  occoc 17445  Posetcpo 18490  lubclub 18492  glbcglb 18493  joincjn 18494  meetcmee 18495  0.cp0 18604  1.cp1 18605  OPcops 40366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-dm 5661  df-iota 6497  df-fv 6551  df-ov 7427  df-oposet 40370

This theorem is used by:  opoccl  40388  opococ  40389  oplecon3  40393  opexmid  40401  opnoncon  40402