Theorem oposlem 39138

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 2740	. . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		eqid 2740	. . . . 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 39136	. . . 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 6920	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑋))
13	12	eleq1d 2829	. . . . . 6 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘𝑥) ∈ 𝐵 ↔ ( ⊥ ‘𝑋) ∈ 𝐵))
14		2fveq3 6925	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘( ⊥ ‘𝑥)) = ( ⊥ ‘( ⊥ ‘𝑋)))
15		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
16	14, 15	eqeq12d 2756	. . . . . 6 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
17		breq1 5169	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
18	12	breq2d 5178	. . . . . . 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 1436	. . . . 5 ⊢ (𝑥 = 𝑋 → ((( ⊥ ‘𝑥) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥 ∧ (𝑥 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑥))) ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋)))))
21	15, 12	oveq12d 7466	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∨ ( ⊥ ‘𝑥)) = (𝑋 ∨ ( ⊥ ‘𝑋)))
22	21	eqeq1d 2742	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ∨ ( ⊥ ‘𝑥)) = 1 ↔ (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 ))
23	15, 12	oveq12d 7466	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ ( ⊥ ‘𝑥)) = (𝑋 ∧ ( ⊥ ‘𝑋)))
24	23	eqeq1d 2742	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ∧ ( ⊥ ‘𝑥)) = 0 ↔ (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 ))
25	20, 22, 24	3anbi123d 1436	. . . 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 5170	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
27		fveq2 6920	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘𝑦) = ( ⊥ ‘𝑌))
28	27	breq1d 5176	. . . . . . 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 1442	. . . . 5 ⊢ (𝑦 = 𝑌 → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑦 → ( ⊥ ‘𝑦) ≤ ( ⊥ ‘𝑋))) ↔ (( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋 ≤ 𝑌 → ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))))
31	30	3anbi1d 1440	. . . 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 3646	. . 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 1115	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 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   class class class wbr 5166  dom cdm 5700  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  lecple 17318  occoc 17319  Posetcpo 18377  lubclub 18379  glbcglb 18380  joincjn 18381  meetcmee 18382  0.cp0 18493  1.cp1 18494  OPcops 39128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-dm 5710  df-iota 6525  df-fv 6581  df-ov 7451  df-oposet 39132

This theorem is referenced by:  opoccl  39150  opococ  39151  oplecon3  39155  opexmid  39163  opnoncon  39164