Theorem opococ 39947

Description: Double negative law for orthoposets. (ococ 31736 analog.) (Contributed by NM, 13-Sep-2011.)

Hypotheses

Ref	Expression
opoccl.b	⊢ 𝐵 = (Base‘𝐾)
opoccl.o	⊢ ⊥ = (oc‘𝐾)

Assertion

Ref	Expression
opococ	⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)

Proof of Theorem opococ

Step	Hyp	Ref	Expression
1		opoccl.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2763	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
3		opoccl.o	. . . . 5 ⊢ ⊥ = (oc‘𝐾)
4		eqid 2763	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
5		eqid 2763	. . . . 5 ⊢ (meet‘𝐾) = (meet‘𝐾)
6		eqid 2763	. . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾)
7		eqid 2763	. . . . 5 ⊢ (1.‘𝐾) = (1.‘𝐾)
8	1, 2, 3, 4, 5, 6, 7	oposlem 39934	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋))) ∧ (𝑋(join‘𝐾)( ⊥ ‘𝑋)) = (1.‘𝐾) ∧ (𝑋(meet‘𝐾)( ⊥ ‘𝑋)) = (0.‘𝐾)))
9	8	3anidm23 1448	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋))) ∧ (𝑋(join‘𝐾)( ⊥ ‘𝑋)) = (1.‘𝐾) ∧ (𝑋(meet‘𝐾)( ⊥ ‘𝑋)) = (0.‘𝐾)))
10	9	simp1d 1160	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋))))
11	10	simp2d 1161	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  lecple 17318  occoc 17319  joincjn 18368  meetcmee 18369  0.cp0 18478  1.cp1 18479  OPcops 39924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-dm 5673  df-iota 6494  df-fv 6546  df-ov 7415  df-oposet 39928

This theorem is referenced by:  opcon3b  39948  opcon2b  39949  oplecon3b  39952  oplecon1b  39953  opltcon1b  39957  opltcon2b  39958  oldmm2  39970  oldmm3N  39971  oldmm4  39972  oldmj1  39973  oldmj2  39974  oldmj3  39975  oldmj4  39976  olm11  39979  omllaw4  39998  cmt2N  40002  glbconN  40129  1cvratex  40225  1cvrjat  40227  polval2N  40658  2polpmapN  40665  2polvalN  40666  2polatN  40684  lhpoc2N  40767  doch2val2  42116  dochocss  42118  dochoc  42119