Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  oplecon3 Structured version   Visualization version   GIF version

Theorem oplecon3 34804
 Description: Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)
Hypotheses
Ref Expression
opcon3.b 𝐵 = (Base‘𝐾)
opcon3.l = (le‘𝐾)
opcon3.o = (oc‘𝐾)
Assertion
Ref Expression
oplecon3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ( 𝑌) ( 𝑋)))

Proof of Theorem oplecon3
StepHypRef Expression
1 opcon3.b . . . 4 𝐵 = (Base‘𝐾)
2 opcon3.l . . . 4 = (le‘𝐾)
3 opcon3.o . . . 4 = (oc‘𝐾)
4 eqid 2651 . . . 4 (join‘𝐾) = (join‘𝐾)
5 eqid 2651 . . . 4 (meet‘𝐾) = (meet‘𝐾)
6 eqid 2651 . . . 4 (0.‘𝐾) = (0.‘𝐾)
7 eqid 2651 . . . 4 (1.‘𝐾) = (1.‘𝐾)
81, 2, 3, 4, 5, 6, 7oposlem 34787 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ((( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋))) ∧ (𝑋(join‘𝐾)( 𝑋)) = (1.‘𝐾) ∧ (𝑋(meet‘𝐾)( 𝑋)) = (0.‘𝐾)))
98simp1d 1093 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) ∈ 𝐵 ∧ ( ‘( 𝑋)) = 𝑋 ∧ (𝑋 𝑌 → ( 𝑌) ( 𝑋))))
109simp3d 1095 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ( 𝑌) ( 𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  occoc 15996  joincjn 16991  meetcmee 16992  0.cp0 17084  1.cp1 17085  OPcops 34777 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-dm 5153  df-iota 5889  df-fv 5934  df-ov 6693  df-oposet 34781 This theorem is referenced by:  oplecon3b  34805
 Copyright terms: Public domain W3C validator