Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  oplecon3b Structured version   Visualization version   GIF version

Theorem oplecon3b 34313
Description: Contraposition law for orthoposets. (chsscon3 28343 analog.) (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
opcon3.b 𝐵 = (Base‘𝐾)
opcon3.l = (le‘𝐾)
opcon3.o = (oc‘𝐾)
Assertion
Ref Expression
oplecon3b ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))

Proof of Theorem oplecon3b
StepHypRef Expression
1 opcon3.b . . 3 𝐵 = (Base‘𝐾)
2 opcon3.l . . 3 = (le‘𝐾)
3 opcon3.o . . 3 = (oc‘𝐾)
41, 2, 3oplecon3 34312 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ( 𝑌) ( 𝑋)))
5 simp1 1060 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
61, 3opoccl 34307 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
763adant2 1079 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
81, 3opoccl 34307 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
983adant3 1080 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
101, 2, 3oplecon3 34312 . . . 4 ((𝐾 ∈ OP ∧ ( 𝑌) ∈ 𝐵 ∧ ( 𝑋) ∈ 𝐵) → (( 𝑌) ( 𝑋) → ( ‘( 𝑋)) ( ‘( 𝑌))))
115, 7, 9, 10syl3anc 1325 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( 𝑋) → ( ‘( 𝑋)) ( ‘( 𝑌))))
121, 3opococ 34308 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)
13123adant3 1080 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑋)) = 𝑋)
141, 3opococ 34308 . . . . 5 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
15143adant2 1079 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
1613, 15breq12d 4664 . . 3 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( ‘( 𝑋)) ( ‘( 𝑌)) ↔ 𝑋 𝑌))
1711, 16sylibd 229 . 2 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑌) ( 𝑋) → 𝑋 𝑌))
184, 17impbid 202 1 ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1037   = wceq 1482  wcel 1989   class class class wbr 4651  cfv 5886  Basecbs 15851  lecple 15942  occoc 15943  OPcops 34285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-nul 4787
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-dm 5122  df-iota 5849  df-fv 5894  df-ov 6650  df-oposet 34289
This theorem is referenced by:  oplecon1b  34314  opltcon3b  34317  oldmm1  34330  omllaw4  34359  cvrcmp2  34397  glbconN  34489  lhpmod2i2  35150  lhpmod6i1  35151  lhprelat3N  35152  dochss  36480
  Copyright terms: Public domain W3C validator