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 37665
Description: Contraposition law for orthoposets. (chsscon3 30445 analog.) (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
opcon3.b 𝐡 = (Baseβ€˜πΎ)
opcon3.l ≀ = (leβ€˜πΎ)
opcon3.o βŠ₯ = (ocβ€˜πΎ)
Assertion
Ref Expression
oplecon3b ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ↔ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹)))

Proof of Theorem oplecon3b
StepHypRef Expression
1 opcon3.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 opcon3.l . . 3 ≀ = (leβ€˜πΎ)
3 opcon3.o . . 3 βŠ₯ = (ocβ€˜πΎ)
41, 2, 3oplecon3 37664 . 2 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹)))
5 simp1 1137 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OP)
61, 3opoccl 37659 . . . . 5 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
763adant2 1132 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
81, 3opoccl 37659 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
983adant3 1133 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
101, 2, 3oplecon3 37664 . . . 4 ((𝐾 ∈ OP ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘‹) ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) ≀ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))))
115, 7, 9, 10syl3anc 1372 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) ≀ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))))
121, 3opococ 37660 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
13123adant3 1133 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
141, 3opococ 37660 . . . . 5 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) = π‘Œ)
15143adant2 1132 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) = π‘Œ)
1613, 15breq12d 5119 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) ≀ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) ↔ 𝑋 ≀ π‘Œ))
1711, 16sylibd 238 . 2 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹) β†’ 𝑋 ≀ π‘Œ))
184, 17impbid 211 1 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ ↔ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5106  β€˜cfv 6497  Basecbs 17084  lecple 17141  occoc 17142  OPcops 37637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-dm 5644  df-iota 6449  df-fv 6505  df-ov 7361  df-oposet 37641
This theorem is referenced by:  oplecon1b  37666  opltcon3b  37669  oldmm1  37682  omllaw4  37711  cvrcmp2  37749  glbconN  37842  glbconNOLD  37843  lhpmod2i2  38504  lhpmod6i1  38505  lhprelat3N  38506  dochss  39831
  Copyright terms: Public domain W3C validator