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Theorem oplecon3b 38583
Description: Contraposition law for orthoposets. (chsscon3 31262 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 38582 . 2 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ ( βŠ₯ β€˜π‘Œ) ≀ ( βŠ₯ β€˜π‘‹)))
5 simp1 1133 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OP)
61, 3opoccl 38577 . . . . 5 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
763adant2 1128 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
81, 3opoccl 38577 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
983adant3 1129 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
101, 2, 3oplecon3 38582 . . . 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 1368 . . 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 38578 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
13123adant3 1129 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = 𝑋)
141, 3opococ 38578 . . . . 5 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) = π‘Œ)
15143adant2 1128 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) = π‘Œ)
1613, 15breq12d 5154 . . 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 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  β€˜cfv 6537  Basecbs 17153  lecple 17213  occoc 17214  OPcops 38555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-dm 5679  df-iota 6489  df-fv 6545  df-ov 7408  df-oposet 38559
This theorem is referenced by:  oplecon1b  38584  opltcon3b  38587  oldmm1  38600  omllaw4  38629  cvrcmp2  38667  glbconN  38760  glbconNOLD  38761  lhpmod2i2  39422  lhpmod6i1  39423  lhprelat3N  39424  dochss  40749
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