MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  plttr Structured version   Visualization version   GIF version

Theorem plttr 17582
Description: The less-than relation is transitive. (psstr 4083 analog.) (Contributed by NM, 2-Dec-2011.)
Hypotheses
Ref Expression
pltnlt.b 𝐵 = (Base‘𝐾)
pltnlt.s < = (lt‘𝐾)
Assertion
Ref Expression
plttr ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))

Proof of Theorem plttr
StepHypRef Expression
1 eqid 2823 . . . . . 6 (le‘𝐾) = (le‘𝐾)
2 pltnlt.s . . . . . 6 < = (lt‘𝐾)
31, 2pltle 17573 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌𝑋(le‘𝐾)𝑌))
433adant3r3 1180 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑌𝑋(le‘𝐾)𝑌))
51, 2pltle 17573 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑍𝐵) → (𝑌 < 𝑍𝑌(le‘𝐾)𝑍))
653adant3r1 1178 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 < 𝑍𝑌(le‘𝐾)𝑍))
7 pltnlt.b . . . . 5 𝐵 = (Base‘𝐾)
87, 1postr 17565 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋(le‘𝐾)𝑌𝑌(le‘𝐾)𝑍) → 𝑋(le‘𝐾)𝑍))
94, 6, 8syl2and 609 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋(le‘𝐾)𝑍))
107, 2pltn2lp 17581 . . . . . 6 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → ¬ (𝑋 < 𝑌𝑌 < 𝑋))
11103adant3r3 1180 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ¬ (𝑋 < 𝑌𝑌 < 𝑋))
12 breq2 5072 . . . . . . 7 (𝑋 = 𝑍 → (𝑌 < 𝑋𝑌 < 𝑍))
1312anbi2d 630 . . . . . 6 (𝑋 = 𝑍 → ((𝑋 < 𝑌𝑌 < 𝑋) ↔ (𝑋 < 𝑌𝑌 < 𝑍)))
1413notbid 320 . . . . 5 (𝑋 = 𝑍 → (¬ (𝑋 < 𝑌𝑌 < 𝑋) ↔ ¬ (𝑋 < 𝑌𝑌 < 𝑍)))
1511, 14syl5ibcom 247 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 = 𝑍 → ¬ (𝑋 < 𝑌𝑌 < 𝑍)))
1615necon2ad 3033 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋𝑍))
179, 16jcad 515 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → (𝑋(le‘𝐾)𝑍𝑋𝑍)))
181, 2pltval 17572 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑍𝐵) → (𝑋 < 𝑍 ↔ (𝑋(le‘𝐾)𝑍𝑋𝑍)))
19183adant3r2 1179 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑍 ↔ (𝑋(le‘𝐾)𝑍𝑋𝑍)))
2017, 19sylibrd 261 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wne 3018   class class class wbr 5068  cfv 6357  Basecbs 16485  lecple 16574  Posetcpo 17552  ltcplt 17553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-proset 17540  df-poset 17558  df-plt 17570
This theorem is referenced by:  pltletr  17583  plelttr  17584  pospo  17585  archiabllem2c  30826  ofldchr  30889  hlhgt2  36527  hl0lt1N  36528  lhp0lt  37141
  Copyright terms: Public domain W3C validator