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Theorem pltletr 16794
 Description: Transitive law for chained less-than and less-than-or-equal. (psssstr 3675 analog.) (Contributed by NM, 2-Dec-2011.)
Hypotheses
Ref Expression
pltletr.b 𝐵 = (Base‘𝐾)
pltletr.l = (le‘𝐾)
pltletr.s < = (lt‘𝐾)
Assertion
Ref Expression
pltletr ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 𝑍) → 𝑋 < 𝑍))

Proof of Theorem pltletr
StepHypRef Expression
1 pltletr.b . . . . . 6 𝐵 = (Base‘𝐾)
2 pltletr.l . . . . . 6 = (le‘𝐾)
3 pltletr.s . . . . . 6 < = (lt‘𝐾)
41, 2, 3pleval2 16788 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
543adant3r1 1266 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
65adantr 480 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
71, 3plttr 16793 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
87expdimp 452 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 < 𝑍𝑋 < 𝑍))
9 breq2 4587 . . . . . 6 (𝑌 = 𝑍 → (𝑋 < 𝑌𝑋 < 𝑍))
109biimpcd 238 . . . . 5 (𝑋 < 𝑌 → (𝑌 = 𝑍𝑋 < 𝑍))
1110adantl 481 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 = 𝑍𝑋 < 𝑍))
128, 11jaod 394 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → ((𝑌 < 𝑍𝑌 = 𝑍) → 𝑋 < 𝑍))
136, 12sylbid 229 . 2 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 𝑍𝑋 < 𝑍))
1413expimpd 627 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 𝑍) → 𝑋 < 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775  Posetcpo 16763  ltcplt 16764 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-preset 16751  df-poset 16769  df-plt 16781 This theorem is referenced by:  cvrletrN  33578  atlen0  33615  atlelt  33742  2atlt  33743  ps-2  33782  llnnleat  33817  lplnnle2at  33845  lvolnle3at  33886  dalemcea  33964  2atm2atN  34089  dia2dimlem2  35372  dia2dimlem3  35373
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