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Theorem pltletr 16887
Description: Transitive law for chained less-than and less-than-or-equal. (psssstr 3696 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 16881 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
543adant3r1 1271 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
65adantr 481 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
71, 3plttr 16886 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
87expdimp 453 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 < 𝑍𝑋 < 𝑍))
9 breq2 4622 . . . . . 6 (𝑌 = 𝑍 → (𝑋 < 𝑌𝑋 < 𝑍))
109biimpcd 239 . . . . 5 (𝑋 < 𝑌 → (𝑌 = 𝑍𝑋 < 𝑍))
1110adantl 482 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 = 𝑍𝑋 < 𝑍))
128, 11jaod 395 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → ((𝑌 < 𝑍𝑌 = 𝑍) → 𝑋 < 𝑍))
136, 12sylbid 230 . 2 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 𝑍𝑋 < 𝑍))
1413expimpd 628 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 𝑍) → 𝑋 < 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1992   class class class wbr 4618  cfv 5850  Basecbs 15776  lecple 15864  Posetcpo 16856  ltcplt 16857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-preset 16844  df-poset 16862  df-plt 16874
This theorem is referenced by:  cvrletrN  34026  atlen0  34063  atlelt  34190  2atlt  34191  ps-2  34230  llnnleat  34265  lplnnle2at  34293  lvolnle3at  34334  dalemcea  34412  2atm2atN  34537  dia2dimlem2  35820  dia2dimlem3  35821
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