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Theorem pltletr 18401
Description: Transitive law for chained "less than" and "less than or equal to". (psssstr 4119 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 18395 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
543adant3r1 1181 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
65adantr 480 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
71, 3plttr 18400 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
87expdimp 452 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 < 𝑍𝑋 < 𝑍))
9 breq2 5152 . . . . . 6 (𝑌 = 𝑍 → (𝑋 < 𝑌𝑋 < 𝑍))
109biimpcd 249 . . . . 5 (𝑋 < 𝑌 → (𝑌 = 𝑍𝑋 < 𝑍))
1110adantl 481 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 = 𝑍𝑋 < 𝑍))
128, 11jaod 859 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → ((𝑌 < 𝑍𝑌 = 𝑍) → 𝑋 < 𝑍))
136, 12sylbid 240 . 2 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 𝑍𝑋 < 𝑍))
1413expimpd 453 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 𝑍) → 𝑋 < 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  Basecbs 17245  lecple 17305  Posetcpo 18365  ltcplt 18366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-proset 18352  df-poset 18371  df-plt 18388
This theorem is referenced by:  cvrletrN  39255  atlen0  39292  atlelt  39421  2atlt  39422  ps-2  39461  llnnleat  39496  lplnnle2at  39524  lvolnle3at  39565  dalemcea  39643  2atm2atN  39768  dia2dimlem2  41048  dia2dimlem3  41049
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