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Theorem pltletr 18061
Description: Transitive law for chained "less than" and "less than or equal to". (psssstr 4041 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 18055 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
543adant3r1 1181 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
65adantr 481 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
71, 3plttr 18060 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
87expdimp 453 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 < 𝑍𝑋 < 𝑍))
9 breq2 5078 . . . . . 6 (𝑌 = 𝑍 → (𝑋 < 𝑌𝑋 < 𝑍))
109biimpcd 248 . . . . 5 (𝑋 < 𝑌 → (𝑌 = 𝑍𝑋 < 𝑍))
1110adantl 482 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 = 𝑍𝑋 < 𝑍))
128, 11jaod 856 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → ((𝑌 < 𝑍𝑌 = 𝑍) → 𝑋 < 𝑍))
136, 12sylbid 239 . 2 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 𝑍𝑋 < 𝑍))
1413expimpd 454 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 𝑍) → 𝑋 < 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844  w3a 1086   = wceq 1539  wcel 2106   class class class wbr 5074  cfv 6433  Basecbs 16912  lecple 16969  Posetcpo 18025  ltcplt 18026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-proset 18013  df-poset 18031  df-plt 18048
This theorem is referenced by:  cvrletrN  37287  atlen0  37324  atlelt  37452  2atlt  37453  ps-2  37492  llnnleat  37527  lplnnle2at  37555  lvolnle3at  37596  dalemcea  37674  2atm2atN  37799  dia2dimlem2  39079  dia2dimlem3  39080
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