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Theorem plelttr 17019
Description: Transitive law for chained less-than-or-equal and less-than. (sspsstr 3745 analog.) (Contributed by NM, 2-May-2012.)
Hypotheses
Ref Expression
pltletr.b 𝐵 = (Base‘𝐾)
pltletr.l = (le‘𝐾)
pltletr.s < = (lt‘𝐾)
Assertion
Ref Expression
plelttr ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))

Proof of Theorem plelttr
StepHypRef Expression
1 pltletr.b . . . . 5 𝐵 = (Base‘𝐾)
2 pltletr.l . . . . 5 = (le‘𝐾)
3 pltletr.s . . . . 5 < = (lt‘𝐾)
41, 2, 3pleval2 17012 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
543adant3r3 1297 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 ↔ (𝑋 < 𝑌𝑋 = 𝑌)))
61, 3plttr 17017 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
76expd 451 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑌 → (𝑌 < 𝑍𝑋 < 𝑍)))
8 breq1 4688 . . . . . 6 (𝑋 = 𝑌 → (𝑋 < 𝑍𝑌 < 𝑍))
98biimprd 238 . . . . 5 (𝑋 = 𝑌 → (𝑌 < 𝑍𝑋 < 𝑍))
109a1i 11 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 = 𝑌 → (𝑌 < 𝑍𝑋 < 𝑍)))
117, 10jaod 394 . . 3 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑋 = 𝑌) → (𝑌 < 𝑍𝑋 < 𝑍)))
125, 11sylbid 230 . 2 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 → (𝑌 < 𝑍𝑋 < 𝑍)))
1312impd 446 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030   class class class wbr 4685  cfv 5926  Basecbs 15904  lecple 15995  Posetcpo 16987  ltcplt 16988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-preset 16975  df-poset 16993  df-plt 17005
This theorem is referenced by:  isarchi3  29869  archiabllem2c  29877  athgt  35060  1cvratex  35077
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