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Theorem pltletr 17172
Description: Transitive law for chained less-than and less-than-or-equal. (psssstr 3855 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 17166 . . . . 5 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
543adant3r1 1198 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
65adantr 472 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 𝑍 ↔ (𝑌 < 𝑍𝑌 = 𝑍)))
71, 3plttr 17171 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 < 𝑍) → 𝑋 < 𝑍))
87expdimp 452 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 < 𝑍𝑋 < 𝑍))
9 breq2 4808 . . . . . 6 (𝑌 = 𝑍 → (𝑋 < 𝑌𝑋 < 𝑍))
109biimpcd 239 . . . . 5 (𝑋 < 𝑌 → (𝑌 = 𝑍𝑋 < 𝑍))
1110adantl 473 . . . 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 230 . 2 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 𝑍𝑋 < 𝑍))
1413expimpd 630 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 1072   = wceq 1632  wcel 2139   class class class wbr 4804  cfv 6049  Basecbs 16059  lecple 16150  Posetcpo 17141  ltcplt 17142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-preset 17129  df-poset 17147  df-plt 17159
This theorem is referenced by:  cvrletrN  35063  atlen0  35100  atlelt  35227  2atlt  35228  ps-2  35267  llnnleat  35302  lplnnle2at  35330  lvolnle3at  35371  dalemcea  35449  2atm2atN  35574  dia2dimlem2  36856  dia2dimlem3  36857
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