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Theorem joinle 16837
 Description: A join is less than or equal to a third value iff each argument is less than or equal to the third value. (Contributed by NM, 16-Sep-2011.)
Hypotheses
Ref Expression
joinle.b 𝐵 = (Base‘𝐾)
joinle.l = (le‘𝐾)
joinle.j = (join‘𝐾)
joinle.k (𝜑𝐾 ∈ Poset)
joinle.x (𝜑𝑋𝐵)
joinle.y (𝜑𝑌𝐵)
joinle.z (𝜑𝑍𝐵)
joinle.e (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
Assertion
Ref Expression
joinle (𝜑 → ((𝑋 𝑍𝑌 𝑍) ↔ (𝑋 𝑌) 𝑍))

Proof of Theorem joinle
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 joinle.z . . 3 (𝜑𝑍𝐵)
2 joinle.b . . . . 5 𝐵 = (Base‘𝐾)
3 joinle.l . . . . 5 = (le‘𝐾)
4 joinle.j . . . . 5 = (join‘𝐾)
5 joinle.k . . . . 5 (𝜑𝐾 ∈ Poset)
6 joinle.x . . . . 5 (𝜑𝑋𝐵)
7 joinle.y . . . . 5 (𝜑𝑌𝐵)
8 joinle.e . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
92, 3, 4, 5, 6, 7, 8joinlem 16834 . . . 4 (𝜑 → ((𝑋 (𝑋 𝑌) ∧ 𝑌 (𝑋 𝑌)) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧)))
109simprd 478 . . 3 (𝜑 → ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧))
11 breq2 4587 . . . . . 6 (𝑧 = 𝑍 → (𝑋 𝑧𝑋 𝑍))
12 breq2 4587 . . . . . 6 (𝑧 = 𝑍 → (𝑌 𝑧𝑌 𝑍))
1311, 12anbi12d 743 . . . . 5 (𝑧 = 𝑍 → ((𝑋 𝑧𝑌 𝑧) ↔ (𝑋 𝑍𝑌 𝑍)))
14 breq2 4587 . . . . 5 (𝑧 = 𝑍 → ((𝑋 𝑌) 𝑧 ↔ (𝑋 𝑌) 𝑍))
1513, 14imbi12d 333 . . . 4 (𝑧 = 𝑍 → (((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧) ↔ ((𝑋 𝑍𝑌 𝑍) → (𝑋 𝑌) 𝑍)))
1615rspcva 3280 . . 3 ((𝑍𝐵 ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧)) → ((𝑋 𝑍𝑌 𝑍) → (𝑋 𝑌) 𝑍))
171, 10, 16syl2anc 691 . 2 (𝜑 → ((𝑋 𝑍𝑌 𝑍) → (𝑋 𝑌) 𝑍))
182, 3, 4, 5, 6, 7, 8lejoin1 16835 . . . 4 (𝜑𝑋 (𝑋 𝑌))
192, 4, 5, 6, 7, 8joincl 16829 . . . . 5 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
202, 3postr 16776 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵)) → ((𝑋 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑋 𝑍))
215, 6, 19, 1, 20syl13anc 1320 . . . 4 (𝜑 → ((𝑋 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑋 𝑍))
2218, 21mpand 707 . . 3 (𝜑 → ((𝑋 𝑌) 𝑍𝑋 𝑍))
232, 3, 4, 5, 6, 7, 8lejoin2 16836 . . . 4 (𝜑𝑌 (𝑋 𝑌))
242, 3postr 16776 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑌𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵)) → ((𝑌 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑌 𝑍))
255, 7, 19, 1, 24syl13anc 1320 . . . 4 (𝜑 → ((𝑌 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑌 𝑍))
2623, 25mpand 707 . . 3 (𝜑 → ((𝑋 𝑌) 𝑍𝑌 𝑍))
2722, 26jcad 554 . 2 (𝜑 → ((𝑋 𝑌) 𝑍 → (𝑋 𝑍𝑌 𝑍)))
2817, 27impbid 201 1 (𝜑 → ((𝑋 𝑍𝑌 𝑍) ↔ (𝑋 𝑌) 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⟨cop 4131   class class class wbr 4583  dom cdm 5038  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  Posetcpo 16763  joincjn 16767 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-poset 16769  df-lub 16797  df-join 16799 This theorem is referenced by:  latjle12  16885
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