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Theorem joinle 17616
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 breq2 5034 . . . . 5 (𝑧 = 𝑍 → (𝑋 𝑧𝑋 𝑍))
2 breq2 5034 . . . . 5 (𝑧 = 𝑍 → (𝑌 𝑧𝑌 𝑍))
31, 2anbi12d 633 . . . 4 (𝑧 = 𝑍 → ((𝑋 𝑧𝑌 𝑧) ↔ (𝑋 𝑍𝑌 𝑍)))
4 breq2 5034 . . . 4 (𝑧 = 𝑍 → ((𝑋 𝑌) 𝑧 ↔ (𝑋 𝑌) 𝑍))
53, 4imbi12d 348 . . 3 (𝑧 = 𝑍 → (((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧) ↔ ((𝑋 𝑍𝑌 𝑍) → (𝑋 𝑌) 𝑍)))
6 joinle.b . . . . 5 𝐵 = (Base‘𝐾)
7 joinle.l . . . . 5 = (le‘𝐾)
8 joinle.j . . . . 5 = (join‘𝐾)
9 joinle.k . . . . 5 (𝜑𝐾 ∈ Poset)
10 joinle.x . . . . 5 (𝜑𝑋𝐵)
11 joinle.y . . . . 5 (𝜑𝑌𝐵)
12 joinle.e . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
136, 7, 8, 9, 10, 11, 12joinlem 17613 . . . 4 (𝜑 → ((𝑋 (𝑋 𝑌) ∧ 𝑌 (𝑋 𝑌)) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧)))
1413simprd 499 . . 3 (𝜑 → ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧))
15 joinle.z . . 3 (𝜑𝑍𝐵)
165, 14, 15rspcdva 3573 . 2 (𝜑 → ((𝑋 𝑍𝑌 𝑍) → (𝑋 𝑌) 𝑍))
176, 7, 8, 9, 10, 11, 12lejoin1 17614 . . . 4 (𝜑𝑋 (𝑋 𝑌))
186, 8, 9, 10, 11, 12joincl 17608 . . . . 5 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
196, 7postr 17555 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵)) → ((𝑋 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑋 𝑍))
209, 10, 18, 15, 19syl13anc 1369 . . . 4 (𝜑 → ((𝑋 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑋 𝑍))
2117, 20mpand 694 . . 3 (𝜑 → ((𝑋 𝑌) 𝑍𝑋 𝑍))
226, 7, 8, 9, 10, 11, 12lejoin2 17615 . . . 4 (𝜑𝑌 (𝑋 𝑌))
236, 7postr 17555 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑌𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵)) → ((𝑌 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑌 𝑍))
249, 11, 18, 15, 23syl13anc 1369 . . . 4 (𝜑 → ((𝑌 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑌 𝑍))
2522, 24mpand 694 . . 3 (𝜑 → ((𝑋 𝑌) 𝑍𝑌 𝑍))
2621, 25jcad 516 . 2 (𝜑 → ((𝑋 𝑌) 𝑍 → (𝑋 𝑍𝑌 𝑍)))
2716, 26impbid 215 1 (𝜑 → ((𝑋 𝑍𝑌 𝑍) ↔ (𝑋 𝑌) 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wral 3106  cop 4531   class class class wbr 5030  dom cdm 5519  cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  Posetcpo 17542  joincjn 17546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-poset 17548  df-lub 17576  df-join 17578
This theorem is referenced by:  latjle12  17664
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