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Theorem joinle 18326
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 5148 . . . . 5 (𝑧 = 𝑍 → (𝑋 𝑧𝑋 𝑍))
2 breq2 5148 . . . . 5 (𝑧 = 𝑍 → (𝑌 𝑧𝑌 𝑍))
31, 2anbi12d 632 . . . 4 (𝑧 = 𝑍 → ((𝑋 𝑧𝑌 𝑧) ↔ (𝑋 𝑍𝑌 𝑍)))
4 breq2 5148 . . . 4 (𝑧 = 𝑍 → ((𝑋 𝑌) 𝑧 ↔ (𝑋 𝑌) 𝑍))
53, 4imbi12d 345 . . 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 18323 . . . 4 (𝜑 → ((𝑋 (𝑋 𝑌) ∧ 𝑌 (𝑋 𝑌)) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧)))
1413simprd 497 . . 3 (𝜑 → ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧))
15 joinle.z . . 3 (𝜑𝑍𝐵)
165, 14, 15rspcdva 3612 . 2 (𝜑 → ((𝑋 𝑍𝑌 𝑍) → (𝑋 𝑌) 𝑍))
176, 7, 8, 9, 10, 11, 12lejoin1 18324 . . . 4 (𝜑𝑋 (𝑋 𝑌))
186, 8, 9, 10, 11, 12joincl 18318 . . . . 5 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
196, 7postr 18260 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵)) → ((𝑋 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑋 𝑍))
209, 10, 18, 15, 19syl13anc 1373 . . . 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 18325 . . . 4 (𝜑𝑌 (𝑋 𝑌))
236, 7postr 18260 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑌𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵)) → ((𝑌 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑌 𝑍))
249, 11, 18, 15, 23syl13anc 1373 . . . 4 (𝜑 → ((𝑌 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑌 𝑍))
2522, 24mpand 694 . . 3 (𝜑 → ((𝑋 𝑌) 𝑍𝑌 𝑍))
2621, 25jcad 514 . 2 (𝜑 → ((𝑋 𝑌) 𝑍 → (𝑋 𝑍𝑌 𝑍)))
2716, 26impbid 211 1 (𝜑 → ((𝑋 𝑍𝑌 𝑍) ↔ (𝑋 𝑌) 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  cop 4630   class class class wbr 5144  dom cdm 5672  cfv 6535  (class class class)co 7396  Basecbs 17131  lecple 17191  Posetcpo 18247  joincjn 18251
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7352  df-ov 7399  df-oprab 7400  df-poset 18253  df-lub 18286  df-join 18288
This theorem is referenced by:  latjle12  18390
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