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Theorem meetle 17638
Description: A meet 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.) (Revised by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
meetle.b 𝐵 = (Base‘𝐾)
meetle.l = (le‘𝐾)
meetle.m = (meet‘𝐾)
meetle.k (𝜑𝐾 ∈ Poset)
meetle.x (𝜑𝑋𝐵)
meetle.y (𝜑𝑌𝐵)
meetle.z (𝜑𝑍𝐵)
meetle.e (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
Assertion
Ref Expression
meetle (𝜑 → ((𝑍 𝑋𝑍 𝑌) ↔ 𝑍 (𝑋 𝑌)))

Proof of Theorem meetle
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 breq1 5055 . . . . 5 (𝑧 = 𝑍 → (𝑧 𝑋𝑍 𝑋))
2 breq1 5055 . . . . 5 (𝑧 = 𝑍 → (𝑧 𝑌𝑍 𝑌))
31, 2anbi12d 633 . . . 4 (𝑧 = 𝑍 → ((𝑧 𝑋𝑧 𝑌) ↔ (𝑍 𝑋𝑍 𝑌)))
4 breq1 5055 . . . 4 (𝑧 = 𝑍 → (𝑧 (𝑋 𝑌) ↔ 𝑍 (𝑋 𝑌)))
53, 4imbi12d 348 . . 3 (𝑧 = 𝑍 → (((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌)) ↔ ((𝑍 𝑋𝑍 𝑌) → 𝑍 (𝑋 𝑌))))
6 meetle.b . . . . 5 𝐵 = (Base‘𝐾)
7 meetle.l . . . . 5 = (le‘𝐾)
8 meetle.m . . . . 5 = (meet‘𝐾)
9 meetle.k . . . . 5 (𝜑𝐾 ∈ Poset)
10 meetle.x . . . . 5 (𝜑𝑋𝐵)
11 meetle.y . . . . 5 (𝜑𝑌𝐵)
12 meetle.e . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
136, 7, 8, 9, 10, 11, 12meetlem 17635 . . . 4 (𝜑 → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
1413simprd 499 . . 3 (𝜑 → ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌)))
15 meetle.z . . 3 (𝜑𝑍𝐵)
165, 14, 15rspcdva 3611 . 2 (𝜑 → ((𝑍 𝑋𝑍 𝑌) → 𝑍 (𝑋 𝑌)))
176, 7, 8, 9, 10, 11, 12lemeet1 17636 . . . 4 (𝜑 → (𝑋 𝑌) 𝑋)
186, 8, 9, 10, 11, 12meetcl 17630 . . . . 5 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
196, 7postr 17563 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑍𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑋𝐵)) → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑋) → 𝑍 𝑋))
209, 15, 18, 10, 19syl13anc 1369 . . . 4 (𝜑 → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑋) → 𝑍 𝑋))
2117, 20mpan2d 693 . . 3 (𝜑 → (𝑍 (𝑋 𝑌) → 𝑍 𝑋))
226, 7, 8, 9, 10, 11, 12lemeet2 17637 . . . 4 (𝜑 → (𝑋 𝑌) 𝑌)
236, 7postr 17563 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑍𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑌) → 𝑍 𝑌))
249, 15, 18, 11, 23syl13anc 1369 . . . 4 (𝜑 → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑌) → 𝑍 𝑌))
2522, 24mpan2d 693 . . 3 (𝜑 → (𝑍 (𝑋 𝑌) → 𝑍 𝑌))
2621, 25jcad 516 . 2 (𝜑 → (𝑍 (𝑋 𝑌) → (𝑍 𝑋𝑍 𝑌)))
2716, 26impbid 215 1 (𝜑 → ((𝑍 𝑋𝑍 𝑌) ↔ 𝑍 (𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  cop 4556   class class class wbr 5052  dom cdm 5542  cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572  Posetcpo 17550  meetcmee 17555
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-poset 17556  df-glb 17585  df-meet 17587
This theorem is referenced by:  latlem12  17688
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