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Theorem meetle 18349
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 5150 . . . . 5 (𝑧 = 𝑍 → (𝑧 𝑋𝑍 𝑋))
2 breq1 5150 . . . . 5 (𝑧 = 𝑍 → (𝑧 𝑌𝑍 𝑌))
31, 2anbi12d 632 . . . 4 (𝑧 = 𝑍 → ((𝑧 𝑋𝑧 𝑌) ↔ (𝑍 𝑋𝑍 𝑌)))
4 breq1 5150 . . . 4 (𝑧 = 𝑍 → (𝑧 (𝑋 𝑌) ↔ 𝑍 (𝑋 𝑌)))
53, 4imbi12d 345 . . 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 18346 . . . 4 (𝜑 → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
1413simprd 497 . . 3 (𝜑 → ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌)))
15 meetle.z . . 3 (𝜑𝑍𝐵)
165, 14, 15rspcdva 3613 . 2 (𝜑 → ((𝑍 𝑋𝑍 𝑌) → 𝑍 (𝑋 𝑌)))
176, 7, 8, 9, 10, 11, 12lemeet1 18347 . . . 4 (𝜑 → (𝑋 𝑌) 𝑋)
186, 8, 9, 10, 11, 12meetcl 18341 . . . . 5 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
196, 7postr 18269 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑍𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑋𝐵)) → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑋) → 𝑍 𝑋))
209, 15, 18, 10, 19syl13anc 1373 . . . 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 18348 . . . 4 (𝜑 → (𝑋 𝑌) 𝑌)
236, 7postr 18269 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑍𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑌) → 𝑍 𝑌))
249, 15, 18, 11, 23syl13anc 1373 . . . 4 (𝜑 → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑌) → 𝑍 𝑌))
2522, 24mpan2d 693 . . 3 (𝜑 → (𝑍 (𝑋 𝑌) → 𝑍 𝑌))
2621, 25jcad 514 . 2 (𝜑 → (𝑍 (𝑋 𝑌) → (𝑍 𝑋𝑍 𝑌)))
2716, 26impbid 211 1 (𝜑 → ((𝑍 𝑋𝑍 𝑌) ↔ 𝑍 (𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  cop 4633   class class class wbr 5147  dom cdm 5675  cfv 6540  (class class class)co 7404  Basecbs 17140  lecple 17200  Posetcpo 18256  meetcmee 18261
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-poset 18262  df-glb 18296  df-meet 18298
This theorem is referenced by:  latlem12  18415
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