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Theorem meetle 17343
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 4846 . . . . 5 (𝑧 = 𝑍 → (𝑧 𝑋𝑍 𝑋))
2 breq1 4846 . . . . 5 (𝑧 = 𝑍 → (𝑧 𝑌𝑍 𝑌))
31, 2anbi12d 625 . . . 4 (𝑧 = 𝑍 → ((𝑧 𝑋𝑧 𝑌) ↔ (𝑍 𝑋𝑍 𝑌)))
4 breq1 4846 . . . 4 (𝑧 = 𝑍 → (𝑧 (𝑋 𝑌) ↔ 𝑍 (𝑋 𝑌)))
53, 4imbi12d 336 . . 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 17340 . . . 4 (𝜑 → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
1413simprd 490 . . 3 (𝜑 → ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌)))
15 meetle.z . . 3 (𝜑𝑍𝐵)
165, 14, 15rspcdva 3503 . 2 (𝜑 → ((𝑍 𝑋𝑍 𝑌) → 𝑍 (𝑋 𝑌)))
176, 7, 8, 9, 10, 11, 12lemeet1 17341 . . . 4 (𝜑 → (𝑋 𝑌) 𝑋)
186, 8, 9, 10, 11, 12meetcl 17335 . . . . 5 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
196, 7postr 17268 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑍𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑋𝐵)) → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑋) → 𝑍 𝑋))
209, 15, 18, 10, 19syl13anc 1492 . . . 4 (𝜑 → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑋) → 𝑍 𝑋))
2117, 20mpan2d 686 . . 3 (𝜑 → (𝑍 (𝑋 𝑌) → 𝑍 𝑋))
226, 7, 8, 9, 10, 11, 12lemeet2 17342 . . . 4 (𝜑 → (𝑋 𝑌) 𝑌)
236, 7postr 17268 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑍𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑌) → 𝑍 𝑌))
249, 15, 18, 11, 23syl13anc 1492 . . . 4 (𝜑 → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑌) → 𝑍 𝑌))
2522, 24mpan2d 686 . . 3 (𝜑 → (𝑍 (𝑋 𝑌) → 𝑍 𝑌))
2621, 25jcad 509 . 2 (𝜑 → (𝑍 (𝑋 𝑌) → (𝑍 𝑋𝑍 𝑌)))
2716, 26impbid 204 1 (𝜑 → ((𝑍 𝑋𝑍 𝑌) ↔ 𝑍 (𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  cop 4374   class class class wbr 4843  dom cdm 5312  cfv 6101  (class class class)co 6878  Basecbs 16184  lecple 16274  Posetcpo 17255  meetcmee 17260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-poset 17261  df-glb 17290  df-meet 17292
This theorem is referenced by:  latlem12  17393
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