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Theorem meetle 17229
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 4807 . . . . 5 (𝑧 = 𝑍 → (𝑧 𝑋𝑍 𝑋))
2 breq1 4807 . . . . 5 (𝑧 = 𝑍 → (𝑧 𝑌𝑍 𝑌))
31, 2anbi12d 749 . . . 4 (𝑧 = 𝑍 → ((𝑧 𝑋𝑧 𝑌) ↔ (𝑍 𝑋𝑍 𝑌)))
4 breq1 4807 . . . 4 (𝑧 = 𝑍 → (𝑧 (𝑋 𝑌) ↔ 𝑍 (𝑋 𝑌)))
53, 4imbi12d 333 . . 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 17226 . . . 4 (𝜑 → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
1413simprd 482 . . 3 (𝜑 → ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌)))
15 meetle.z . . 3 (𝜑𝑍𝐵)
165, 14, 15rspcdva 3455 . 2 (𝜑 → ((𝑍 𝑋𝑍 𝑌) → 𝑍 (𝑋 𝑌)))
176, 7, 8, 9, 10, 11, 12lemeet1 17227 . . . 4 (𝜑 → (𝑋 𝑌) 𝑋)
186, 8, 9, 10, 11, 12meetcl 17221 . . . . 5 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
196, 7postr 17154 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑍𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑋𝐵)) → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑋) → 𝑍 𝑋))
209, 15, 18, 10, 19syl13anc 1479 . . . 4 (𝜑 → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑋) → 𝑍 𝑋))
2117, 20mpan2d 712 . . 3 (𝜑 → (𝑍 (𝑋 𝑌) → 𝑍 𝑋))
226, 7, 8, 9, 10, 11, 12lemeet2 17228 . . . 4 (𝜑 → (𝑋 𝑌) 𝑌)
236, 7postr 17154 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑍𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑌) → 𝑍 𝑌))
249, 15, 18, 11, 23syl13anc 1479 . . . 4 (𝜑 → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑌) → 𝑍 𝑌))
2522, 24mpan2d 712 . . 3 (𝜑 → (𝑍 (𝑋 𝑌) → 𝑍 𝑌))
2621, 25jcad 556 . 2 (𝜑 → (𝑍 (𝑋 𝑌) → (𝑍 𝑋𝑍 𝑌)))
2716, 26impbid 202 1 (𝜑 → ((𝑍 𝑋𝑍 𝑌) ↔ 𝑍 (𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  cop 4327   class class class wbr 4804  dom cdm 5266  cfv 6049  (class class class)co 6813  Basecbs 16059  lecple 16150  Posetcpo 17141  meetcmee 17146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-poset 17147  df-glb 17176  df-meet 17178
This theorem is referenced by:  latlem12  17279
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