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Theorem meetle 16952
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 meetle.z . . 3 (𝜑𝑍𝐵)
2 meetle.b . . . . 5 𝐵 = (Base‘𝐾)
3 meetle.l . . . . 5 = (le‘𝐾)
4 meetle.m . . . . 5 = (meet‘𝐾)
5 meetle.k . . . . 5 (𝜑𝐾 ∈ Poset)
6 meetle.x . . . . 5 (𝜑𝑋𝐵)
7 meetle.y . . . . 5 (𝜑𝑌𝐵)
8 meetle.e . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
92, 3, 4, 5, 6, 7, 8meetlem 16949 . . . 4 (𝜑 → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
109simprd 479 . . 3 (𝜑 → ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌)))
11 breq1 4618 . . . . . 6 (𝑧 = 𝑍 → (𝑧 𝑋𝑍 𝑋))
12 breq1 4618 . . . . . 6 (𝑧 = 𝑍 → (𝑧 𝑌𝑍 𝑌))
1311, 12anbi12d 746 . . . . 5 (𝑧 = 𝑍 → ((𝑧 𝑋𝑧 𝑌) ↔ (𝑍 𝑋𝑍 𝑌)))
14 breq1 4618 . . . . 5 (𝑧 = 𝑍 → (𝑧 (𝑋 𝑌) ↔ 𝑍 (𝑋 𝑌)))
1513, 14imbi12d 334 . . . 4 (𝑧 = 𝑍 → (((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌)) ↔ ((𝑍 𝑋𝑍 𝑌) → 𝑍 (𝑋 𝑌))))
1615rspcva 3293 . . 3 ((𝑍𝐵 ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))) → ((𝑍 𝑋𝑍 𝑌) → 𝑍 (𝑋 𝑌)))
171, 10, 16syl2anc 692 . 2 (𝜑 → ((𝑍 𝑋𝑍 𝑌) → 𝑍 (𝑋 𝑌)))
182, 3, 4, 5, 6, 7, 8lemeet1 16950 . . . 4 (𝜑 → (𝑋 𝑌) 𝑋)
192, 4, 5, 6, 7, 8meetcl 16944 . . . . 5 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
202, 3postr 16877 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑍𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑋𝐵)) → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑋) → 𝑍 𝑋))
215, 1, 19, 6, 20syl13anc 1325 . . . 4 (𝜑 → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑋) → 𝑍 𝑋))
2218, 21mpan2d 709 . . 3 (𝜑 → (𝑍 (𝑋 𝑌) → 𝑍 𝑋))
232, 3, 4, 5, 6, 7, 8lemeet2 16951 . . . 4 (𝜑 → (𝑋 𝑌) 𝑌)
242, 3postr 16877 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑍𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑌) → 𝑍 𝑌))
255, 1, 19, 7, 24syl13anc 1325 . . . 4 (𝜑 → ((𝑍 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑌) → 𝑍 𝑌))
2623, 25mpan2d 709 . . 3 (𝜑 → (𝑍 (𝑋 𝑌) → 𝑍 𝑌))
2722, 26jcad 555 . 2 (𝜑 → (𝑍 (𝑋 𝑌) → (𝑍 𝑋𝑍 𝑌)))
2817, 27impbid 202 1 (𝜑 → ((𝑍 𝑋𝑍 𝑌) ↔ 𝑍 (𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  cop 4156   class class class wbr 4615  dom cdm 5076  cfv 5849  (class class class)co 6607  Basecbs 15784  lecple 15872  Posetcpo 16864  meetcmee 16869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-poset 16870  df-glb 16899  df-meet 16901
This theorem is referenced by:  latlem12  17002
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