Theorem meetle 18453

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	⊢ (𝜑 → 𝐾 ∈ Poset)
meetle.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
meetle.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
meetle.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
meetle.e	⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∧ )

Assertion

Ref	Expression
meetle	⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 ≤ (𝑋 ∧ 𝑌)))

Proof of Theorem meetle

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5116	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ 𝑋 ↔ 𝑍 ≤ 𝑋))
2		breq1 5116	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ 𝑌 ↔ 𝑍 ≤ 𝑌))
3	1, 2	anbi12d 643	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) ↔ (𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌)))
4		breq1 5116	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ (𝑋 ∧ 𝑌) ↔ 𝑍 ≤ (𝑋 ∧ 𝑌)))
5	3, 4	imbi12d 347	. . 3 ⊢ (𝑧 = 𝑍 → (((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌)) ↔ ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 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 ⊢ (𝜑 → 𝐾 ∈ Poset)
10		meetle.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11		meetle.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
12		meetle.e	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∧ )
13	6, 7, 8, 9, 10, 11, 12	meetlem 18450	. . . 4 ⊢ (𝜑 → (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌))))
14	13	simprd 500	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌)))
15		meetle.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
16	5, 14, 15	rspcdva 3591	. 2 ⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) → 𝑍 ≤ (𝑋 ∧ 𝑌)))
17	6, 7, 8, 9, 10, 11, 12	lemeet1 18451	. . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑋)
18	6, 8, 9, 10, 11, 12	meetcl 18445	. . . . 5 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ∈ 𝐵)
19	6, 7	postr 18375	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑍 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑋) → 𝑍 ≤ 𝑋))
20	9, 15, 18, 10, 19	syl13anc 1397	. . . 4 ⊢ (𝜑 → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑋) → 𝑍 ≤ 𝑋))
21	17, 20	mpan2d 706	. . 3 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → 𝑍 ≤ 𝑋))
22	6, 7, 8, 9, 10, 11, 12	lemeet2 18452	. . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑌)
23	6, 7	postr 18375	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑍 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) → 𝑍 ≤ 𝑌))
24	9, 15, 18, 11, 23	syl13anc 1397	. . . 4 ⊢ (𝜑 → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) → 𝑍 ≤ 𝑌))
25	22, 24	mpan2d 706	. . 3 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → 𝑍 ≤ 𝑌))
26	21, 25	jcad 521	. 2 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → (𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌)))
27	16, 26	impbid 215	1 ⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 ≤ (𝑋 ∧ 𝑌)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ⟨cop 4600   class class class wbr 5113  dom cdm 5662  ‘cfv 6537  (class class class)co 7411  Basecbs 17268  lecple 17316  Posetcpo 18362  meetcmee 18367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-poset 18368  df-glb 18400  df-meet 18402

This theorem is referenced by:  latlem12  18521