Theorem meetle 18304

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 5092	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ 𝑋 ↔ 𝑍 ≤ 𝑋))
2		breq1 5092	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ 𝑌 ↔ 𝑍 ≤ 𝑌))
3	1, 2	anbi12d 632	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) ↔ (𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌)))
4		breq1 5092	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ (𝑋 ∧ 𝑌) ↔ 𝑍 ≤ (𝑋 ∧ 𝑌)))
5	3, 4	imbi12d 344	. . 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 18301	. . . 4 ⊢ (𝜑 → (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌))))
14	13	simprd 495	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌)))
15		meetle.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
16	5, 14, 15	rspcdva 3573	. 2 ⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) → 𝑍 ≤ (𝑋 ∧ 𝑌)))
17	6, 7, 8, 9, 10, 11, 12	lemeet1 18302	. . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑋)
18	6, 8, 9, 10, 11, 12	meetcl 18296	. . . . 5 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ∈ 𝐵)
19	6, 7	postr 18226	. . . . 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 1374	. . . 4 ⊢ (𝜑 → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑋) → 𝑍 ≤ 𝑋))
21	17, 20	mpan2d 694	. . 3 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → 𝑍 ≤ 𝑋))
22	6, 7, 8, 9, 10, 11, 12	lemeet2 18303	. . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑌)
23	6, 7	postr 18226	. . . . 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 1374	. . . 4 ⊢ (𝜑 → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) → 𝑍 ≤ 𝑌))
25	22, 24	mpan2d 694	. . 3 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → 𝑍 ≤ 𝑌))
26	21, 25	jcad 512	. 2 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → (𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌)))
27	16, 26	impbid 212	1 ⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 ≤ (𝑋 ∧ 𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ⟨cop 4579   class class class wbr 5089  dom cdm 5614  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  lecple 17168  Posetcpo 18213  meetcmee 18218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-poset 18219  df-glb 18251  df-meet 18253

This theorem is referenced by:  latlem12  18372