Theorem meetle 17028

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		meetle.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
2		meetle.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
3		meetle.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		meetle.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
5		meetle.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Poset)
6		meetle.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		meetle.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8		meetle.e	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∧ )
9	2, 3, 4, 5, 6, 7, 8	meetlem 17025	. . . 4 ⊢ (𝜑 → (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌))))
10	9	simprd 479	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌)))
11		breq1 4656	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ 𝑋 ↔ 𝑍 ≤ 𝑋))
12		breq1 4656	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ 𝑌 ↔ 𝑍 ≤ 𝑌))
13	11, 12	anbi12d 747	. . . . 5 ⊢ (𝑧 = 𝑍 → ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) ↔ (𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌)))
14		breq1 4656	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 ≤ (𝑋 ∧ 𝑌) ↔ 𝑍 ≤ (𝑋 ∧ 𝑌)))
15	13, 14	imbi12d 334	. . . 4 ⊢ (𝑧 = 𝑍 → (((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌)) ↔ ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) → 𝑍 ≤ (𝑋 ∧ 𝑌))))
16	15	rspcva 3307	. . 3 ⊢ ((𝑍 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌))) → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) → 𝑍 ≤ (𝑋 ∧ 𝑌)))
17	1, 10, 16	syl2anc 693	. 2 ⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) → 𝑍 ≤ (𝑋 ∧ 𝑌)))
18	2, 3, 4, 5, 6, 7, 8	lemeet1 17026	. . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑋)
19	2, 4, 5, 6, 7, 8	meetcl 17020	. . . . 5 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ∈ 𝐵)
20	2, 3	postr 16953	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑍 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑋) → 𝑍 ≤ 𝑋))
21	5, 1, 19, 6, 20	syl13anc 1328	. . . 4 ⊢ (𝜑 → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑋) → 𝑍 ≤ 𝑋))
22	18, 21	mpan2d 710	. . 3 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → 𝑍 ≤ 𝑋))
23	2, 3, 4, 5, 6, 7, 8	lemeet2 17027	. . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑌)
24	2, 3	postr 16953	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑍 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) → 𝑍 ≤ 𝑌))
25	5, 1, 19, 7, 24	syl13anc 1328	. . . 4 ⊢ (𝜑 → ((𝑍 ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) → 𝑍 ≤ 𝑌))
26	23, 25	mpan2d 710	. . 3 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → 𝑍 ≤ 𝑌))
27	22, 26	jcad 555	. 2 ⊢ (𝜑 → (𝑍 ≤ (𝑋 ∧ 𝑌) → (𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌)))
28	17, 27	impbid 202	1 ⊢ (𝜑 → ((𝑍 ≤ 𝑋 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 ≤ (𝑋 ∧ 𝑌)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ⟨cop 4183   class class class wbr 4653  dom cdm 5114  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  lecple 15948  Posetcpo 16940  meetcmee 16945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-poset 16946  df-glb 16975  df-meet 16977

This theorem is referenced by:  latlem12  17078