Theorem joinle 18345

Description: A join 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.)

Hypotheses

Ref	Expression
joinle.b	⊢ 𝐵 = (Base‘𝐾)
joinle.l	⊢ ≤ = (le‘𝐾)
joinle.j	⊢ ∨ = (join‘𝐾)
joinle.k	⊢ (𝜑 → 𝐾 ∈ Poset)
joinle.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
joinle.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
joinle.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
joinle.e	⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )

Assertion

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

Proof of Theorem joinle

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5111	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋 ≤ 𝑧 ↔ 𝑋 ≤ 𝑍))
2		breq2 5111	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑌 ≤ 𝑧 ↔ 𝑌 ≤ 𝑍))
3	1, 2	anbi12d 632	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) ↔ (𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍)))
4		breq2 5111	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑋 ∨ 𝑌) ≤ 𝑧 ↔ (𝑋 ∨ 𝑌) ≤ 𝑍))
5	3, 4	imbi12d 344	. . 3 ⊢ (𝑧 = 𝑍 → (((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) → (𝑋 ∨ 𝑌) ≤ 𝑍)))
6		joinle.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
7		joinle.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
8		joinle.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
9		joinle.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Poset)
10		joinle.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11		joinle.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
12		joinle.e	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )
13	6, 7, 8, 9, 10, 11, 12	joinlem 18342	. . . 4 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)))
14	13	simprd 495	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧))
15		joinle.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
16	5, 14, 15	rspcdva 3589	. 2 ⊢ (𝜑 → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) → (𝑋 ∨ 𝑌) ≤ 𝑍))
17	6, 7, 8, 9, 10, 11, 12	lejoin1 18343	. . . 4 ⊢ (𝜑 → 𝑋 ≤ (𝑋 ∨ 𝑌))
18	6, 8, 9, 10, 11, 12	joincl 18337	. . . . 5 ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ 𝐵)
19	6, 7	postr 18281	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑋 ≤ 𝑍))
20	9, 10, 18, 15, 19	syl13anc 1374	. . . 4 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑋 ≤ 𝑍))
21	17, 20	mpand 695	. . 3 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → 𝑋 ≤ 𝑍))
22	6, 7, 8, 9, 10, 11, 12	lejoin2 18344	. . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑋 ∨ 𝑌))
23	6, 7	postr 18281	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑌 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑌 ≤ 𝑍))
24	9, 11, 18, 15, 23	syl13anc 1374	. . . 4 ⊢ (𝜑 → ((𝑌 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑌 ≤ 𝑍))
25	22, 24	mpand 695	. . 3 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → 𝑌 ≤ 𝑍))
26	21, 25	jcad 512	. 2 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → (𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍)))
27	16, 26	impbid 212	1 ⊢ (𝜑 → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) ↔ (𝑋 ∨ 𝑌) ≤ 𝑍))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ⟨cop 4595   class class class wbr 5107  dom cdm 5638  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  lecple 17227  Posetcpo 18268  joincjn 18272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-poset 18274  df-lub 18305  df-join 18307

This theorem is referenced by:  latjle12  18409