Theorem joinle 18307

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 5102	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋 ≤ 𝑧 ↔ 𝑋 ≤ 𝑍))
2		breq2 5102	. . . . 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 5102	. . . 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 18304	. . . 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 3577	. 2 ⊢ (𝜑 → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) → (𝑋 ∨ 𝑌) ≤ 𝑍))
17	6, 7, 8, 9, 10, 11, 12	lejoin1 18305	. . . 4 ⊢ (𝜑 → 𝑋 ≤ (𝑋 ∨ 𝑌))
18	6, 8, 9, 10, 11, 12	joincl 18299	. . . . 5 ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ 𝐵)
19	6, 7	postr 18243	. . . . 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 18306	. . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑋 ∨ 𝑌))
23	6, 7	postr 18243	. . . . 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 1541   ∈ wcel 2113  ∀wral 3051  ⟨cop 4586   class class class wbr 5098  dom cdm 5624  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  lecple 17184  Posetcpo 18230  joincjn 18234

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-poset 18236  df-lub 18267  df-join 18269

This theorem is referenced by:  latjle12  18373