Theorem joinle 17616

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 5061	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋 ≤ 𝑧 ↔ 𝑋 ≤ 𝑍))
2		breq2 5061	. . . . 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 5061	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑋 ∨ 𝑌) ≤ 𝑧 ↔ (𝑋 ∨ 𝑌) ≤ 𝑍))
5	3, 4	imbi12d 347	. . 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 17613	. . . 4 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)))
14	13	simprd 498	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧))
15		joinle.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
16	5, 14, 15	rspcdva 3623	. 2 ⊢ (𝜑 → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) → (𝑋 ∨ 𝑌) ≤ 𝑍))
17	6, 7, 8, 9, 10, 11, 12	lejoin1 17614	. . . 4 ⊢ (𝜑 → 𝑋 ≤ (𝑋 ∨ 𝑌))
18	6, 8, 9, 10, 11, 12	joincl 17608	. . . . 5 ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ 𝐵)
19	6, 7	postr 17555	. . . . 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 1367	. . . 4 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑋 ≤ 𝑍))
21	17, 20	mpand 693	. . 3 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → 𝑋 ≤ 𝑍))
22	6, 7, 8, 9, 10, 11, 12	lejoin2 17615	. . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑋 ∨ 𝑌))
23	6, 7	postr 17555	. . . . 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 1367	. . . 4 ⊢ (𝜑 → ((𝑌 ≤ (𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑌) ≤ 𝑍) → 𝑌 ≤ 𝑍))
25	22, 24	mpand 693	. . 3 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → 𝑌 ≤ 𝑍))
26	21, 25	jcad 515	. 2 ⊢ (𝜑 → ((𝑋 ∨ 𝑌) ≤ 𝑍 → (𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍)))
27	16, 26	impbid 214	1 ⊢ (𝜑 → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) ↔ (𝑋 ∨ 𝑌) ≤ 𝑍))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1531   ∈ wcel 2108  ∀wral 3136  ⟨cop 4565   class class class wbr 5057  dom cdm 5548  ‘cfv 6348  (class class class)co 7148  Basecbs 16475  lecple 16564  Posetcpo 17542  joincjn 17546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-poset 17548  df-lub 17576  df-join 17578

This theorem is referenced by:  latjle12  17664