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Theorem joinle 16995
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 (𝜑𝐾 ∈ Poset)
joinle.x (𝜑𝑋𝐵)
joinle.y (𝜑𝑌𝐵)
joinle.z (𝜑𝑍𝐵)
joinle.e (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
Assertion
Ref Expression
joinle (𝜑 → ((𝑋 𝑍𝑌 𝑍) ↔ (𝑋 𝑌) 𝑍))

Proof of Theorem joinle
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 joinle.z . . 3 (𝜑𝑍𝐵)
2 joinle.b . . . . 5 𝐵 = (Base‘𝐾)
3 joinle.l . . . . 5 = (le‘𝐾)
4 joinle.j . . . . 5 = (join‘𝐾)
5 joinle.k . . . . 5 (𝜑𝐾 ∈ Poset)
6 joinle.x . . . . 5 (𝜑𝑋𝐵)
7 joinle.y . . . . 5 (𝜑𝑌𝐵)
8 joinle.e . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
92, 3, 4, 5, 6, 7, 8joinlem 16992 . . . 4 (𝜑 → ((𝑋 (𝑋 𝑌) ∧ 𝑌 (𝑋 𝑌)) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧)))
109simprd 479 . . 3 (𝜑 → ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧))
11 breq2 4648 . . . . . 6 (𝑧 = 𝑍 → (𝑋 𝑧𝑋 𝑍))
12 breq2 4648 . . . . . 6 (𝑧 = 𝑍 → (𝑌 𝑧𝑌 𝑍))
1311, 12anbi12d 746 . . . . 5 (𝑧 = 𝑍 → ((𝑋 𝑧𝑌 𝑧) ↔ (𝑋 𝑍𝑌 𝑍)))
14 breq2 4648 . . . . 5 (𝑧 = 𝑍 → ((𝑋 𝑌) 𝑧 ↔ (𝑋 𝑌) 𝑍))
1513, 14imbi12d 334 . . . 4 (𝑧 = 𝑍 → (((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧) ↔ ((𝑋 𝑍𝑌 𝑍) → (𝑋 𝑌) 𝑍)))
1615rspcva 3302 . . 3 ((𝑍𝐵 ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧)) → ((𝑋 𝑍𝑌 𝑍) → (𝑋 𝑌) 𝑍))
171, 10, 16syl2anc 692 . 2 (𝜑 → ((𝑋 𝑍𝑌 𝑍) → (𝑋 𝑌) 𝑍))
182, 3, 4, 5, 6, 7, 8lejoin1 16993 . . . 4 (𝜑𝑋 (𝑋 𝑌))
192, 4, 5, 6, 7, 8joincl 16987 . . . . 5 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
202, 3postr 16934 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵)) → ((𝑋 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑋 𝑍))
215, 6, 19, 1, 20syl13anc 1326 . . . 4 (𝜑 → ((𝑋 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑋 𝑍))
2218, 21mpand 710 . . 3 (𝜑 → ((𝑋 𝑌) 𝑍𝑋 𝑍))
232, 3, 4, 5, 6, 7, 8lejoin2 16994 . . . 4 (𝜑𝑌 (𝑋 𝑌))
242, 3postr 16934 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑌𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵)) → ((𝑌 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑌 𝑍))
255, 7, 19, 1, 24syl13anc 1326 . . . 4 (𝜑 → ((𝑌 (𝑋 𝑌) ∧ (𝑋 𝑌) 𝑍) → 𝑌 𝑍))
2623, 25mpand 710 . . 3 (𝜑 → ((𝑋 𝑌) 𝑍𝑌 𝑍))
2722, 26jcad 555 . 2 (𝜑 → ((𝑋 𝑌) 𝑍 → (𝑋 𝑍𝑌 𝑍)))
2817, 27impbid 202 1 (𝜑 → ((𝑋 𝑍𝑌 𝑍) ↔ (𝑋 𝑌) 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  cop 4174   class class class wbr 4644  dom cdm 5104  cfv 5876  (class class class)co 6635  Basecbs 15838  lecple 15929  Posetcpo 16921  joincjn 16925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-poset 16927  df-lub 16955  df-join 16957
This theorem is referenced by:  latjle12  17043
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