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Theorem joinle 18339
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 𝐡 = (Baseβ€˜πΎ)
joinle.l ≀ = (leβ€˜πΎ)
joinle.j ∨ = (joinβ€˜πΎ)
joinle.k (πœ‘ β†’ 𝐾 ∈ Poset)
joinle.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
joinle.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
joinle.z (πœ‘ β†’ 𝑍 ∈ 𝐡)
joinle.e (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ )
Assertion
Ref Expression
joinle (πœ‘ β†’ ((𝑋 ≀ 𝑍 ∧ π‘Œ ≀ 𝑍) ↔ (𝑋 ∨ π‘Œ) ≀ 𝑍))

Proof of Theorem joinle
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 breq2 5153 . . . . 5 (𝑧 = 𝑍 β†’ (𝑋 ≀ 𝑧 ↔ 𝑋 ≀ 𝑍))
2 breq2 5153 . . . . 5 (𝑧 = 𝑍 β†’ (π‘Œ ≀ 𝑧 ↔ π‘Œ ≀ 𝑍))
31, 2anbi12d 632 . . . 4 (𝑧 = 𝑍 β†’ ((𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧) ↔ (𝑋 ≀ 𝑍 ∧ π‘Œ ≀ 𝑍)))
4 breq2 5153 . . . 4 (𝑧 = 𝑍 β†’ ((𝑋 ∨ π‘Œ) ≀ 𝑧 ↔ (𝑋 ∨ π‘Œ) ≀ 𝑍))
53, 4imbi12d 345 . . 3 (𝑧 = 𝑍 β†’ (((𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧) β†’ (𝑋 ∨ π‘Œ) ≀ 𝑧) ↔ ((𝑋 ≀ 𝑍 ∧ π‘Œ ≀ 𝑍) β†’ (𝑋 ∨ π‘Œ) ≀ 𝑍)))
6 joinle.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
7 joinle.l . . . . 5 ≀ = (leβ€˜πΎ)
8 joinle.j . . . . 5 ∨ = (joinβ€˜πΎ)
9 joinle.k . . . . 5 (πœ‘ β†’ 𝐾 ∈ Poset)
10 joinle.x . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝐡)
11 joinle.y . . . . 5 (πœ‘ β†’ π‘Œ ∈ 𝐡)
12 joinle.e . . . . 5 (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ )
136, 7, 8, 9, 10, 11, 12joinlem 18336 . . . 4 (πœ‘ β†’ ((𝑋 ≀ (𝑋 ∨ π‘Œ) ∧ π‘Œ ≀ (𝑋 ∨ π‘Œ)) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧) β†’ (𝑋 ∨ π‘Œ) ≀ 𝑧)))
1413simprd 497 . . 3 (πœ‘ β†’ βˆ€π‘§ ∈ 𝐡 ((𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧) β†’ (𝑋 ∨ π‘Œ) ≀ 𝑧))
15 joinle.z . . 3 (πœ‘ β†’ 𝑍 ∈ 𝐡)
165, 14, 15rspcdva 3614 . 2 (πœ‘ β†’ ((𝑋 ≀ 𝑍 ∧ π‘Œ ≀ 𝑍) β†’ (𝑋 ∨ π‘Œ) ≀ 𝑍))
176, 7, 8, 9, 10, 11, 12lejoin1 18337 . . . 4 (πœ‘ β†’ 𝑋 ≀ (𝑋 ∨ π‘Œ))
186, 8, 9, 10, 11, 12joincl 18331 . . . . 5 (πœ‘ β†’ (𝑋 ∨ π‘Œ) ∈ 𝐡)
196, 7postr 18273 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ (𝑋 ∨ π‘Œ) ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 ≀ (𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ π‘Œ) ≀ 𝑍) β†’ 𝑋 ≀ 𝑍))
209, 10, 18, 15, 19syl13anc 1373 . . . 4 (πœ‘ β†’ ((𝑋 ≀ (𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ π‘Œ) ≀ 𝑍) β†’ 𝑋 ≀ 𝑍))
2117, 20mpand 694 . . 3 (πœ‘ β†’ ((𝑋 ∨ π‘Œ) ≀ 𝑍 β†’ 𝑋 ≀ 𝑍))
226, 7, 8, 9, 10, 11, 12lejoin2 18338 . . . 4 (πœ‘ β†’ π‘Œ ≀ (𝑋 ∨ π‘Œ))
236, 7postr 18273 . . . . 5 ((𝐾 ∈ Poset ∧ (π‘Œ ∈ 𝐡 ∧ (𝑋 ∨ π‘Œ) ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((π‘Œ ≀ (𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ π‘Œ) ≀ 𝑍) β†’ π‘Œ ≀ 𝑍))
249, 11, 18, 15, 23syl13anc 1373 . . . 4 (πœ‘ β†’ ((π‘Œ ≀ (𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ π‘Œ) ≀ 𝑍) β†’ π‘Œ ≀ 𝑍))
2522, 24mpand 694 . . 3 (πœ‘ β†’ ((𝑋 ∨ π‘Œ) ≀ 𝑍 β†’ π‘Œ ≀ 𝑍))
2621, 25jcad 514 . 2 (πœ‘ β†’ ((𝑋 ∨ π‘Œ) ≀ 𝑍 β†’ (𝑋 ≀ 𝑍 ∧ π‘Œ ≀ 𝑍)))
2716, 26impbid 211 1 (πœ‘ β†’ ((𝑋 ≀ 𝑍 ∧ π‘Œ ≀ 𝑍) ↔ (𝑋 ∨ π‘Œ) ≀ 𝑍))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βŸ¨cop 4635   class class class wbr 5149  dom cdm 5677  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  Posetcpo 18260  joincjn 18264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-poset 18266  df-lub 18299  df-join 18301
This theorem is referenced by:  latjle12  18403
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