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Theorem joinle 18341
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 5142 . . . . 5 (𝑧 = 𝑍 β†’ (𝑋 ≀ 𝑧 ↔ 𝑋 ≀ 𝑍))
2 breq2 5142 . . . . 5 (𝑧 = 𝑍 β†’ (π‘Œ ≀ 𝑧 ↔ π‘Œ ≀ 𝑍))
31, 2anbi12d 630 . . . 4 (𝑧 = 𝑍 β†’ ((𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧) ↔ (𝑋 ≀ 𝑍 ∧ π‘Œ ≀ 𝑍)))
4 breq2 5142 . . . 4 (𝑧 = 𝑍 β†’ ((𝑋 ∨ π‘Œ) ≀ 𝑧 ↔ (𝑋 ∨ π‘Œ) ≀ 𝑍))
53, 4imbi12d 344 . . 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 18338 . . . 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 495 . . 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 3605 . 2 (πœ‘ β†’ ((𝑋 ≀ 𝑍 ∧ π‘Œ ≀ 𝑍) β†’ (𝑋 ∨ π‘Œ) ≀ 𝑍))
176, 7, 8, 9, 10, 11, 12lejoin1 18339 . . . 4 (πœ‘ β†’ 𝑋 ≀ (𝑋 ∨ π‘Œ))
186, 8, 9, 10, 11, 12joincl 18333 . . . . 5 (πœ‘ β†’ (𝑋 ∨ π‘Œ) ∈ 𝐡)
196, 7postr 18275 . . . . 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 1369 . . . 4 (πœ‘ β†’ ((𝑋 ≀ (𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ π‘Œ) ≀ 𝑍) β†’ 𝑋 ≀ 𝑍))
2117, 20mpand 692 . . 3 (πœ‘ β†’ ((𝑋 ∨ π‘Œ) ≀ 𝑍 β†’ 𝑋 ≀ 𝑍))
226, 7, 8, 9, 10, 11, 12lejoin2 18340 . . . 4 (πœ‘ β†’ π‘Œ ≀ (𝑋 ∨ π‘Œ))
236, 7postr 18275 . . . . 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 1369 . . . 4 (πœ‘ β†’ ((π‘Œ ≀ (𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ π‘Œ) ≀ 𝑍) β†’ π‘Œ ≀ 𝑍))
2522, 24mpand 692 . . 3 (πœ‘ β†’ ((𝑋 ∨ π‘Œ) ≀ 𝑍 β†’ π‘Œ ≀ 𝑍))
2621, 25jcad 512 . 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 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βŸ¨cop 4626   class class class wbr 5138  dom cdm 5666  β€˜cfv 6533  (class class class)co 7401  Basecbs 17143  lecple 17203  Posetcpo 18262  joincjn 18266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-poset 18268  df-lub 18301  df-join 18303
This theorem is referenced by:  latjle12  18405
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