MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  meetle Structured version   Visualization version   GIF version

Theorem meetle 18358
Description: A meet 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.) (Revised by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
meetle.b 𝐡 = (Baseβ€˜πΎ)
meetle.l ≀ = (leβ€˜πΎ)
meetle.m ∧ = (meetβ€˜πΎ)
meetle.k (πœ‘ β†’ 𝐾 ∈ Poset)
meetle.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
meetle.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
meetle.z (πœ‘ β†’ 𝑍 ∈ 𝐡)
meetle.e (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ )
Assertion
Ref Expression
meetle (πœ‘ β†’ ((𝑍 ≀ 𝑋 ∧ 𝑍 ≀ π‘Œ) ↔ 𝑍 ≀ (𝑋 ∧ π‘Œ)))

Proof of Theorem meetle
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 breq1 5151 . . . . 5 (𝑧 = 𝑍 β†’ (𝑧 ≀ 𝑋 ↔ 𝑍 ≀ 𝑋))
2 breq1 5151 . . . . 5 (𝑧 = 𝑍 β†’ (𝑧 ≀ π‘Œ ↔ 𝑍 ≀ π‘Œ))
31, 2anbi12d 630 . . . 4 (𝑧 = 𝑍 β†’ ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) ↔ (𝑍 ≀ 𝑋 ∧ 𝑍 ≀ π‘Œ)))
4 breq1 5151 . . . 4 (𝑧 = 𝑍 β†’ (𝑧 ≀ (𝑋 ∧ π‘Œ) ↔ 𝑍 ≀ (𝑋 ∧ π‘Œ)))
53, 4imbi12d 344 . . 3 (𝑧 = 𝑍 β†’ (((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ (𝑋 ∧ π‘Œ)) ↔ ((𝑍 ≀ 𝑋 ∧ 𝑍 ≀ π‘Œ) β†’ 𝑍 ≀ (𝑋 ∧ π‘Œ))))
6 meetle.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
7 meetle.l . . . . 5 ≀ = (leβ€˜πΎ)
8 meetle.m . . . . 5 ∧ = (meetβ€˜πΎ)
9 meetle.k . . . . 5 (πœ‘ β†’ 𝐾 ∈ Poset)
10 meetle.x . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝐡)
11 meetle.y . . . . 5 (πœ‘ β†’ π‘Œ ∈ 𝐡)
12 meetle.e . . . . 5 (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ )
136, 7, 8, 9, 10, 11, 12meetlem 18355 . . . 4 (πœ‘ β†’ (((𝑋 ∧ π‘Œ) ≀ 𝑋 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ (𝑋 ∧ π‘Œ))))
1413simprd 495 . . 3 (πœ‘ β†’ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ (𝑋 ∧ π‘Œ)))
15 meetle.z . . 3 (πœ‘ β†’ 𝑍 ∈ 𝐡)
165, 14, 15rspcdva 3613 . 2 (πœ‘ β†’ ((𝑍 ≀ 𝑋 ∧ 𝑍 ≀ π‘Œ) β†’ 𝑍 ≀ (𝑋 ∧ π‘Œ)))
176, 7, 8, 9, 10, 11, 12lemeet1 18356 . . . 4 (πœ‘ β†’ (𝑋 ∧ π‘Œ) ≀ 𝑋)
186, 8, 9, 10, 11, 12meetcl 18350 . . . . 5 (πœ‘ β†’ (𝑋 ∧ π‘Œ) ∈ 𝐡)
196, 7postr 18278 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑍 ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡)) β†’ ((𝑍 ≀ (𝑋 ∧ π‘Œ) ∧ (𝑋 ∧ π‘Œ) ≀ 𝑋) β†’ 𝑍 ≀ 𝑋))
209, 15, 18, 10, 19syl13anc 1371 . . . 4 (πœ‘ β†’ ((𝑍 ≀ (𝑋 ∧ π‘Œ) ∧ (𝑋 ∧ π‘Œ) ≀ 𝑋) β†’ 𝑍 ≀ 𝑋))
2117, 20mpan2d 691 . . 3 (πœ‘ β†’ (𝑍 ≀ (𝑋 ∧ π‘Œ) β†’ 𝑍 ≀ 𝑋))
226, 7, 8, 9, 10, 11, 12lemeet2 18357 . . . 4 (πœ‘ β†’ (𝑋 ∧ π‘Œ) ≀ π‘Œ)
236, 7postr 18278 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑍 ∈ 𝐡 ∧ (𝑋 ∧ π‘Œ) ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ((𝑍 ≀ (𝑋 ∧ π‘Œ) ∧ (𝑋 ∧ π‘Œ) ≀ π‘Œ) β†’ 𝑍 ≀ π‘Œ))
249, 15, 18, 11, 23syl13anc 1371 . . . 4 (πœ‘ β†’ ((𝑍 ≀ (𝑋 ∧ π‘Œ) ∧ (𝑋 ∧ π‘Œ) ≀ π‘Œ) β†’ 𝑍 ≀ π‘Œ))
2522, 24mpan2d 691 . . 3 (πœ‘ β†’ (𝑍 ≀ (𝑋 ∧ π‘Œ) β†’ 𝑍 ≀ π‘Œ))
2621, 25jcad 512 . 2 (πœ‘ β†’ (𝑍 ≀ (𝑋 ∧ π‘Œ) β†’ (𝑍 ≀ 𝑋 ∧ 𝑍 ≀ π‘Œ)))
2716, 26impbid 211 1 (πœ‘ β†’ ((𝑍 ≀ 𝑋 ∧ 𝑍 ≀ π‘Œ) ↔ 𝑍 ≀ (𝑋 ∧ π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βŸ¨cop 4634   class class class wbr 5148  dom cdm 5676  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149  lecple 17209  Posetcpo 18265  meetcmee 18270
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-poset 18271  df-glb 18305  df-meet 18307
This theorem is referenced by:  latlem12  18424
  Copyright terms: Public domain W3C validator