MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  joinval2lem Structured version   Visualization version   GIF version

Theorem joinval2lem 18332
Description: Lemma for joinval2 18333 and joineu 18334. (Contributed by NM, 12-Sep-2018.) TODO: combine this through joineu 18334 into joinlem 18335?
Hypotheses
Ref Expression
joinval2.b 𝐡 = (Baseβ€˜πΎ)
joinval2.l ≀ = (leβ€˜πΎ)
joinval2.j ∨ = (joinβ€˜πΎ)
joinval2.k (πœ‘ β†’ 𝐾 ∈ 𝑉)
joinval2.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
joinval2.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
Assertion
Ref Expression
joinval2lem ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑦 ≀ π‘₯ ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧)) ↔ ((𝑋 ≀ π‘₯ ∧ π‘Œ ≀ π‘₯) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧) β†’ π‘₯ ≀ 𝑧))))
Distinct variable groups:   π‘₯,𝑧,𝐡   π‘₯, ∨ ,𝑧   π‘₯,𝑦,𝐾,𝑧   𝑦, ≀   π‘₯,𝑋,𝑦,𝑧   π‘₯,π‘Œ,𝑦,𝑧
Allowed substitution hints:   πœ‘(π‘₯,𝑦,𝑧)   𝐡(𝑦)   ∨ (𝑦)   ≀ (π‘₯,𝑧)   𝑉(π‘₯,𝑦,𝑧)

Proof of Theorem joinval2lem
StepHypRef Expression
1 breq1 5151 . . 3 (𝑦 = 𝑋 β†’ (𝑦 ≀ π‘₯ ↔ 𝑋 ≀ π‘₯))
2 breq1 5151 . . 3 (𝑦 = π‘Œ β†’ (𝑦 ≀ π‘₯ ↔ π‘Œ ≀ π‘₯))
31, 2ralprg 4698 . 2 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑦 ≀ π‘₯ ↔ (𝑋 ≀ π‘₯ ∧ π‘Œ ≀ π‘₯)))
4 breq1 5151 . . . . 5 (𝑦 = 𝑋 β†’ (𝑦 ≀ 𝑧 ↔ 𝑋 ≀ 𝑧))
5 breq1 5151 . . . . 5 (𝑦 = π‘Œ β†’ (𝑦 ≀ 𝑧 ↔ π‘Œ ≀ 𝑧))
64, 5ralprg 4698 . . . 4 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑦 ≀ 𝑧 ↔ (𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧)))
76imbi1d 341 . . 3 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧) ↔ ((𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧) β†’ π‘₯ ≀ 𝑧)))
87ralbidv 3177 . 2 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧) ↔ βˆ€π‘§ ∈ 𝐡 ((𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧) β†’ π‘₯ ≀ 𝑧)))
93, 8anbi12d 631 1 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑦 ≀ π‘₯ ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑦 ≀ 𝑧 β†’ π‘₯ ≀ 𝑧)) ↔ ((𝑋 ≀ π‘₯ ∧ π‘Œ ≀ π‘₯) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧) β†’ π‘₯ ≀ 𝑧))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {cpr 4630   class class class wbr 5148  β€˜cfv 6543  Basecbs 17143  lecple 17203  joincjn 18263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149
This theorem is referenced by:  joinval2  18333  joineu  18334  joindm3  47592
  Copyright terms: Public domain W3C validator