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Theorem joinval2lem 18333
Description: Lemma for joinval2 18334 and joineu 18335. (Contributed by NM, 12-Sep-2018.) TODO: combine this through joineu 18335 into joinlem 18336?
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 5152 . . 3 (𝑦 = 𝑋 β†’ (𝑦 ≀ π‘₯ ↔ 𝑋 ≀ π‘₯))
2 breq1 5152 . . 3 (𝑦 = π‘Œ β†’ (𝑦 ≀ π‘₯ ↔ π‘Œ ≀ π‘₯))
31, 2ralprg 4699 . 2 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑦 ≀ π‘₯ ↔ (𝑋 ≀ π‘₯ ∧ π‘Œ ≀ π‘₯)))
4 breq1 5152 . . . . 5 (𝑦 = 𝑋 β†’ (𝑦 ≀ 𝑧 ↔ 𝑋 ≀ 𝑧))
5 breq1 5152 . . . . 5 (𝑦 = π‘Œ β†’ (𝑦 ≀ 𝑧 ↔ π‘Œ ≀ 𝑧))
64, 5ralprg 4699 . . . 4 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑦 ≀ 𝑧 ↔ (𝑋 ≀ 𝑧 ∧ π‘Œ ≀ 𝑧)))
76imbi1d 342 . . 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 3178 . 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 632 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {cpr 4631   class class class wbr 5149  β€˜cfv 6544  Basecbs 17144  lecple 17204  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-ext 2704
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150
This theorem is referenced by:  joinval2  18334  joineu  18335  joindm3  47602
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