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

Theorem joinval2lem 17606
Description: Lemma for joinval2 17607 and joineu 17608. (Contributed by NM, 12-Sep-2018.) TODO: combine this through joineu 17608 into joinlem 17609?
Hypotheses
Ref Expression
joinval2.b 𝐵 = (Base‘𝐾)
joinval2.l = (le‘𝐾)
joinval2.j = (join‘𝐾)
joinval2.k (𝜑𝐾𝑉)
joinval2.x (𝜑𝑋𝐵)
joinval2.y (𝜑𝑌𝐵)
Assertion
Ref Expression
joinval2lem ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)) ↔ ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
Distinct variable groups:   𝑥,𝑧,𝐵   𝑥, ,𝑧   𝑥,𝑦,𝐾,𝑧   𝑦,   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐵(𝑦)   (𝑦)   (𝑥,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem joinval2lem
StepHypRef Expression
1 breq1 5060 . . 3 (𝑦 = 𝑋 → (𝑦 𝑥𝑋 𝑥))
2 breq1 5060 . . 3 (𝑦 = 𝑌 → (𝑦 𝑥𝑌 𝑥))
31, 2ralprg 4624 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ↔ (𝑋 𝑥𝑌 𝑥)))
4 breq1 5060 . . . . 5 (𝑦 = 𝑋 → (𝑦 𝑧𝑋 𝑧))
5 breq1 5060 . . . . 5 (𝑦 = 𝑌 → (𝑦 𝑧𝑌 𝑧))
64, 5ralprg 4624 . . . 4 ((𝑋𝐵𝑌𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧 ↔ (𝑋 𝑧𝑌 𝑧)))
76imbi1d 343 . . 3 ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧) ↔ ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧)))
87ralbidv 3194 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧) ↔ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧)))
93, 8anbi12d 630 1 ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)) ↔ ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  {cpr 4559   class class class wbr 5057  cfv 6348  Basecbs 16471  lecple 16560  joincjn 17542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058
This theorem is referenced by:  joinval2  17607  joineu  17608
  Copyright terms: Public domain W3C validator