Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  3dim1lem5 Structured version   Visualization version   GIF version

Theorem 3dim1lem5 36644
 Description: Lemma for 3dim1 36645. (Contributed by NM, 26-Jul-2012.)
Hypotheses
Ref Expression
3dim0.j = (join‘𝐾)
3dim0.l = (le‘𝐾)
3dim0.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
3dim1lem5 (((𝑢𝐴𝑣𝐴𝑤𝐴) ∧ (𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑤 ((𝑃 𝑢) 𝑣))) → ∃𝑞𝐴𝑟𝐴𝑠𝐴 (𝑃𝑞 ∧ ¬ 𝑟 (𝑃 𝑞) ∧ ¬ 𝑠 ((𝑃 𝑞) 𝑟)))
Distinct variable groups:   𝑟,𝑞,𝑠,𝐴   ,𝑟,𝑠   𝑣,𝑢,𝑤,𝐴,𝑞   ,𝑞,𝑢,𝑣,𝑤   𝑢,𝐾,𝑣,𝑤   ,𝑞   𝑢,𝑟,𝑣,𝑤, ,𝑠   𝑃,𝑞,𝑟,𝑠,𝑢,𝑣,𝑤
Allowed substitution hints:   𝐾(𝑠,𝑟,𝑞)

Proof of Theorem 3dim1lem5
StepHypRef Expression
1 neeq2 3070 . . 3 (𝑞 = 𝑢 → (𝑃𝑞𝑃𝑢))
2 oveq2 7138 . . . . 5 (𝑞 = 𝑢 → (𝑃 𝑞) = (𝑃 𝑢))
32breq2d 5051 . . . 4 (𝑞 = 𝑢 → (𝑟 (𝑃 𝑞) ↔ 𝑟 (𝑃 𝑢)))
43notbid 321 . . 3 (𝑞 = 𝑢 → (¬ 𝑟 (𝑃 𝑞) ↔ ¬ 𝑟 (𝑃 𝑢)))
52oveq1d 7145 . . . . 5 (𝑞 = 𝑢 → ((𝑃 𝑞) 𝑟) = ((𝑃 𝑢) 𝑟))
65breq2d 5051 . . . 4 (𝑞 = 𝑢 → (𝑠 ((𝑃 𝑞) 𝑟) ↔ 𝑠 ((𝑃 𝑢) 𝑟)))
76notbid 321 . . 3 (𝑞 = 𝑢 → (¬ 𝑠 ((𝑃 𝑞) 𝑟) ↔ ¬ 𝑠 ((𝑃 𝑢) 𝑟)))
81, 4, 73anbi123d 1433 . 2 (𝑞 = 𝑢 → ((𝑃𝑞 ∧ ¬ 𝑟 (𝑃 𝑞) ∧ ¬ 𝑠 ((𝑃 𝑞) 𝑟)) ↔ (𝑃𝑢 ∧ ¬ 𝑟 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑟))))
9 breq1 5042 . . . 4 (𝑟 = 𝑣 → (𝑟 (𝑃 𝑢) ↔ 𝑣 (𝑃 𝑢)))
109notbid 321 . . 3 (𝑟 = 𝑣 → (¬ 𝑟 (𝑃 𝑢) ↔ ¬ 𝑣 (𝑃 𝑢)))
11 oveq2 7138 . . . . 5 (𝑟 = 𝑣 → ((𝑃 𝑢) 𝑟) = ((𝑃 𝑢) 𝑣))
1211breq2d 5051 . . . 4 (𝑟 = 𝑣 → (𝑠 ((𝑃 𝑢) 𝑟) ↔ 𝑠 ((𝑃 𝑢) 𝑣)))
1312notbid 321 . . 3 (𝑟 = 𝑣 → (¬ 𝑠 ((𝑃 𝑢) 𝑟) ↔ ¬ 𝑠 ((𝑃 𝑢) 𝑣)))
1410, 133anbi23d 1436 . 2 (𝑟 = 𝑣 → ((𝑃𝑢 ∧ ¬ 𝑟 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑟)) ↔ (𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑣))))
15 breq1 5042 . . . 4 (𝑠 = 𝑤 → (𝑠 ((𝑃 𝑢) 𝑣) ↔ 𝑤 ((𝑃 𝑢) 𝑣)))
1615notbid 321 . . 3 (𝑠 = 𝑤 → (¬ 𝑠 ((𝑃 𝑢) 𝑣) ↔ ¬ 𝑤 ((𝑃 𝑢) 𝑣)))
17163anbi3d 1439 . 2 (𝑠 = 𝑤 → ((𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑣)) ↔ (𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑤 ((𝑃 𝑢) 𝑣))))
188, 14, 17rspc3ev 3614 1 (((𝑢𝐴𝑣𝐴𝑤𝐴) ∧ (𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑤 ((𝑃 𝑢) 𝑣))) → ∃𝑞𝐴𝑟𝐴𝑠𝐴 (𝑃𝑞 ∧ ¬ 𝑟 (𝑃 𝑞) ∧ ¬ 𝑠 ((𝑃 𝑞) 𝑟)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ≠ wne 3007  ∃wrex 3127   class class class wbr 5039  ‘cfv 6328  (class class class)co 7130  lecple 16551  joincjn 17533  Atomscatm 36441 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336  df-ov 7133 This theorem is referenced by:  3dim1  36645
 Copyright terms: Public domain W3C validator