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Theorem 3dim1lem5 35070
Description: Lemma for 3dim1 35071. (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 2886 . . 3 (𝑞 = 𝑢 → (𝑃𝑞𝑃𝑢))
2 oveq2 6698 . . . . 5 (𝑞 = 𝑢 → (𝑃 𝑞) = (𝑃 𝑢))
32breq2d 4697 . . . 4 (𝑞 = 𝑢 → (𝑟 (𝑃 𝑞) ↔ 𝑟 (𝑃 𝑢)))
43notbid 307 . . 3 (𝑞 = 𝑢 → (¬ 𝑟 (𝑃 𝑞) ↔ ¬ 𝑟 (𝑃 𝑢)))
52oveq1d 6705 . . . . 5 (𝑞 = 𝑢 → ((𝑃 𝑞) 𝑟) = ((𝑃 𝑢) 𝑟))
65breq2d 4697 . . . 4 (𝑞 = 𝑢 → (𝑠 ((𝑃 𝑞) 𝑟) ↔ 𝑠 ((𝑃 𝑢) 𝑟)))
76notbid 307 . . 3 (𝑞 = 𝑢 → (¬ 𝑠 ((𝑃 𝑞) 𝑟) ↔ ¬ 𝑠 ((𝑃 𝑢) 𝑟)))
81, 4, 73anbi123d 1439 . 2 (𝑞 = 𝑢 → ((𝑃𝑞 ∧ ¬ 𝑟 (𝑃 𝑞) ∧ ¬ 𝑠 ((𝑃 𝑞) 𝑟)) ↔ (𝑃𝑢 ∧ ¬ 𝑟 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑟))))
9 breq1 4688 . . . 4 (𝑟 = 𝑣 → (𝑟 (𝑃 𝑢) ↔ 𝑣 (𝑃 𝑢)))
109notbid 307 . . 3 (𝑟 = 𝑣 → (¬ 𝑟 (𝑃 𝑢) ↔ ¬ 𝑣 (𝑃 𝑢)))
11 oveq2 6698 . . . . 5 (𝑟 = 𝑣 → ((𝑃 𝑢) 𝑟) = ((𝑃 𝑢) 𝑣))
1211breq2d 4697 . . . 4 (𝑟 = 𝑣 → (𝑠 ((𝑃 𝑢) 𝑟) ↔ 𝑠 ((𝑃 𝑢) 𝑣)))
1312notbid 307 . . 3 (𝑟 = 𝑣 → (¬ 𝑠 ((𝑃 𝑢) 𝑟) ↔ ¬ 𝑠 ((𝑃 𝑢) 𝑣)))
1410, 133anbi23d 1442 . 2 (𝑟 = 𝑣 → ((𝑃𝑢 ∧ ¬ 𝑟 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑟)) ↔ (𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑣))))
15 breq1 4688 . . . 4 (𝑠 = 𝑤 → (𝑠 ((𝑃 𝑢) 𝑣) ↔ 𝑤 ((𝑃 𝑢) 𝑣)))
1615notbid 307 . . 3 (𝑠 = 𝑤 → (¬ 𝑠 ((𝑃 𝑢) 𝑣) ↔ ¬ 𝑤 ((𝑃 𝑢) 𝑣)))
17163anbi3d 1445 . 2 (𝑠 = 𝑤 → ((𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑠 ((𝑃 𝑢) 𝑣)) ↔ (𝑃𝑢 ∧ ¬ 𝑣 (𝑃 𝑢) ∧ ¬ 𝑤 ((𝑃 𝑢) 𝑣))))
188, 14, 17rspc3ev 3357 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 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wrex 2942   class class class wbr 4685  cfv 5926  (class class class)co 6690  lecple 15995  joincjn 16991  Atomscatm 34868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693
This theorem is referenced by:  3dim1  35071
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