Theorem 3dim1lem5 39467

Description: Lemma for 3dim1 39468. (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	⊢ (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑃 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣))) → ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑃 ≠ 𝑞 ∧ ¬ 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

Step	Hyp	Ref	Expression
1		neeq2 2989	. . 3 ⊢ (𝑞 = 𝑢 → (𝑃 ≠ 𝑞 ↔ 𝑃 ≠ 𝑢))
2		oveq2 7398	. . . . 5 ⊢ (𝑞 = 𝑢 → (𝑃 ∨ 𝑞) = (𝑃 ∨ 𝑢))
3	2	breq2d 5122	. . . 4 ⊢ (𝑞 = 𝑢 → (𝑟 ≤ (𝑃 ∨ 𝑞) ↔ 𝑟 ≤ (𝑃 ∨ 𝑢)))
4	3	notbid 318	. . 3 ⊢ (𝑞 = 𝑢 → (¬ 𝑟 ≤ (𝑃 ∨ 𝑞) ↔ ¬ 𝑟 ≤ (𝑃 ∨ 𝑢)))
5	2	oveq1d 7405	. . . . 5 ⊢ (𝑞 = 𝑢 → ((𝑃 ∨ 𝑞) ∨ 𝑟) = ((𝑃 ∨ 𝑢) ∨ 𝑟))
6	5	breq2d 5122	. . . 4 ⊢ (𝑞 = 𝑢 → (𝑠 ≤ ((𝑃 ∨ 𝑞) ∨ 𝑟) ↔ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑟)))
7	6	notbid 318	. . 3 ⊢ (𝑞 = 𝑢 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑞) ∨ 𝑟) ↔ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑟)))
8	1, 4, 7	3anbi123d 1438	. 2 ⊢ (𝑞 = 𝑢 → ((𝑃 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑞) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑞) ∨ 𝑟)) ↔ (𝑃 ≠ 𝑢 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑢) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑟))))
9		breq1 5113	. . . 4 ⊢ (𝑟 = 𝑣 → (𝑟 ≤ (𝑃 ∨ 𝑢) ↔ 𝑣 ≤ (𝑃 ∨ 𝑢)))
10	9	notbid 318	. . 3 ⊢ (𝑟 = 𝑣 → (¬ 𝑟 ≤ (𝑃 ∨ 𝑢) ↔ ¬ 𝑣 ≤ (𝑃 ∨ 𝑢)))
11		oveq2 7398	. . . . 5 ⊢ (𝑟 = 𝑣 → ((𝑃 ∨ 𝑢) ∨ 𝑟) = ((𝑃 ∨ 𝑢) ∨ 𝑣))
12	11	breq2d 5122	. . . 4 ⊢ (𝑟 = 𝑣 → (𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑟) ↔ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣)))
13	12	notbid 318	. . 3 ⊢ (𝑟 = 𝑣 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑟) ↔ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣)))
14	10, 13	3anbi23d 1441	. 2 ⊢ (𝑟 = 𝑣 → ((𝑃 ≠ 𝑢 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑢) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑟)) ↔ (𝑃 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑢) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣))))
15		breq1 5113	. . . 4 ⊢ (𝑠 = 𝑤 → (𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣) ↔ 𝑤 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣)))
16	15	notbid 318	. . 3 ⊢ (𝑠 = 𝑤 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣) ↔ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣)))
17	16	3anbi3d 1444	. 2 ⊢ (𝑠 = 𝑤 → ((𝑃 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑢) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣)) ↔ (𝑃 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣))))
18	8, 14, 17	rspc3ev 3608	1 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑃 ≠ 𝑢 ∧ ¬ 𝑣 ≤ (𝑃 ∨ 𝑢) ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑢) ∨ 𝑣))) → ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑃 ≠ 𝑞 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑞) ∧ ¬ 𝑠 ≤ ((𝑃 ∨ 𝑞) ∨ 𝑟)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  lecple 17234  joincjn 18279  Atomscatm 39263

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393

This theorem is referenced by:  3dim1  39468