Theorem cdleme0moN 41059

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme0.l	⊢ ≤ = (le‘𝐾)
cdleme0.j	⊢ ∨ = (join‘𝐾)
cdleme0.m	⊢ ∧ = (meet‘𝐾)
cdleme0.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme0.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme0.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)

Assertion

Ref	Expression
cdleme0moN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝑅 = 𝑃 ∨ 𝑅 = 𝑄))

Distinct variable groups:   𝐴,𝑟   ∨ ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑅,𝑟   𝑈,𝑟

Allowed substitution hints:   𝐻(𝑟)   𝐾(𝑟)   ≤ (𝑟)   ∧ (𝑟)   𝑊(𝑟)

Proof of Theorem cdleme0moN

Step	Hyp	Ref	Expression
1		simp23r 1314	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ¬ 𝑅 ≤ 𝑊)
2		neanior 3053	. . 3 ⊢ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) ↔ ¬ (𝑅 = 𝑃 ∨ 𝑅 = 𝑄))
3		simpl33 1275	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
4		simp23l 1313	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑅 ∈ 𝐴)
5	4	adantr 486	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → 𝑅 ∈ 𝐴)
6		simprl 783	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → 𝑅 ≠ 𝑃)
7		simprr 785	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → 𝑅 ≠ 𝑄)
8		simpl32 1274	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → 𝑅 ≤ (𝑃 ∨ 𝑄))
9		simpl1l 1243	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → 𝐾 ∈ HL)
10		hlcvl 40193	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)
11	9, 10	syl 18	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → 𝐾 ∈ CvLat)
12		simp21l 1309	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑃 ∈ 𝐴)
13	12	adantr 486	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → 𝑃 ∈ 𝐴)
14		simp22l 1311	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑄 ∈ 𝐴)
15	14	adantr 486	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → 𝑄 ∈ 𝐴)
16		simpl31 1273	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → 𝑃 ≠ 𝑄)
17		cdleme0.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
18		cdleme0.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
19		cdleme0.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
20	17, 18, 19	cvlsupr2 40177	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
21	11, 13, 15, 5, 16, 20	syl131anc 1410	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → ((𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅) ↔ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))))
22	6, 7, 8, 21	mpbir3and 1361	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
23		simp1l 1216	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝐾 ∈ HL)
24		simp1r 1217	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑊 ∈ 𝐻)
25		simp21r 1310	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ¬ 𝑃 ≤ 𝑊)
26		simp31 1228	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑃 ≠ 𝑄)
27		cdleme0.m	. . . . . . . . 9 ⊢ ∧ = (meet‘𝐾)
28		cdleme0.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
29		cdleme0.u	. . . . . . . . 9 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
30	18, 19, 27, 17, 28, 29	lhpat2 40879	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑈 ∈ 𝐴)
31	23, 24, 12, 25, 14, 26, 30	syl222anc 1413	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → 𝑈 ∈ 𝐴)
32	31	adantr 486	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → 𝑈 ∈ 𝐴)
33		simpl1 1210	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
34		simpl21 1270	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
35		simpl22 1271	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
36	18, 19, 27, 17, 28, 29	cdleme02N 41056	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ((𝑃 ∨ 𝑈) = (𝑄 ∨ 𝑈) ∧ 𝑈 ≤ 𝑊))
37	36	simpld 500	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∨ 𝑈) = (𝑄 ∨ 𝑈))
38	33, 34, 35, 16, 37	syl121anc 1402	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → (𝑃 ∨ 𝑈) = (𝑄 ∨ 𝑈))
39		df-rmo 3371	. . . . . . 7 ⊢ (∃*𝑟 ∈ 𝐴 (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) ↔ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
40		oveq2 7427	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑅))
41		oveq2 7427	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑅))
42	40, 41	eqeq12d 2781	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
43		oveq2 7427	. . . . . . . . 9 ⊢ (𝑟 = 𝑈 → (𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑈))
44		oveq2 7427	. . . . . . . . 9 ⊢ (𝑟 = 𝑈 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑈))
45	43, 44	eqeq12d 2781	. . . . . . . 8 ⊢ (𝑟 = 𝑈 → ((𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) ↔ (𝑃 ∨ 𝑈) = (𝑄 ∨ 𝑈)))
46	42, 45	rmoi 3845	. . . . . . 7 ⊢ ((∃*𝑟 ∈ 𝐴 (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) ∧ (𝑅 ∈ 𝐴 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑈 ∈ 𝐴 ∧ (𝑃 ∨ 𝑈) = (𝑄 ∨ 𝑈))) → 𝑅 = 𝑈)
47	39, 46	syl3an1br 1433	. . . . . 6 ⊢ ((∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)) ∧ (𝑅 ∈ 𝐴 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑈 ∈ 𝐴 ∧ (𝑃 ∨ 𝑈) = (𝑄 ∨ 𝑈))) → 𝑅 = 𝑈)
48	3, 5, 22, 32, 38, 47	syl122anc 1406	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → 𝑅 = 𝑈)
49	36	simprd 501	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → 𝑈 ≤ 𝑊)
50	33, 34, 35, 16, 49	syl121anc 1402	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → 𝑈 ≤ 𝑊)
51	48, 50	eqbrtrd 5135	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) ∧ (𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄)) → 𝑅 ≤ 𝑊)
52	51	ex 418	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑅 ≠ 𝑃 ∧ 𝑅 ≠ 𝑄) → 𝑅 ≤ 𝑊))
53	2, 52	biimtrrid 246	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (¬ (𝑅 = 𝑃 ∨ 𝑅 = 𝑄) → 𝑅 ≤ 𝑊))
54	1, 53	mt3d 149	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝑅 = 𝑃 ∨ 𝑅 = 𝑄))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃*wmo 2567   ≠ wne 2960  ∃*wrmo 3370   class class class wbr 5111  ‘cfv 6540  (class class class)co 7419  lecple 17341  joincjn 18391  meetcmee 18392  Atomscatm 40097  CvLatclc 40099  HLchlt 40184  LHypclh 40818

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-proset 18374  df-poset 18393  df-plt 18408  df-lub 18424  df-glb 18425  df-join 18426  df-meet 18427  df-p0 18503  df-p1 18504  df-lat 18512  df-clat 18579  df-oposet 40010  df-ol 40012  df-oml 40013  df-covers 40100  df-ats 40101  df-atl 40132  df-cvlat 40156  df-hlat 40185  df-lhyp 40822

This theorem is used by: (None)