Theorem cdleme0moN 41419

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 3048	. . 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 40553	. . . . . . . . 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 40537	. . . . . . . 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 41239	. . . . . . . 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 41416	. . . . . . . 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 3365	. . . . . . 7 ⊢ (∃*𝑟 ∈ 𝐴 (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) ↔ ∃*𝑟(𝑟 ∈ 𝐴 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))
40		oveq2 7432	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑅))
41		oveq2 7432	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑅))
42	40, 41	eqeq12d 2776	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)))
43		oveq2 7432	. . . . . . . . 9 ⊢ (𝑟 = 𝑈 → (𝑃 ∨ 𝑟) = (𝑃 ∨ 𝑈))
44		oveq2 7432	. . . . . . . . 9 ⊢ (𝑟 = 𝑈 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑈))
45	43, 44	eqeq12d 2776	. . . . . . . 8 ⊢ (𝑟 = 𝑈 → ((𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟) ↔ (𝑃 ∨ 𝑈) = (𝑄 ∨ 𝑈)))
46	42, 45	rmoi 3838	. . . . . . 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 5126	. . . 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 1569   ∈ wcel 2144  ∃*wmo 2562   ≠ wne 2955  ∃*wrmo 3364   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424  lecple 17444  joincjn 18494  meetcmee 18495  Atomscatm 40457  CvLatclc 40459  HLchlt 40544  LHypclh 41178

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-proset 18477  df-poset 18496  df-plt 18511  df-lub 18527  df-glb 18528  df-join 18529  df-meet 18530  df-p0 18606  df-p1 18607  df-lat 18615  df-clat 18682  df-oposet 40370  df-ol 40372  df-oml 40373  df-covers 40460  df-ats 40461  df-atl 40492  df-cvlat 40516  df-hlat 40545  df-lhyp 41182

This theorem is used by: (None)