Theorem cdlemg31d 40683

Description: Eliminate (𝐹‘𝑃) ≠ 𝑃 from cdlemg31c 40682. TODO: Prove directly. TODO: do we need to eliminate (𝐹‘𝑃) ≠ 𝑃? It might be better to do this all at once at the end. See also cdlemg29 40688 versus cdlemg28 40687. (Contributed by NM, 29-May-2013.)

Hypotheses

Ref	Expression
cdlemg12.l	⊢ ≤ = (le‘𝐾)
cdlemg12.j	⊢ ∨ = (join‘𝐾)
cdlemg12.m	⊢ ∧ = (meet‘𝐾)
cdlemg12.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg12.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg12.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg12b.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemg31.n	⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))

Assertion

Ref	Expression
cdlemg31d	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) → ¬ 𝑁 ≤ 𝑊)

Proof of Theorem cdlemg31d

Step	Hyp	Ref	Expression
1		simp22r 1292	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) → ¬ 𝑄 ≤ 𝑊)
2	1	adantr 480	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → ¬ 𝑄 ≤ 𝑊)
3		simpl1 1190	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4		simp21l 1289	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
5	4	adantr 480	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → 𝑃 ∈ 𝐴)
6		simp22l 1291	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
7	6	adantr 480	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → 𝑄 ∈ 𝐴)
8		simp23l 1293	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) → 𝑣 ∈ 𝐴)
9	8	adantr 480	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → 𝑣 ∈ 𝐴)
10		simpl31 1253	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → 𝐹 ∈ 𝑇)
11		cdlemg12.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
12		cdlemg12.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
13		cdlemg12.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
14		cdlemg12.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
15		cdlemg12.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
16		cdlemg12.t	. . . . . . . 8 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
17		cdlemg12b.r	. . . . . . . 8 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
18		cdlemg31.n	. . . . . . . 8 ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))
19	11, 12, 13, 14, 15, 16, 17, 18	cdlemg31b 40681	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑣 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇)) → 𝑁 ≤ (𝑄 ∨ (𝑅‘𝐹)))
20	3, 5, 7, 9, 10, 19	syl122anc 1378	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → 𝑁 ≤ (𝑄 ∨ (𝑅‘𝐹)))
21		simpl21 1250	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
22		simpr 484	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → (𝐹‘𝑃) = 𝑃)
23		eqid 2735	. . . . . . . . . 10 ⊢ (0.‘𝐾) = (0.‘𝐾)
24	11, 23, 14, 15, 16, 17	trl0 40153	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑅‘𝐹) = (0.‘𝐾))
25	3, 21, 10, 22, 24	syl112anc 1373	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → (𝑅‘𝐹) = (0.‘𝐾))
26	25	oveq2d 7447	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → (𝑄 ∨ (𝑅‘𝐹)) = (𝑄 ∨ (0.‘𝐾)))
27		simp1l 1196	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) → 𝐾 ∈ HL)
28		hlol 39343	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
29	27, 28	syl 17	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) → 𝐾 ∈ OL)
30	29	adantr 480	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → 𝐾 ∈ OL)
31		eqid 2735	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
32	31, 14	atbase 39271	. . . . . . . . 9 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
33	7, 32	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → 𝑄 ∈ (Base‘𝐾))
34	31, 12, 23	olj01 39207	. . . . . . . 8 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 ∨ (0.‘𝐾)) = 𝑄)
35	30, 33, 34	syl2anc 584	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → (𝑄 ∨ (0.‘𝐾)) = 𝑄)
36	26, 35	eqtrd 2775	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → (𝑄 ∨ (𝑅‘𝐹)) = 𝑄)
37	20, 36	breqtrd 5174	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → 𝑁 ≤ 𝑄)
38		hlatl 39342	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
39	27, 38	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) → 𝐾 ∈ AtLat)
40	39	adantr 480	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → 𝐾 ∈ AtLat)
41		simpl33 1255	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → 𝑁 ∈ 𝐴)
42	11, 14	atcmp 39293	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑁 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑁 ≤ 𝑄 ↔ 𝑁 = 𝑄))
43	40, 41, 7, 42	syl3anc 1370	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → (𝑁 ≤ 𝑄 ↔ 𝑁 = 𝑄))
44	37, 43	mpbid 232	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → 𝑁 = 𝑄)
45	44	breq1d 5158	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → (𝑁 ≤ 𝑊 ↔ 𝑄 ≤ 𝑊))
46	2, 45	mtbird 325	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) = 𝑃) → ¬ 𝑁 ≤ 𝑊)
47		simpl1 1190	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
48		simpl21 1250	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
49		simpl22 1251	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
50		simpl23 1252	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊))
51		simpl31 1253	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) ≠ 𝑃) → 𝐹 ∈ 𝑇)
52		simpl32 1254	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) ≠ 𝑃) → 𝑣 ≠ (𝑅‘𝐹))
53		simpr 484	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝐹‘𝑃) ≠ 𝑃)
54		simpl33 1255	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) ≠ 𝑃) → 𝑁 ∈ 𝐴)
55	11, 12, 13, 14, 15, 16, 17, 18	cdlemg31c 40682	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃 ∧ 𝑁 ∈ 𝐴)) → ¬ 𝑁 ≤ 𝑊)
56	47, 48, 49, 50, 51, 52, 53, 54, 55	syl323anc 1399	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) ∧ (𝐹‘𝑃) ≠ 𝑃) → ¬ 𝑁 ≤ 𝑊)
57	46, 56	pm2.61dane 3027	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) → ¬ 𝑁 ≤ 𝑊)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  0.cp0 18481  OLcol 39156  Atomscatm 39245  AtLatcal 39246  HLchlt 39332  LHypclh 39967  LTrncltrn 40084  trLctrl 40141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142

This theorem is referenced by:  cdlemg33b0  40684  cdlemg33a  40689