Theorem cdlemg31b0N 38353

Description: TODO: Fix comment. (Contributed by NM, 30-May-2013.) (New usage is discouraged.)

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
cdlemg31b0N	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑁 ∈ 𝐴 ∨ 𝑁 = (0.‘𝐾)))

Proof of Theorem cdlemg31b0N

Step	Hyp	Ref	Expression
1		simp11 1204	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝐾 ∈ HL)
2		simp2ll 1241	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑃 ∈ 𝐴)
3		simp31l 1297	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑣 ∈ 𝐴)
4		simp2rl 1243	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑄 ∈ 𝐴)
5		simp12 1205	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑊 ∈ 𝐻)
6	1, 5	jca 515	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7		simp2l 1200	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
8		simp13 1206	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝐹 ∈ 𝑇)
9		simp33 1212	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑃) ≠ 𝑃)
10		cdlemg12.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
11		cdlemg12.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
12		cdlemg12.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
13		cdlemg12.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
14		cdlemg12b.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
15	10, 11, 12, 13, 14	trlat 37828	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
16	6, 7, 8, 9, 15	syl112anc 1375	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
17		simp2r 1201	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
18	10, 12, 13, 14	trlle 37843	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊)
19	6, 8, 18	syl2anc 587	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ≤ 𝑊)
20	16, 19	jca 515	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝑅‘𝐹) ∈ 𝐴 ∧ (𝑅‘𝐹) ≤ 𝑊))
21		simp31 1210	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊))
22		simp32 1211	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑣 ≠ (𝑅‘𝐹))
23	22	necomd 2989	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ≠ 𝑣)
24		cdlemg12.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
25	10, 24, 11, 12	lhp2atne 37693	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ∈ 𝐴) ∧ (((𝑅‘𝐹) ∈ 𝐴 ∧ (𝑅‘𝐹) ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝑅‘𝐹) ≠ 𝑣) → (𝑄 ∨ (𝑅‘𝐹)) ≠ (𝑃 ∨ 𝑣))
26	6, 17, 2, 20, 21, 23, 25	syl321anc 1393	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑄 ∨ (𝑅‘𝐹)) ≠ (𝑃 ∨ 𝑣))
27	26	necomd 2989	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑃 ∨ 𝑣) ≠ (𝑄 ∨ (𝑅‘𝐹)))
28		cdlemg12.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
29		eqid 2738	. . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾)
30	24, 28, 29, 11	2atmat0 37185	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑄 ∈ 𝐴 ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ (𝑃 ∨ 𝑣) ≠ (𝑄 ∨ (𝑅‘𝐹)))) → (((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹))) ∈ 𝐴 ∨ ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹))) = (0.‘𝐾)))
31	1, 2, 3, 4, 16, 27, 30	syl33anc 1386	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹))) ∈ 𝐴 ∨ ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹))) = (0.‘𝐾)))
32		cdlemg31.n	. . . 4 ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))
33	32	eleq1i 2823	. . 3 ⊢ (𝑁 ∈ 𝐴 ↔ ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹))) ∈ 𝐴)
34	32	eqeq1i 2743	. . 3 ⊢ (𝑁 = (0.‘𝐾) ↔ ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹))) = (0.‘𝐾))
35	33, 34	orbi12i 914	. 2 ⊢ ((𝑁 ∈ 𝐴 ∨ 𝑁 = (0.‘𝐾)) ↔ (((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹))) ∈ 𝐴 ∨ ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹))) = (0.‘𝐾)))
36	31, 35	sylibr 237	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑁 ∈ 𝐴 ∨ 𝑁 = (0.‘𝐾)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114   ≠ wne 2934   class class class wbr 5030  ‘cfv 6339  (class class class)co 7172  lecple 16677  joincjn 17672  meetcmee 17673  0.cp0 17765  Atomscatm 36922  HLchlt 37009  LHypclh 37643  LTrncltrn 37760  trLctrl 37817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7129  df-ov 7175  df-oprab 7176  df-mpo 7177  df-1st 7716  df-2nd 7717  df-map 8441  df-proset 17656  df-poset 17674  df-plt 17686  df-lub 17702  df-glb 17703  df-join 17704  df-meet 17705  df-p0 17767  df-p1 17768  df-lat 17774  df-clat 17836  df-oposet 36835  df-ol 36837  df-oml 36838  df-covers 36925  df-ats 36926  df-atl 36957  df-cvlat 36981  df-hlat 37010  df-llines 37157  df-psubsp 37162  df-pmap 37163  df-padd 37455  df-lhyp 37647  df-laut 37648  df-ldil 37763  df-ltrn 37764  df-trl 37818

This theorem is referenced by: (None)