Theorem cdlemg31b0N 40680

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 511	. . . 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 40155	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
16	6, 7, 8, 9, 15	syl112anc 1376	. . 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 40170	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊)
19	6, 8, 18	syl2anc 584	. . . . . 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 511	. . . . 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 2982	. . . . 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 40020	. . . . 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 1394	. . . 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 2982	. . 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 2730	. . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾)
30	24, 28, 29, 11	2atmat0 39512	. . 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 1387	. 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 2820	. . 3 ⊢ (𝑁 ∈ 𝐴 ↔ ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹))) ∈ 𝐴)
34	32	eqeq1i 2735	. . 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 234	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 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2927   class class class wbr 5115  ‘cfv 6519  (class class class)co 7394  lecple 17233  joincjn 18278  meetcmee 18279  0.cp0 18388  Atomscatm 39248  HLchlt 39335  LHypclh 39970  LTrncltrn 40087  trLctrl 40144

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-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

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-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rmo 3357  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-iin 4966  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-riota 7351  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-map 8805  df-proset 18261  df-poset 18280  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-p1 18391  df-lat 18397  df-clat 18464  df-oposet 39161  df-ol 39163  df-oml 39164  df-covers 39251  df-ats 39252  df-atl 39283  df-cvlat 39307  df-hlat 39336  df-llines 39484  df-psubsp 39489  df-pmap 39490  df-padd 39782  df-lhyp 39974  df-laut 39975  df-ldil 40090  df-ltrn 40091  df-trl 40145

This theorem is referenced by: (None)