Theorem cdleme3c 40603

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 40609 and cdleme3 40610. (Contributed by NM, 6-Jun-2012.)

Hypotheses

Ref	Expression
cdleme1.l	⊢ ≤ = (le‘𝐾)
cdleme1.j	⊢ ∨ = (join‘𝐾)
cdleme1.m	⊢ ∧ = (meet‘𝐾)
cdleme1.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme1.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme1.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme1.f	⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
cdleme3c.z	⊢ 0 = (0.‘𝐾)

Assertion

Ref	Expression
cdleme3c	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝐹 ≠ 0 )

Proof of Theorem cdleme3c

Step	Hyp	Ref	Expression
1		simpll 767	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝐾 ∈ HL)
2		hllat 39736	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
3	2	ad2antrr 727	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝐾 ∈ Lat)
4		simpr3l 1236	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝑅 ∈ 𝐴)
5		eqid 2737	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		cdleme1.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
7	5, 6	atbase 39662	. . . . . 6 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
8	4, 7	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝑅 ∈ (Base‘𝐾))
9		hlop 39735	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
10	9	ad2antrr 727	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝐾 ∈ OP)
11		cdleme3c.z	. . . . . . 7 ⊢ 0 = (0.‘𝐾)
12	5, 11	op0cl 39557	. . . . . 6 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾))
13	10, 12	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 0 ∈ (Base‘𝐾))
14		cdleme1.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
15	5, 14	latjcl 18374	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 0 ∈ (Base‘𝐾)) → (𝑅 ∨ 0 ) ∈ (Base‘𝐾))
16	3, 8, 13, 15	syl3anc 1374	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → (𝑅 ∨ 0 ) ∈ (Base‘𝐾))
17		simpl 482	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
18		simpr1l 1232	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝑃 ∈ 𝐴)
19		simpr2l 1234	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝑄 ∈ 𝐴)
20		cdleme1.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
21		cdleme1.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
22		cdleme1.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
23		cdleme1.u	. . . . . . 7 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
24		cdleme1.f	. . . . . . 7 ⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
25	20, 14, 21, 6, 22, 23, 24, 5	cdleme1b 40599	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝐹 ∈ (Base‘𝐾))
26	17, 18, 19, 4, 25	syl13anc 1375	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝐹 ∈ (Base‘𝐾))
27	5, 14	latjcl 18374	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝐹 ∈ (Base‘𝐾)) → (𝑅 ∨ 𝐹) ∈ (Base‘𝐾))
28	3, 8, 26, 27	syl3anc 1374	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → (𝑅 ∨ 𝐹) ∈ (Base‘𝐾))
29	5, 6	atbase 39662	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
30	18, 29	syl 17	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝑃 ∈ (Base‘𝐾))
31	5, 6	atbase 39662	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
32	19, 31	syl 17	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝑄 ∈ (Base‘𝐾))
33	5, 14	latjcl 18374	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
34	3, 30, 32, 33	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
35	5, 22	lhpbase 40371	. . . . . . . . . . 11 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
36	35	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝑊 ∈ (Base‘𝐾))
37	5, 20, 21	latmle2 18400	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
38	3, 34, 36, 37	syl3anc 1374	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
39	23, 38	eqbrtrid 5135	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝑈 ≤ 𝑊)
40		simpr3r 1237	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → ¬ 𝑅 ≤ 𝑊)
41		nbrne2 5120	. . . . . . . 8 ⊢ ((𝑈 ≤ 𝑊 ∧ ¬ 𝑅 ≤ 𝑊) → 𝑈 ≠ 𝑅)
42	39, 40, 41	syl2anc 585	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝑈 ≠ 𝑅)
43	42	necomd 2988	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝑅 ≠ 𝑈)
44	20, 14, 21, 6, 22, 23	lhpat2 40418	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑈 ∈ 𝐴)
45	44	3adant3r3 1186	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝑈 ∈ 𝐴)
46		eqid 2737	. . . . . . . 8 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
47	14, 46, 6	atcvr1 39790	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑅 ≠ 𝑈 ↔ 𝑅( ⋖ ‘𝐾)(𝑅 ∨ 𝑈)))
48	1, 4, 45, 47	syl3anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → (𝑅 ≠ 𝑈 ↔ 𝑅( ⋖ ‘𝐾)(𝑅 ∨ 𝑈)))
49	43, 48	mpbid 232	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝑅( ⋖ ‘𝐾)(𝑅 ∨ 𝑈))
50		hlol 39734	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
51	50	ad2antrr 727	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝐾 ∈ OL)
52	5, 14, 11	olj01 39598	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑅 ∨ 0 ) = 𝑅)
53	51, 8, 52	syl2anc 585	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → (𝑅 ∨ 0 ) = 𝑅)
54		simpr3 1198	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
55	20, 14, 21, 6, 22, 23, 24	cdleme1 40600	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → (𝑅 ∨ 𝐹) = (𝑅 ∨ 𝑈))
56	17, 18, 19, 54, 55	syl13anc 1375	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → (𝑅 ∨ 𝐹) = (𝑅 ∨ 𝑈))
57	49, 53, 56	3brtr4d 5132	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → (𝑅 ∨ 0 )( ⋖ ‘𝐾)(𝑅 ∨ 𝐹))
58	5, 46	cvrne 39654	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑅 ∨ 0 ) ∈ (Base‘𝐾) ∧ (𝑅 ∨ 𝐹) ∈ (Base‘𝐾)) ∧ (𝑅 ∨ 0 )( ⋖ ‘𝐾)(𝑅 ∨ 𝐹)) → (𝑅 ∨ 0 ) ≠ (𝑅 ∨ 𝐹))
59	1, 16, 28, 57, 58	syl31anc 1376	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → (𝑅 ∨ 0 ) ≠ (𝑅 ∨ 𝐹))
60		oveq2 7376	. . . 4 ⊢ ( 0 = 𝐹 → (𝑅 ∨ 0 ) = (𝑅 ∨ 𝐹))
61	60	necon3i 2965	. . 3 ⊢ ((𝑅 ∨ 0 ) ≠ (𝑅 ∨ 𝐹) → 0 ≠ 𝐹)
62	59, 61	syl 17	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 0 ≠ 𝐹)
63	62	necomd 2988	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → 𝐹 ≠ 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  lecple 17196  joincjn 18246  meetcmee 18247  0.cp0 18356  Latclat 18366  OPcops 39545  OLcol 39547   ⋖ ccvr 39635  Atomscatm 39636  HLchlt 39723  LHypclh 40357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-proset 18229  df-poset 18248  df-plt 18263  df-lub 18279  df-glb 18280  df-join 18281  df-meet 18282  df-p0 18358  df-p1 18359  df-lat 18367  df-clat 18434  df-oposet 39549  df-ol 39551  df-oml 39552  df-covers 39639  df-ats 39640  df-atl 39671  df-cvlat 39695  df-hlat 39724  df-psubsp 39876  df-pmap 39877  df-padd 40169  df-lhyp 40361

This theorem is referenced by:  cdleme3h  40608