Theorem cdlemf 41062

Description: Lemma F in [Crawley] p. 116. If u is an atom under w, there exists a translation whose trace is u. (Contributed by NM, 12-Apr-2013.)

Hypotheses

Ref	Expression
cdlemf.l	⊢ ≤ = (le‘𝐾)
cdlemf.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemf.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemf.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemf.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

Assertion

Ref	Expression
cdlemf	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈)

Distinct variable groups:   𝐴,𝑓   𝑓,𝐻   𝑓,𝐾   ≤ ,𝑓   𝑇,𝑓   𝑈,𝑓   𝑓,𝑊

Allowed substitution hint:   𝑅(𝑓)

Proof of Theorem cdlemf

Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdlemf.l	. . 3 ⊢ ≤ = (le‘𝐾)
2		eqid 2740	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		cdlemf.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
4		cdlemf.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2740	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
6	1, 2, 3, 4, 5	cdlemf2 41061	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)))
7		simp1l 1204	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8		simp2l 1206	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → 𝑝 ∈ 𝐴)
9		simp3ll 1251	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ¬ 𝑝 ≤ 𝑊)
10		simp2r 1207	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → 𝑞 ∈ 𝐴)
11		simp3lr 1252	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ¬ 𝑞 ≤ 𝑊)
12		cdlemf.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
13	1, 3, 4, 12	cdleme50ex 41058	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)
14	7, 8, 9, 10, 11, 13	syl122anc 1387	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ∃𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)
15		simp3r 1209	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → (𝑓‘𝑝) = 𝑞)
16	15	oveq2d 7379	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → (𝑝(join‘𝐾)(𝑓‘𝑝)) = (𝑝(join‘𝐾)𝑞))
17	16	oveq1d 7378	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → ((𝑝(join‘𝐾)(𝑓‘𝑝))(meet‘𝐾)𝑊) = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))
18		simp11 1210	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
19		simp3l 1208	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → 𝑓 ∈ 𝑇)
20		simp13l 1295	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → 𝑝 ∈ 𝐴)
21		simp2ll 1247	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → ¬ 𝑝 ≤ 𝑊)
22		cdlemf.r	. . . . . . . . . . . . 13 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
23	1, 2, 5, 3, 4, 12, 22	trlval2 40662	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑅‘𝑓) = ((𝑝(join‘𝐾)(𝑓‘𝑝))(meet‘𝐾)𝑊))
24	18, 19, 20, 21, 23	syl112anc 1382	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → (𝑅‘𝑓) = ((𝑝(join‘𝐾)(𝑓‘𝑝))(meet‘𝐾)𝑊))
25		simp2r 1207	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))
26	17, 24, 25	3eqtr4d 2785	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞)) → (𝑅‘𝑓) = 𝑈)
27	26	3exp 1125	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) → (((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) → ((𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞) → (𝑅‘𝑓) = 𝑈)))
28	27	3expia 1127	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ((𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) → (((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) → ((𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞) → (𝑅‘𝑓) = 𝑈))))
29	28	3imp 1116	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ((𝑓 ∈ 𝑇 ∧ (𝑓‘𝑝) = 𝑞) → (𝑅‘𝑓) = 𝑈))
30	29	expd 416	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → (𝑓 ∈ 𝑇 → ((𝑓‘𝑝) = 𝑞 → (𝑅‘𝑓) = 𝑈)))
31	30	reximdvai 3151	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → (∃𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞 → ∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈))
32	14, 31	mpd 15	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊))) → ∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈)
33	32	3exp 1125	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ((𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) → (((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) → ∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈)))
34	33	rexlimdvv 3196	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝑈 = ((𝑝(join‘𝐾)𝑞)(meet‘𝐾)𝑊)) → ∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈))
35	6, 34	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∃wrex 3064   class class class wbr 5079  ‘cfv 6492  (class class class)co 7363  lecple 17225  joincjn 18275  meetcmee 18276  Atomscatm 39762  HLchlt 39849  LHypclh 40483  LTrncltrn 40600  trLctrl 40657

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-riotaBAD 39452

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-undef 8220  df-map 8772  df-proset 18258  df-poset 18277  df-plt 18292  df-lub 18308  df-glb 18309  df-join 18310  df-meet 18311  df-p0 18387  df-p1 18388  df-lat 18396  df-clat 18463  df-oposet 39675  df-ol 39677  df-oml 39678  df-covers 39765  df-ats 39766  df-atl 39797  df-cvlat 39821  df-hlat 39850  df-llines 39997  df-lplanes 39998  df-lvols 39999  df-lines 40000  df-psubsp 40002  df-pmap 40003  df-padd 40295  df-lhyp 40487  df-laut 40488  df-ldil 40603  df-ltrn 40604  df-trl 40658

This theorem is referenced by:  cdlemfnid  41063  trlord  41068  dih1dimb2  41740