Theorem cdlemf 40582

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 2735	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		cdlemf.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
4		cdlemf.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2735	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
6	1, 2, 3, 4, 5	cdlemf2 40581	. 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 1198	. . . . . 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 1200	. . . . . 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 1245	. . . . . 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 1201	. . . . . 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 1246	. . . . . 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 40578	. . . . . 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 1381	. . . . 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 1203	. . . . . . . . . . . . 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 7421	. . . . . . . . . . . 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 7420	. . . . . . . . . . 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 1204	. . . . . . . . . . . 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 1202	. . . . . . . . . . . 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 1289	. . . . . . . . . . . 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 1241	. . . . . . . . . . . 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 40182	. . . . . . . . . . . 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 1376	. . . . . . . . . . 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 1201	. . . . . . . . . . 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 2780	. . . . . . . . . 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 1119	. . . . . . . . 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 1121	. . . . . . . 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 1110	. . . . . . 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 415	. . . . . 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 1119	. . 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 3197	. 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∃wrex 3060   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  lecple 17278  joincjn 18323  meetcmee 18324  Atomscatm 39281  HLchlt 39368  LHypclh 40003  LTrncltrn 40120  trLctrl 40177

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-riotaBAD 38971

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-undef 8272  df-map 8842  df-proset 18306  df-poset 18325  df-plt 18340  df-lub 18356  df-glb 18357  df-join 18358  df-meet 18359  df-p0 18435  df-p1 18436  df-lat 18442  df-clat 18509  df-oposet 39194  df-ol 39196  df-oml 39197  df-covers 39284  df-ats 39285  df-atl 39316  df-cvlat 39340  df-hlat 39369  df-llines 39517  df-lplanes 39518  df-lvols 39519  df-lines 39520  df-psubsp 39522  df-pmap 39523  df-padd 39815  df-lhyp 40007  df-laut 40008  df-ldil 40123  df-ltrn 40124  df-trl 40178

This theorem is referenced by:  cdlemfnid  40583  trlord  40588  dih1dimb2  41260