Theorem cdlemf 40819

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 2736	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		cdlemf.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
4		cdlemf.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2736	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
6	1, 2, 3, 4, 5	cdlemf2 40818	. 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 40815	. . . . . 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 7374	. . . . . . . . . . . 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 7373	. . . . . . . . . . 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 40419	. . . . . . . . . . . 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 2781	. . . . . . . . . 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 3147	. . . . 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 3192	. 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 1541   ∈ wcel 2113  ∃wrex 3060   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  lecple 17184  joincjn 18234  meetcmee 18235  Atomscatm 39519  HLchlt 39606  LHypclh 40240  LTrncltrn 40357  trLctrl 40414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-riotaBAD 39209

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-undef 8215  df-map 8765  df-proset 18217  df-poset 18236  df-plt 18251  df-lub 18267  df-glb 18268  df-join 18269  df-meet 18270  df-p0 18346  df-p1 18347  df-lat 18355  df-clat 18422  df-oposet 39432  df-ol 39434  df-oml 39435  df-covers 39522  df-ats 39523  df-atl 39554  df-cvlat 39578  df-hlat 39607  df-llines 39754  df-lplanes 39755  df-lvols 39756  df-lines 39757  df-psubsp 39759  df-pmap 39760  df-padd 40052  df-lhyp 40244  df-laut 40245  df-ldil 40360  df-ltrn 40361  df-trl 40415

This theorem is referenced by:  cdlemfnid  40820  trlord  40825  dih1dimb2  41497