Theorem cdlemf 41444

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 2762	. . 3 ⊢ (join‘𝐾) = (join‘𝐾)
3		cdlemf.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
4		cdlemf.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2762	. . 3 ⊢ (meet‘𝐾) = (meet‘𝐾)
6	1, 2, 3, 4, 5	cdlemf2 41443	. 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 1216	. . . . . 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 1218	. . . . . 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 1263	. . . . . 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 1219	. . . . . 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 1264	. . . . . 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 41440	. . . . . 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 1406	. . . . 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 1221	. . . . . . . . . . . . 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 7433	. . . . . . . . . . . 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 7432	. . . . . . . . . . 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 1222	. . . . . . . . . . . 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 1220	. . . . . . . . . . . 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 1307	. . . . . . . . . . . 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 1259	. . . . . . . . . . . 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 41044	. . . . . . . . . . . 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 1401	. . . . . . . . . . 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 1219	. . . . . . . . . . 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 2807	. . . . . . . . . 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 1137	. . . . . . . . 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 1139	. . . . . . . 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 1128	. . . . . . 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 421	. . . . . 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 3175	. . . . 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 16	. . . 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 1137	. . 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 3220	. 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 16	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3088   class class class wbr 5107  ‘cfv 6537  (class class class)co 7417  lecple 17355  joincjn 18405  meetcmee 18406  Atomscatm 40144  HLchlt 40231  LHypclh 40865  LTrncltrn 40982  trLctrl 41039

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-riotaBAD 39834

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-undef 8275  df-map 8832  df-proset 18388  df-poset 18407  df-plt 18422  df-lub 18438  df-glb 18439  df-join 18440  df-meet 18441  df-p0 18517  df-p1 18518  df-lat 18526  df-clat 18593  df-oposet 40057  df-ol 40059  df-oml 40060  df-covers 40147  df-ats 40148  df-atl 40179  df-cvlat 40203  df-hlat 40232  df-llines 40379  df-lplanes 40380  df-lvols 40381  df-lines 40382  df-psubsp 40384  df-pmap 40385  df-padd 40677  df-lhyp 40869  df-laut 40870  df-ldil 40985  df-ltrn 40986  df-trl 41040

This theorem is used by:  cdlemfnid  41445  trlord  41450  dih1dimb2  42122