Theorem ltrncnv 40770

Description: The converse of a lattice translation is a lattice translation. (Contributed by NM, 10-May-2013.)

Hypotheses

Ref	Expression
ltrncnv.h	⊢ 𝐻 = (LHyp‘𝐾)
ltrncnv.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
ltrncnv	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)

Proof of Theorem ltrncnv

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

Step	Hyp	Ref	Expression
1		ltrncnv.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		eqid 2762	. . . 4 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
3		ltrncnv.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4	1, 2, 3	ltrnldil 40746	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
5	1, 2	ldilcnv 40739	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) → ◡𝐹 ∈ ((LDil‘𝐾)‘𝑊))
6	4, 5	syldan 600	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ ((LDil‘𝐾)‘𝑊))
7		simp1 1149	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇))
8		simp1l 1211	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		simp1r 1212	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐹 ∈ 𝑇)
10		simp2l 1213	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
11		simp3l 1215	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ¬ 𝑝(le‘𝐾)𝑊)
12		eqid 2762	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
13		eqid 2762	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
14	12, 13, 1, 3	ltrncnvel 40766	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ((◡𝐹‘𝑝) ∈ (Atoms‘𝐾) ∧ ¬ (◡𝐹‘𝑝)(le‘𝐾)𝑊))
15	8, 9, 10, 11, 14	syl112anc 1393	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((◡𝐹‘𝑝) ∈ (Atoms‘𝐾) ∧ ¬ (◡𝐹‘𝑝)(le‘𝐾)𝑊))
16		simp2r 1214	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑞 ∈ (Atoms‘𝐾))
17		simp3r 1216	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ¬ 𝑞(le‘𝐾)𝑊)
18	12, 13, 1, 3	ltrncnvel 40766	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((◡𝐹‘𝑞) ∈ (Atoms‘𝐾) ∧ ¬ (◡𝐹‘𝑞)(le‘𝐾)𝑊))
19	8, 9, 16, 17, 18	syl112anc 1393	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((◡𝐹‘𝑞) ∈ (Atoms‘𝐾) ∧ ¬ (◡𝐹‘𝑞)(le‘𝐾)𝑊))
20		eqid 2762	. . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾)
21		eqid 2762	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
22	12, 20, 21, 13, 1, 3	ltrnu 40745	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ((◡𝐹‘𝑝) ∈ (Atoms‘𝐾) ∧ ¬ (◡𝐹‘𝑝)(le‘𝐾)𝑊) ∧ ((◡𝐹‘𝑞) ∈ (Atoms‘𝐾) ∧ ¬ (◡𝐹‘𝑞)(le‘𝐾)𝑊)) → (((◡𝐹‘𝑝)(join‘𝐾)(𝐹‘(◡𝐹‘𝑝)))(meet‘𝐾)𝑊) = (((◡𝐹‘𝑞)(join‘𝐾)(𝐹‘(◡𝐹‘𝑞)))(meet‘𝐾)𝑊))
23	7, 15, 19, 22	syl3anc 1390	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (((◡𝐹‘𝑝)(join‘𝐾)(𝐹‘(◡𝐹‘𝑝)))(meet‘𝐾)𝑊) = (((◡𝐹‘𝑞)(join‘𝐾)(𝐹‘(◡𝐹‘𝑞)))(meet‘𝐾)𝑊))
24		eqid 2762	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
25	24, 1, 3	ltrn1o 40748	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
26	25	3ad2ant1 1146	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
27	24, 13	atbase 39913	. . . . . . . . . 10 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
28	10, 27	syl 17	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑝 ∈ (Base‘𝐾))
29		f1ocnvfv2 7261	. . . . . . . . 9 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾)) → (𝐹‘(◡𝐹‘𝑝)) = 𝑝)
30	26, 28, 29	syl2anc 593	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐹‘(◡𝐹‘𝑝)) = 𝑝)
31	30	oveq2d 7412	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((◡𝐹‘𝑝)(join‘𝐾)(𝐹‘(◡𝐹‘𝑝))) = ((◡𝐹‘𝑝)(join‘𝐾)𝑝))
32		simp1ll 1250	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐾 ∈ HL)
33	12, 13, 1, 3	ltrncnvat 40765	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑝 ∈ (Atoms‘𝐾)) → (◡𝐹‘𝑝) ∈ (Atoms‘𝐾))
34	8, 9, 10, 33	syl3anc 1390	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (◡𝐹‘𝑝) ∈ (Atoms‘𝐾))
35	20, 13	hlatjcom 39992	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (◡𝐹‘𝑝) ∈ (Atoms‘𝐾) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((◡𝐹‘𝑝)(join‘𝐾)𝑝) = (𝑝(join‘𝐾)(◡𝐹‘𝑝)))
36	32, 34, 10, 35	syl3anc 1390	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((◡𝐹‘𝑝)(join‘𝐾)𝑝) = (𝑝(join‘𝐾)(◡𝐹‘𝑝)))
37	31, 36	eqtrd 2797	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((◡𝐹‘𝑝)(join‘𝐾)(𝐹‘(◡𝐹‘𝑝))) = (𝑝(join‘𝐾)(◡𝐹‘𝑝)))
38	37	oveq1d 7411	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (((◡𝐹‘𝑝)(join‘𝐾)(𝐹‘(◡𝐹‘𝑝)))(meet‘𝐾)𝑊) = ((𝑝(join‘𝐾)(◡𝐹‘𝑝))(meet‘𝐾)𝑊))
39	24, 13	atbase 39913	. . . . . . . . . 10 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
40	16, 39	syl 17	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑞 ∈ (Base‘𝐾))
41		f1ocnvfv2 7261	. . . . . . . . 9 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝐹‘(◡𝐹‘𝑞)) = 𝑞)
42	26, 40, 41	syl2anc 593	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐹‘(◡𝐹‘𝑞)) = 𝑞)
43	42	oveq2d 7412	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((◡𝐹‘𝑞)(join‘𝐾)(𝐹‘(◡𝐹‘𝑞))) = ((◡𝐹‘𝑞)(join‘𝐾)𝑞))
44	12, 13, 1, 3	ltrncnvat 40765	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑞 ∈ (Atoms‘𝐾)) → (◡𝐹‘𝑞) ∈ (Atoms‘𝐾))
45	8, 9, 16, 44	syl3anc 1390	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (◡𝐹‘𝑞) ∈ (Atoms‘𝐾))
46	20, 13	hlatjcom 39992	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (◡𝐹‘𝑞) ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((◡𝐹‘𝑞)(join‘𝐾)𝑞) = (𝑞(join‘𝐾)(◡𝐹‘𝑞)))
47	32, 45, 16, 46	syl3anc 1390	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((◡𝐹‘𝑞)(join‘𝐾)𝑞) = (𝑞(join‘𝐾)(◡𝐹‘𝑞)))
48	43, 47	eqtrd 2797	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((◡𝐹‘𝑞)(join‘𝐾)(𝐹‘(◡𝐹‘𝑞))) = (𝑞(join‘𝐾)(◡𝐹‘𝑞)))
49	48	oveq1d 7411	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (((◡𝐹‘𝑞)(join‘𝐾)(𝐹‘(◡𝐹‘𝑞)))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(◡𝐹‘𝑞))(meet‘𝐾)𝑊))
50	23, 38, 49	3eqtr3d 2805	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑝(join‘𝐾)(◡𝐹‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(◡𝐹‘𝑞))(meet‘𝐾)𝑊))
51	50	3exp 1132	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(◡𝐹‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(◡𝐹‘𝑞))(meet‘𝐾)𝑊))))
52	51	ralrimivv 3203	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(◡𝐹‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(◡𝐹‘𝑞))(meet‘𝐾)𝑊)))
53	12, 20, 21, 13, 1, 2, 3	isltrn 40743	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (◡𝐹 ∈ 𝑇 ↔ (◡𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(◡𝐹‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(◡𝐹‘𝑞))(meet‘𝐾)𝑊)))))
54	53	adantr 484	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (◡𝐹 ∈ 𝑇 ↔ (◡𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(◡𝐹‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(◡𝐹‘𝑞))(meet‘𝐾)𝑊)))))
55	6, 52, 54	mpbir2and 723	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076   class class class wbr 5100  ^◡ccnv 5646  –1-1-onto→wf1o 6520  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  lecple 17293  joincjn 18343  meetcmee 18344  Atomscatm 39887  HLchlt 39974  LHypclh 40608  LDilcldil 40724  LTrncltrn 40725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-map 8810  df-proset 18326  df-poset 18345  df-plt 18360  df-lub 18376  df-glb 18377  df-join 18378  df-p0 18455  df-lat 18464  df-oposet 39800  df-ol 39802  df-oml 39803  df-covers 39890  df-ats 39891  df-atl 39922  df-cvlat 39946  df-hlat 39975  df-lhyp 40612  df-laut 40613  df-ldil 40728  df-ltrn 40729

This theorem is referenced by:  trlcnv  40789  trlcocnv  41344  trlcoabs2N  41346  trlcoat  41347  trlcocnvat  41348  trlcone  41352  cdlemg46  41359  tgrpgrplem  41373  tendoicl  41420  cdlemh1  41439  cdlemh2  41440  cdlemh  41441  cdlemi2  41443  cdlemi  41444  cdlemk2  41456  cdlemk3  41457  cdlemk4  41458  cdlemk8  41462  cdlemk9  41463  cdlemk9bN  41464  cdlemkvcl  41466  cdlemk10  41467  cdlemk11  41473  cdlemk12  41474  cdlemk14  41478  cdlemk11u  41495  cdlemk12u  41496  cdlemk37  41538  cdlemkfid1N  41545  cdlemkid1  41546  cdlemkid2  41548  tendocnv  41645  tendospcanN  41647  dvhgrp  41731  cdlemn8  41828  dihopelvalcpre  41872  dih1  41910  dihglbcpreN  41924  dihjatcclem3  42044  dihjatcclem4  42045