Theorem ltrncnv 41340

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 2760	. . . 4 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
3		ltrncnv.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4	1, 2, 3	ltrnldil 41316	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
5	1, 2	ldilcnv 41309	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) → ◡𝐹 ∈ ((LDil‘𝐾)‘𝑊))
6	4, 5	syldan 603	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ ((LDil‘𝐾)‘𝑊))
7		simp1 1154	. . . . . 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 1216	. . . . . . 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 1217	. . . . . . 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 1218	. . . . . . 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 1220	. . . . . . 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 2760	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
13		eqid 2760	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
14	12, 13, 1, 3	ltrncnvel 41336	. . . . . . 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 1401	. . . . . 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 1219	. . . . . . 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 1221	. . . . . . 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 41336	. . . . . . 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 1401	. . . . . 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 2760	. . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾)
21		eqid 2760	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
22	12, 20, 21, 13, 1, 3	ltrnu 41315	. . . . . 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 1398	. . . . 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 2760	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
25	24, 1, 3	ltrn1o 41318	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
26	25	3ad2ant1 1151	. . . . . . . . 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 40483	. . . . . . . . . 10 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
28	10, 27	syl 18	. . . . . . . . 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 7289	. . . . . . . . 9 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾)) → (𝐹‘(◡𝐹‘𝑝)) = 𝑝)
30	26, 28, 29	syl2anc 596	. . . . . . . 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 7440	. . . . . . 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 1255	. . . . . . . 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 41335	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑝 ∈ (Atoms‘𝐾)) → (◡𝐹‘𝑝) ∈ (Atoms‘𝐾))
34	8, 9, 10, 33	syl3anc 1398	. . . . . . . 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 40562	. . . . . . . 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 1398	. . . . . . 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 2795	. . . . . 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 7439	. . . . 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 40483	. . . . . . . . . 10 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
40	16, 39	syl 18	. . . . . . . . 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 7289	. . . . . . . . 9 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝐹‘(◡𝐹‘𝑞)) = 𝑞)
42	26, 40, 41	syl2anc 596	. . . . . . . 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 7440	. . . . . . 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 41335	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑞 ∈ (Atoms‘𝐾)) → (◡𝐹‘𝑞) ∈ (Atoms‘𝐾))
45	8, 9, 16, 44	syl3anc 1398	. . . . . . . 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 40562	. . . . . . . 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 1398	. . . . . . 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 2795	. . . . . 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 7439	. . . . 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 2803	. . . 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 1137	. . 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 41313	. . 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 486	. 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 726	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   class class class wbr 5102  ^◡ccnv 5650  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  lecple 17444  joincjn 18494  meetcmee 18495  Atomscatm 40457  HLchlt 40544  LHypclh 41178  LDilcldil 41294  LTrncltrn 41295

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8857  df-proset 18477  df-poset 18496  df-plt 18511  df-lub 18527  df-glb 18528  df-join 18529  df-p0 18606  df-lat 18615  df-oposet 40370  df-ol 40372  df-oml 40373  df-covers 40460  df-ats 40461  df-atl 40492  df-cvlat 40516  df-hlat 40545  df-lhyp 41182  df-laut 41183  df-ldil 41298  df-ltrn 41299

This theorem is used by:  trlcnv  41359  trlcocnv  41914  trlcoabs2N  41916  trlcoat  41917  trlcocnvat  41918  trlcone  41922  cdlemg46  41929  tgrpgrplem  41943  tendoicl  41990  cdlemh1  42009  cdlemh2  42010  cdlemh  42011  cdlemi2  42013  cdlemi  42014  cdlemk2  42026  cdlemk3  42027  cdlemk4  42028  cdlemk8  42032  cdlemk9  42033  cdlemk9bN  42034  cdlemkvcl  42036  cdlemk10  42037  cdlemk11  42043  cdlemk12  42044  cdlemk14  42048  cdlemk11u  42065  cdlemk12u  42066  cdlemk37  42108  cdlemkfid1N  42115  cdlemkid1  42116  cdlemkid2  42118  tendocnv  42215  tendospcanN  42217  dvhgrp  42301  cdlemn8  42398  dihopelvalcpre  42442  dih1  42480  dihglbcpreN  42494  dihjatcclem3  42614  dihjatcclem4  42615