Theorem ltrncnv 41203

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 2761	. . . 4 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
3		ltrncnv.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4	1, 2, 3	ltrnldil 41179	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
5	1, 2	ldilcnv 41172	. . 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 2761	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
13		eqid 2761	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
14	12, 13, 1, 3	ltrncnvel 41199	. . . . . . 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 41199	. . . . . . 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 2761	. . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾)
21		eqid 2761	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
22	12, 20, 21, 13, 1, 3	ltrnu 41178	. . . . . 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 2761	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
25	24, 1, 3	ltrn1o 41181	. . . . . . . . . 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 40346	. . . . . . . . . 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 7285	. . . . . . . . 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 7436	. . . . . . 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 41198	. . . . . . . . 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 40425	. . . . . . . 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 2796	. . . . . 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 7435	. . . . 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 40346	. . . . . . . . . 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 7285	. . . . . . . . 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 7436	. . . . . . 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 41198	. . . . . . . . 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 40425	. . . . . . . 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 2796	. . . . . 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 7435	. . . . 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 2804	. . . 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 3204	. 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 41176	. . 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 1570   ∈ wcel 2145  ∀wral 3077   class class class wbr 5103  ^◡ccnv 5650  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  lecple 17435  joincjn 18485  meetcmee 18486  Atomscatm 40320  HLchlt 40407  LHypclh 41041  LDilcldil 41157  LTrncltrn 41158

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  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 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8849  df-proset 18468  df-poset 18487  df-plt 18502  df-lub 18518  df-glb 18519  df-join 18520  df-p0 18597  df-lat 18606  df-oposet 40233  df-ol 40235  df-oml 40236  df-covers 40323  df-ats 40324  df-atl 40355  df-cvlat 40379  df-hlat 40408  df-lhyp 41045  df-laut 41046  df-ldil 41161  df-ltrn 41162

This theorem is used by:  trlcnv  41222  trlcocnv  41777  trlcoabs2N  41779  trlcoat  41780  trlcocnvat  41781  trlcone  41785  cdlemg46  41792  tgrpgrplem  41806  tendoicl  41853  cdlemh1  41872  cdlemh2  41873  cdlemh  41874  cdlemi2  41876  cdlemi  41877  cdlemk2  41889  cdlemk3  41890  cdlemk4  41891  cdlemk8  41895  cdlemk9  41896  cdlemk9bN  41897  cdlemkvcl  41899  cdlemk10  41900  cdlemk11  41906  cdlemk12  41907  cdlemk14  41911  cdlemk11u  41928  cdlemk12u  41929  cdlemk37  41971  cdlemkfid1N  41978  cdlemkid1  41979  cdlemkid2  41981  tendocnv  42078  tendospcanN  42080  dvhgrp  42164  cdlemn8  42261  dihopelvalcpre  42305  dih1  42343  dihglbcpreN  42357  dihjatcclem3  42477  dihjatcclem4  42478