Theorem ltrncnv 40140

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 2729	. . . 4 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
3		ltrncnv.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4	1, 2, 3	ltrnldil 40116	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
5	1, 2	ldilcnv 40109	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) → ◡𝐹 ∈ ((LDil‘𝐾)‘𝑊))
6	4, 5	syldan 591	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ ((LDil‘𝐾)‘𝑊))
7		simp1 1136	. . . . . 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 1198	. . . . . . 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 1199	. . . . . . 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 1200	. . . . . . 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 1202	. . . . . . 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 2729	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
13		eqid 2729	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
14	12, 13, 1, 3	ltrncnvel 40136	. . . . . . 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 1376	. . . . . 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 1201	. . . . . . 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 1203	. . . . . . 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 40136	. . . . . . 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 1376	. . . . . 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 2729	. . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾)
21		eqid 2729	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
22	12, 20, 21, 13, 1, 3	ltrnu 40115	. . . . . 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 1373	. . . . 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 2729	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
25	24, 1, 3	ltrn1o 40118	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
26	25	3ad2ant1 1133	. . . . . . . . 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 39282	. . . . . . . . . 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 7252	. . . . . . . . 9 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾)) → (𝐹‘(◡𝐹‘𝑝)) = 𝑝)
30	26, 28, 29	syl2anc 584	. . . . . . . 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 7403	. . . . . . 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 1237	. . . . . . . 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 40135	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑝 ∈ (Atoms‘𝐾)) → (◡𝐹‘𝑝) ∈ (Atoms‘𝐾))
34	8, 9, 10, 33	syl3anc 1373	. . . . . . . 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 39361	. . . . . . . 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 1373	. . . . . . 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 2764	. . . . . 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 7402	. . . . 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 39282	. . . . . . . . . 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 7252	. . . . . . . . 9 ⊢ ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝐹‘(◡𝐹‘𝑞)) = 𝑞)
42	26, 40, 41	syl2anc 584	. . . . . . . 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 7403	. . . . . . 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 40135	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑞 ∈ (Atoms‘𝐾)) → (◡𝐹‘𝑞) ∈ (Atoms‘𝐾))
45	8, 9, 16, 44	syl3anc 1373	. . . . . . . 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 39361	. . . . . . . 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 1373	. . . . . . 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 2764	. . . . . 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 7402	. . . . 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 2772	. . . 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 1119	. . 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 3178	. 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 40113	. . 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 480	. 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 713	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   class class class wbr 5107  ^◡ccnv 5637  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  lecple 17227  joincjn 18272  meetcmee 18273  Atomscatm 39256  HLchlt 39343  LHypclh 39978  LDilcldil 40094  LTrncltrn 40095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-p0 18384  df-lat 18391  df-oposet 39169  df-ol 39171  df-oml 39172  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315  df-hlat 39344  df-lhyp 39982  df-laut 39983  df-ldil 40098  df-ltrn 40099

This theorem is referenced by:  trlcnv  40159  trlcocnv  40714  trlcoabs2N  40716  trlcoat  40717  trlcocnvat  40718  trlcone  40722  cdlemg46  40729  tgrpgrplem  40743  tendoicl  40790  cdlemh1  40809  cdlemh2  40810  cdlemh  40811  cdlemi2  40813  cdlemi  40814  cdlemk2  40826  cdlemk3  40827  cdlemk4  40828  cdlemk8  40832  cdlemk9  40833  cdlemk9bN  40834  cdlemkvcl  40836  cdlemk10  40837  cdlemk11  40843  cdlemk12  40844  cdlemk14  40848  cdlemk11u  40865  cdlemk12u  40866  cdlemk37  40908  cdlemkfid1N  40915  cdlemkid1  40916  cdlemkid2  40918  tendocnv  41015  tendospcanN  41017  dvhgrp  41101  cdlemn8  41198  dihopelvalcpre  41242  dih1  41280  dihglbcpreN  41294  dihjatcclem3  41414  dihjatcclem4  41415