Theorem ltrncnv 40406

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 2736	. . . 4 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
3		ltrncnv.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4	1, 2, 3	ltrnldil 40382	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
5	1, 2	ldilcnv 40375	. . 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 2736	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
13		eqid 2736	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
14	12, 13, 1, 3	ltrncnvel 40402	. . . . . . 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 40402	. . . . . . 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 2736	. . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾)
21		eqid 2736	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
22	12, 20, 21, 13, 1, 3	ltrnu 40381	. . . . . 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 2736	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
25	24, 1, 3	ltrn1o 40384	. . . . . . . . . 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 39549	. . . . . . . . . 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 7223	. . . . . . . . 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 7374	. . . . . . 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 40401	. . . . . . . . 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 39628	. . . . . . . 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 2771	. . . . . 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 7373	. . . . 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 39549	. . . . . . . . . 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 7223	. . . . . . . . 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 7374	. . . . . . 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 40401	. . . . . . . . 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 39628	. . . . . . . 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 2771	. . . . . 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 7373	. . . . 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 2779	. . . 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 3177	. 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 40379	. . 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 1541   ∈ wcel 2113  ∀wral 3051   class class class wbr 5098  ^◡ccnv 5623  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  lecple 17184  joincjn 18234  meetcmee 18235  Atomscatm 39523  HLchlt 39610  LHypclh 40244  LDilcldil 40360  LTrncltrn 40361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8765  df-proset 18217  df-poset 18236  df-plt 18251  df-lub 18267  df-glb 18268  df-join 18269  df-p0 18346  df-lat 18355  df-oposet 39436  df-ol 39438  df-oml 39439  df-covers 39526  df-ats 39527  df-atl 39558  df-cvlat 39582  df-hlat 39611  df-lhyp 40248  df-laut 40249  df-ldil 40364  df-ltrn 40365

This theorem is referenced by:  trlcnv  40425  trlcocnv  40980  trlcoabs2N  40982  trlcoat  40983  trlcocnvat  40984  trlcone  40988  cdlemg46  40995  tgrpgrplem  41009  tendoicl  41056  cdlemh1  41075  cdlemh2  41076  cdlemh  41077  cdlemi2  41079  cdlemi  41080  cdlemk2  41092  cdlemk3  41093  cdlemk4  41094  cdlemk8  41098  cdlemk9  41099  cdlemk9bN  41100  cdlemkvcl  41102  cdlemk10  41103  cdlemk11  41109  cdlemk12  41110  cdlemk14  41114  cdlemk11u  41131  cdlemk12u  41132  cdlemk37  41174  cdlemkfid1N  41181  cdlemkid1  41182  cdlemkid2  41184  tendocnv  41281  tendospcanN  41283  dvhgrp  41367  cdlemn8  41464  dihopelvalcpre  41508  dih1  41546  dihglbcpreN  41560  dihjatcclem3  41680  dihjatcclem4  41681