Theorem doca2N 41878

Description: Double orthocomplement of partial isomorphism A. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
doca2.h	⊢ 𝐻 = (LHyp‘𝐾)
doca2.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
doca2.n	⊢ ⊥ = ((ocA‘𝐾)‘𝑊)

Assertion

Ref	Expression
doca2N	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ( ⊥ ‘( ⊥ ‘(𝐼‘𝑋))) = (𝐼‘𝑋))

Proof of Theorem doca2N

Step	Hyp	Ref	Expression
1		hlol 40113	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
2	1	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ OL)
3		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		doca2.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (LHyp‘𝐾)
5		doca2.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
6	3, 4, 5	diadmclN 41789	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ (Base‘𝐾))
7	3, 4	lhpbase 40750	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
8	7	ad2antlr 739	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑊 ∈ (Base‘𝐾))
9		eqid 2763	. . . . . . . . . . . . 13 ⊢ (join‘𝐾) = (join‘𝐾)
10		eqid 2763	. . . . . . . . . . . . 13 ⊢ (meet‘𝐾) = (meet‘𝐾)
11		eqid 2763	. . . . . . . . . . . . 13 ⊢ (oc‘𝐾) = (oc‘𝐾)
12	3, 9, 10, 11	oldmm1 39969	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊)) = (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊)))
13	2, 6, 8, 12	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊)) = (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊)))
14	13	oveq1d 7427	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊))(meet‘𝐾)𝑊) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))
15	14	eqcomd 2769	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) = (((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊))(meet‘𝐾)𝑊))
16	15	fveq2d 6887	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = ((oc‘𝐾)‘(((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊))(meet‘𝐾)𝑊)))
17		hllat 40115	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
18	17	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ Lat)
19	3, 10	latmcl 18497	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑋(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
20	18, 6, 8, 19	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
21	3, 9, 10, 11	oldmm2 39970	. . . . . . . . 9 ⊢ ((𝐾 ∈ OL ∧ (𝑋(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊))(meet‘𝐾)𝑊)) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)))
22	2, 20, 8, 21	syl3anc 1398	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘(((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊))(meet‘𝐾)𝑊)) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)))
23	16, 22	eqtrd 2798	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)))
24	23	oveq1d 7427	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊)) = (((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊)))
25		hlop 40114	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
26	25	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ OP)
27	3, 11	opoccl 39946	. . . . . . . . 9 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
28	26, 8, 27	syl2anc 595	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
29	3, 9	latjass 18540	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ ((𝑋(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))) → (((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)(((oc‘𝐾)‘𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))))
30	18, 20, 28, 28, 29	syl13anc 1399	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)(((oc‘𝐾)‘𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))))
31	3, 9	latjidm 18519	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
32	18, 28, 31	syl2anc 595	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((oc‘𝐾)‘𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
33	32	oveq2d 7428	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)(((oc‘𝐾)‘𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)))
34	30, 33	eqtrd 2798	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)))
35	24, 34	eqtrd 2798	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)))
36	35	oveq1d 7427	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) = (((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))
37		hloml 40109	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)
38	37	ad2antrr 738	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ OML)
39		eqid 2763	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
40	3, 39, 10	latmle2 18522	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑋(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
41	18, 6, 8, 40	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
42	3, 39, 9, 10, 11	omlspjN 40013	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ ((𝑋(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ (𝑋(meet‘𝐾)𝑊)(le‘𝐾)𝑊) → (((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) = (𝑋(meet‘𝐾)𝑊))
43	38, 20, 8, 41, 42	syl121anc 1402	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) = (𝑋(meet‘𝐾)𝑊))
44	39, 4, 5	diadmleN 41790	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋(le‘𝐾)𝑊)
45	3, 39, 10	latleeqm1 18524	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋(meet‘𝐾)𝑊) = 𝑋))
46	18, 6, 8, 45	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋(meet‘𝐾)𝑊) = 𝑋))
47	44, 46	mpbid 235	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋(meet‘𝐾)𝑊) = 𝑋)
48	36, 43, 47	3eqtrrd 2803	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 = ((((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))
49	48	fveq2d 6887	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = (𝐼‘((((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
50	3, 11	opoccl 39946	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
51	26, 6, 50	syl2anc 595	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
52	3, 9	latjcl 18496	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
53	18, 51, 28, 52	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
54	3, 10	latmcl 18497	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
55	18, 53, 8, 54	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
56	3, 39, 10	latmle2 18522	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
57	18, 53, 8, 56	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
58	3, 39, 4, 5	diaeldm 41788	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
59	58	adantr 485	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
60	55, 57, 59	mpbir2and 725	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)
61		eqid 2763	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
62		doca2.n	. . . 4 ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)
63	9, 10, 11, 4, 61, 5, 62	diaocN 41877	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼) → (𝐼‘((((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = ( ⊥ ‘(𝐼‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))))
64	60, 63	syldan 602	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘((((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = ( ⊥ ‘(𝐼‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))))
65	9, 10, 11, 4, 61, 5, 62	diaocN 41877	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = ( ⊥ ‘(𝐼‘𝑋)))
66	65	fveq2d 6887	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ( ⊥ ‘(𝐼‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) = ( ⊥ ‘( ⊥ ‘(𝐼‘𝑋))))
67	49, 64, 66	3eqtrrd 2803	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ( ⊥ ‘( ⊥ ‘(𝐼‘𝑋))) = (𝐼‘𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   class class class wbr 5110  dom cdm 5663  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  lecple 17318  occoc 17319  joincjn 18368  meetcmee 18369  Latclat 18488  OPcops 39924  OLcol 39926  OMLcoml 39927  HLchlt 40102  LHypclh 40736  LTrncltrn 40853  DIsoAcdia 41780  ocAcocaN 41871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-riotaBAD 39705

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-undef 8270  df-map 8827  df-proset 18351  df-poset 18370  df-plt 18385  df-lub 18401  df-glb 18402  df-join 18403  df-meet 18404  df-p0 18480  df-p1 18481  df-lat 18489  df-clat 18556  df-oposet 39928  df-cmtN 39929  df-ol 39930  df-oml 39931  df-covers 40018  df-ats 40019  df-atl 40050  df-cvlat 40074  df-hlat 40103  df-llines 40250  df-lplanes 40251  df-lvols 40252  df-lines 40253  df-psubsp 40255  df-pmap 40256  df-padd 40548  df-lhyp 40740  df-laut 40741  df-ldil 40856  df-ltrn 40857  df-trl 40911  df-disoa 41781  df-docaN 41872

This theorem is referenced by:  doca3N  41879