Theorem doca2N 39140

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 37375	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
2	1	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ OL)
3		eqid 2738	. . . . . . . . . . . . 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 39051	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ (Base‘𝐾))
7	3, 4	lhpbase 38012	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
8	7	ad2antlr 724	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑊 ∈ (Base‘𝐾))
9		eqid 2738	. . . . . . . . . . . . 13 ⊢ (join‘𝐾) = (join‘𝐾)
10		eqid 2738	. . . . . . . . . . . . 13 ⊢ (meet‘𝐾) = (meet‘𝐾)
11		eqid 2738	. . . . . . . . . . . . 13 ⊢ (oc‘𝐾) = (oc‘𝐾)
12	3, 9, 10, 11	oldmm1 37231	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 7290	. . . . . . . . . 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 2744	. . . . . . . . 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 6778	. . . . . . . 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 37377	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
18	17	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ Lat)
19	3, 10	latmcl 18158	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑋(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
20	18, 6, 8, 19	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
21	3, 9, 10, 11	oldmm2 37232	. . . . . . . . 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 1370	. . . . . . . 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 2778	. . . . . . 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 7290	. . . . . 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 37376	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
26	25	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ OP)
27	3, 11	opoccl 37208	. . . . . . . . 9 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
28	26, 8, 27	syl2anc 584	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
29	3, 9	latjass 18201	. . . . . . . 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 1371	. . . . . . 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 18180	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
32	18, 28, 31	syl2anc 584	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((oc‘𝐾)‘𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
33	32	oveq2d 7291	. . . . . . 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 2778	. . . . . 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 2778	. . . . 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 7290	. . . 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 37371	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)
38	37	ad2antrr 723	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ OML)
39		eqid 2738	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
40	3, 39, 10	latmle2 18183	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑋(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
41	18, 6, 8, 40	syl3anc 1370	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
42	3, 39, 9, 10, 11	omlspjN 37275	. . . . 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 1374	. . . 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 39052	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋(le‘𝐾)𝑊)
45	3, 39, 10	latleeqm1 18185	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋(meet‘𝐾)𝑊) = 𝑋))
46	18, 6, 8, 45	syl3anc 1370	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋(meet‘𝐾)𝑊) = 𝑋))
47	44, 46	mpbid 231	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋(meet‘𝐾)𝑊) = 𝑋)
48	36, 43, 47	3eqtrrd 2783	. . 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 6778	. 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 37208	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
51	26, 6, 50	syl2anc 584	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
52	3, 9	latjcl 18157	. . . . . 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 1370	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
54	3, 10	latmcl 18158	. . . . 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 1370	. . . 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 18183	. . . . 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 1370	. . . 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 39050	. . . . 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 481	. . . 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 710	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)
61		eqid 2738	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
62		doca2.n	. . . 4 ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)
63	9, 10, 11, 4, 61, 5, 62	diaocN 39139	. . 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 591	. 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 39139	. . 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 6778	. 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 2783	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ( ⊥ ‘( ⊥ ‘(𝐼‘𝑋))) = (𝐼‘𝑋))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   class class class wbr 5074  dom cdm 5589  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  lecple 16969  occoc 16970  joincjn 18029  meetcmee 18030  Latclat 18149  OPcops 37186  OLcol 37188  OMLcoml 37189  HLchlt 37364  LHypclh 37998  LTrncltrn 38115  DIsoAcdia 39042  ocAcocaN 39133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-riotaBAD 36967

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-undef 8089  df-map 8617  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-p1 18144  df-lat 18150  df-clat 18217  df-oposet 37190  df-cmtN 37191  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-llines 37512  df-lplanes 37513  df-lvols 37514  df-lines 37515  df-psubsp 37517  df-pmap 37518  df-padd 37810  df-lhyp 38002  df-laut 38003  df-ldil 38118  df-ltrn 38119  df-trl 38173  df-disoa 39043  df-docaN 39134

This theorem is referenced by:  doca3N  39141