Theorem doca2N 42151

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 40386	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
2	1	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ OL)
3		eqid 2761	. . . . . . . . . . . . 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 42062	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ (Base‘𝐾))
7	3, 4	lhpbase 41023	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
8	7	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑊 ∈ (Base‘𝐾))
9		eqid 2761	. . . . . . . . . . . . 13 ⊢ (join‘𝐾) = (join‘𝐾)
10		eqid 2761	. . . . . . . . . . . . 13 ⊢ (meet‘𝐾) = (meet‘𝐾)
11		eqid 2761	. . . . . . . . . . . . 13 ⊢ (oc‘𝐾) = (oc‘𝐾)
12	3, 9, 10, 11	oldmm1 40242	. . . . . . . . . . . 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 2767	. . . . . . . . 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 6881	. . . . . . . 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 40388	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
18	17	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ Lat)
19	3, 10	latmcl 18594	. . . . . . . . . 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 40243	. . . . . . . . 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 2796	. . . . . . 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 40387	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
26	25	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ OP)
27	3, 11	opoccl 40219	. . . . . . . . 9 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
28	26, 8, 27	syl2anc 596	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
29	3, 9	latjass 18637	. . . . . . . 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 18616	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
32	18, 28, 31	syl2anc 596	. . . . . . . 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 2796	. . . . . 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 2796	. . . . 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 40382	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)
38	37	ad2antrr 739	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ OML)
39		eqid 2761	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
40	3, 39, 10	latmle2 18619	. . . . . 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 40286	. . . . 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 42063	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋(le‘𝐾)𝑊)
45	3, 39, 10	latleeqm1 18621	. . . . . 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 2801	. . 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 6881	. 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 40219	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
51	26, 6, 50	syl2anc 596	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
52	3, 9	latjcl 18593	. . . . . 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 18594	. . . . 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 18619	. . . . 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 42061	. . . . 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 486	. . . 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 726	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)
61		eqid 2761	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
62		doca2.n	. . . 4 ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)
63	9, 10, 11, 4, 61, 5, 62	diaocN 42150	. . 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 603	. 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 42150	. . 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 6881	. 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 2801	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ( ⊥ ‘( ⊥ ‘(𝐼‘𝑋))) = (𝐼‘𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   class class class wbr 5103  dom cdm 5651  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  lecple 17415  occoc 17416  joincjn 18465  meetcmee 18466  Latclat 18585  OPcops 40197  OLcol 40199  OMLcoml 40200  HLchlt 40375  LHypclh 41009  LTrncltrn 41126  DIsoAcdia 42053  ocAcocaN 42144

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 7740  ax-riotaBAD 39978

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  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-int 4908  df-iun 4953  df-iin 4954  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 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-undef 8274  df-map 8833  df-proset 18448  df-poset 18467  df-plt 18482  df-lub 18498  df-glb 18499  df-join 18500  df-meet 18501  df-p0 18577  df-p1 18578  df-lat 18586  df-clat 18653  df-oposet 40201  df-cmtN 40202  df-ol 40203  df-oml 40204  df-covers 40291  df-ats 40292  df-atl 40323  df-cvlat 40347  df-hlat 40376  df-llines 40523  df-lplanes 40524  df-lvols 40525  df-lines 40526  df-psubsp 40528  df-pmap 40529  df-padd 40821  df-lhyp 41013  df-laut 41014  df-ldil 41129  df-ltrn 41130  df-trl 41184  df-disoa 42054  df-docaN 42145

This theorem is used by:  doca3N  42152