djajN - Mathbox for Norm Megill

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Theorem djajN 39996

Description: Transfer lattice join to DVecA partial vector space closed subspace join. Part of Lemma M of [Crawley] p. 120 line 29, with closed subspace join rather than subspace sum. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
djaj.k	⊢ ∨ = (join‘𝐾)
djaj.h	⊢ 𝐻 = (LHyp‘𝐾)
djaj.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
djaj.j	⊢ 𝐽 = ((vA‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
djajN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 ∨ 𝑌)) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌)))

**Proof of Theorem djajN**
Step	Hyp	Ref	Expression
1		hllat 38221	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2	1	ad2antrr 724	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝐾 ∈ Lat)
3		hlop 38220	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
4	3	ad2antrr 724	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝐾 ∈ OP)
5		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		djaj.h	. . . . . . . . . 10 ⊢ 𝐻 = (LHyp‘𝐾)
7		djaj.i	. . . . . . . . . 10 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
8	5, 6, 7	diadmclN 39896	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ (Base‘𝐾))
9	8	adantrr 715	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑋 ∈ (Base‘𝐾))
10		eqid 2732	. . . . . . . . 9 ⊢ (oc‘𝐾) = (oc‘𝐾)
11	5, 10	opoccl 38052	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
12	4, 9, 11	syl2anc 584	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
13	5, 6	lhpbase 38857	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
14	13	ad2antlr 725	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑊 ∈ (Base‘𝐾))
15	5, 10	opoccl 38052	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
16	4, 14, 15	syl2anc 584	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
17		djaj.k	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
18	5, 17	latjcl 18388	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
19	2, 12, 16, 18	syl3anc 1371	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
20		eqid 2732	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
21	5, 20	latmcl 18389	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
22	2, 19, 14, 21	syl3anc 1371	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
23	5, 6, 7	diadmclN 39896	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ dom 𝐼) → 𝑌 ∈ (Base‘𝐾))
24	23	adantrl 714	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑌 ∈ (Base‘𝐾))
25	5, 10	opoccl 38052	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑌) ∈ (Base‘𝐾))
26	4, 24, 25	syl2anc 584	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑌) ∈ (Base‘𝐾))
27	5, 17	latjcl 18388	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑌) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
28	2, 26, 16, 27	syl3anc 1371	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
29	5, 20	latmcl 18389	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
30	2, 28, 14, 29	syl3anc 1371	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
31	5, 20	latmcl 18389	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ ((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾)) → (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))
32	2, 22, 30, 31	syl3anc 1371	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))
33		eqid 2732	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
34	5, 33, 20	latmle2 18414	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾)) → (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(le‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))
35	2, 22, 30, 34	syl3anc 1371	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(le‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))
36	5, 33, 20	latmle2 18414	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
37	2, 28, 14, 36	syl3anc 1371	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
38	5, 33, 2, 32, 30, 14, 35, 37	lattrd 18395	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(le‘𝐾)𝑊)
39	5, 33, 6, 7	diaeldm 39895	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ dom 𝐼 ↔ ((((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ (Base‘𝐾) ∧ (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(le‘𝐾)𝑊)))
40	39	adantr 481	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ dom 𝐼 ↔ ((((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ (Base‘𝐾) ∧ (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(le‘𝐾)𝑊)))
41	32, 38, 40	mpbir2and 711	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ dom 𝐼)
42		eqid 2732	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
43		eqid 2732	. . . 4 ⊢ ((ocA‘𝐾)‘𝑊) = ((ocA‘𝐾)‘𝑊)
44	17, 20, 10, 6, 42, 7, 43	diaocN 39984	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ dom 𝐼) → (𝐼‘((((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = (((ocA‘𝐾)‘𝑊)‘(𝐼‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))))
45	41, 44	syldan 591	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘((((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = (((ocA‘𝐾)‘𝑊)‘(𝐼‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))))
46		hloml 38215	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)
47	46	ad2antrr 724	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝐾 ∈ OML)
48	5, 17	latjcl 18388	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∨ 𝑌) ∈ (Base‘𝐾))
49	2, 9, 24, 48	syl3anc 1371	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋 ∨ 𝑌) ∈ (Base‘𝐾))
50	33, 6, 7	diadmleN 39897	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋(le‘𝐾)𝑊)
51	50	adantrr 715	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑋(le‘𝐾)𝑊)
52	33, 6, 7	diadmleN 39897	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ dom 𝐼) → 𝑌(le‘𝐾)𝑊)
53	52	adantrl 714	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑌(le‘𝐾)𝑊)
54	5, 33, 17	latjle12 18399	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑋(le‘𝐾)𝑊 ∧ 𝑌(le‘𝐾)𝑊) ↔ (𝑋 ∨ 𝑌)(le‘𝐾)𝑊))
55	2, 9, 24, 14, 54	syl13anc 1372	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((𝑋(le‘𝐾)𝑊 ∧ 𝑌(le‘𝐾)𝑊) ↔ (𝑋 ∨ 𝑌)(le‘𝐾)𝑊))
56	51, 53, 55	mpbi2and 710	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋 ∨ 𝑌)(le‘𝐾)𝑊)
57	5, 33, 17, 20, 10	omlspjN 38119	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ ((𝑋 ∨ 𝑌) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ (𝑋 ∨ 𝑌)(le‘𝐾)𝑊) → (((𝑋 ∨ 𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) = (𝑋 ∨ 𝑌))
58	47, 49, 14, 56, 57	syl121anc 1375	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((𝑋 ∨ 𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) = (𝑋 ∨ 𝑌))
59	5, 17	latjidm 18411	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑊) ∨ ((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
60	2, 16, 59	syl2anc 584	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘𝑊) ∨ ((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
61	60	oveq2d 7421	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((𝑋 ∨ 𝑌) ∨ (((oc‘𝐾)‘𝑊) ∨ ((oc‘𝐾)‘𝑊))) = ((𝑋 ∨ 𝑌) ∨ ((oc‘𝐾)‘𝑊)))
62	5, 17	latjass 18432	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ ((𝑋 ∨ 𝑌) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))) → (((𝑋 ∨ 𝑌) ∨ ((oc‘𝐾)‘𝑊)) ∨ ((oc‘𝐾)‘𝑊)) = ((𝑋 ∨ 𝑌) ∨ (((oc‘𝐾)‘𝑊) ∨ ((oc‘𝐾)‘𝑊))))
63	2, 49, 16, 16, 62	syl13anc 1372	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((𝑋 ∨ 𝑌) ∨ ((oc‘𝐾)‘𝑊)) ∨ ((oc‘𝐾)‘𝑊)) = ((𝑋 ∨ 𝑌) ∨ (((oc‘𝐾)‘𝑊) ∨ ((oc‘𝐾)‘𝑊))))
64		hlol 38219	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
65	64	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝐾 ∈ OL)
66	5, 17, 20, 10	oldmm2 38076	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∨ 𝑌) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(((oc‘𝐾)‘(𝑋 ∨ 𝑌))(meet‘𝐾)𝑊)) = ((𝑋 ∨ 𝑌) ∨ ((oc‘𝐾)‘𝑊)))
67	65, 49, 14, 66	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(((oc‘𝐾)‘(𝑋 ∨ 𝑌))(meet‘𝐾)𝑊)) = ((𝑋 ∨ 𝑌) ∨ ((oc‘𝐾)‘𝑊)))
68	5, 17, 20, 10	oldmj1 38079	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(𝑋 ∨ 𝑌)) = (((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)))
69	65, 9, 24, 68	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(𝑋 ∨ 𝑌)) = (((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)))
70	5, 33, 20	latleeqm1 18416	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋(meet‘𝐾)𝑊) = 𝑋))
71	2, 9, 14, 70	syl3anc 1371	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋(meet‘𝐾)𝑊) = 𝑋))
72	51, 71	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋(meet‘𝐾)𝑊) = 𝑋)
73	72	fveq2d 6892	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊)) = ((oc‘𝐾)‘𝑋))
74	5, 17, 20, 10	oldmm1 38075	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊)) = (((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊)))
75	65, 9, 14, 74	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊)) = (((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊)))
76	73, 75	eqtr3d 2774	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑋) = (((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊)))
77	5, 33, 20	latleeqm1 18416	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑌(le‘𝐾)𝑊 ↔ (𝑌(meet‘𝐾)𝑊) = 𝑌))
78	2, 24, 14, 77	syl3anc 1371	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑌(le‘𝐾)𝑊 ↔ (𝑌(meet‘𝐾)𝑊) = 𝑌))
79	53, 78	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑌(meet‘𝐾)𝑊) = 𝑌)
80	79	fveq2d 6892	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(𝑌(meet‘𝐾)𝑊)) = ((oc‘𝐾)‘𝑌))
81	5, 17, 20, 10	oldmm1 38075	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ OL ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(𝑌(meet‘𝐾)𝑊)) = (((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)))
82	65, 24, 14, 81	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(𝑌(meet‘𝐾)𝑊)) = (((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)))
83	80, 82	eqtr3d 2774	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑌) = (((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)))
84	76, 83	oveq12d 7423	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)) = ((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)(((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))))
85	69, 84	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(𝑋 ∨ 𝑌)) = ((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)(((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))))
86	85	oveq1d 7420	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘(𝑋 ∨ 𝑌))(meet‘𝐾)𝑊) = (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)(((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊))
87	5, 20	latmmdir 38093	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ OL ∧ ((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ (((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)(((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊) = (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
88	65, 19, 28, 14, 87	syl13anc 1372	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)(((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊) = (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
89	86, 88	eqtrd 2772	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘(𝑋 ∨ 𝑌))(meet‘𝐾)𝑊) = (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
90	89	fveq2d 6892	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(((oc‘𝐾)‘(𝑋 ∨ 𝑌))(meet‘𝐾)𝑊)) = ((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))))
91	67, 90	eqtr3d 2774	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((𝑋 ∨ 𝑌) ∨ ((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))))
92	91	oveq1d 7420	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((𝑋 ∨ 𝑌) ∨ ((oc‘𝐾)‘𝑊)) ∨ ((oc‘𝐾)‘𝑊)) = (((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ∨ ((oc‘𝐾)‘𝑊)))
93	63, 92	eqtr3d 2774	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((𝑋 ∨ 𝑌) ∨ (((oc‘𝐾)‘𝑊) ∨ ((oc‘𝐾)‘𝑊))) = (((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ∨ ((oc‘𝐾)‘𝑊)))
94	61, 93	eqtr3d 2774	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((𝑋 ∨ 𝑌) ∨ ((oc‘𝐾)‘𝑊)) = (((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ∨ ((oc‘𝐾)‘𝑊)))
95	94	oveq1d 7420	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((𝑋 ∨ 𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) = ((((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))
96	58, 95	eqtr3d 2774	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋 ∨ 𝑌) = ((((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))
97	96	fveq2d 6892	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 ∨ 𝑌)) = (𝐼‘((((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
98		simpl 483	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
99	6, 7	diaclN 39909	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ ran 𝐼)
100	99	adantrr 715	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘𝑋) ∈ ran 𝐼)
101	6, 42, 7	diaelrnN 39904	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑋) ∈ ran 𝐼) → (𝐼‘𝑋) ⊆ ((LTrn‘𝐾)‘𝑊))
102	100, 101	syldan 591	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘𝑋) ⊆ ((LTrn‘𝐾)‘𝑊))
103	6, 7	diaclN 39909	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ dom 𝐼) → (𝐼‘𝑌) ∈ ran 𝐼)
104	103	adantrl 714	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘𝑌) ∈ ran 𝐼)
105	6, 42, 7	diaelrnN 39904	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑌) ∈ ran 𝐼) → (𝐼‘𝑌) ⊆ ((LTrn‘𝐾)‘𝑊))
106	104, 105	syldan 591	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘𝑌) ⊆ ((LTrn‘𝐾)‘𝑊))
107		djaj.j	. . . . 5 ⊢ 𝐽 = ((vA‘𝐾)‘𝑊)
108	6, 42, 7, 43, 107	djavalN 39994	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐼‘𝑋) ⊆ ((LTrn‘𝐾)‘𝑊) ∧ (𝐼‘𝑌) ⊆ ((LTrn‘𝐾)‘𝑊))) → ((𝐼‘𝑋)𝐽(𝐼‘𝑌)) = (((ocA‘𝐾)‘𝑊)‘((((ocA‘𝐾)‘𝑊)‘(𝐼‘𝑋)) ∩ (((ocA‘𝐾)‘𝑊)‘(𝐼‘𝑌)))))
109	98, 102, 106, 108	syl12anc 835	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((𝐼‘𝑋)𝐽(𝐼‘𝑌)) = (((ocA‘𝐾)‘𝑊)‘((((ocA‘𝐾)‘𝑊)‘(𝐼‘𝑋)) ∩ (((ocA‘𝐾)‘𝑊)‘(𝐼‘𝑌)))))
110	5, 33, 20	latmle2 18414	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
111	2, 19, 14, 110	syl3anc 1371	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
112	5, 33, 6, 7	diaeldm 39895	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
113	112	adantr 481	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
114	22, 111, 113	mpbir2and 711	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)
115	5, 33, 6, 7	diaeldm 39895	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
116	115	adantr 481	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
117	30, 37, 116	mpbir2and 711	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)
118	20, 6, 7	diameetN 39915	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ∧ ((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)) → (𝐼‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) = ((𝐼‘((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∩ (𝐼‘((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))))
119	98, 114, 117, 118	syl12anc 835	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) = ((𝐼‘((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∩ (𝐼‘((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))))
120	17, 20, 10, 6, 42, 7, 43	diaocN 39984	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = (((ocA‘𝐾)‘𝑊)‘(𝐼‘𝑋)))
121	120	adantrr 715	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = (((ocA‘𝐾)‘𝑊)‘(𝐼‘𝑋)))
122	17, 20, 10, 6, 42, 7, 43	diaocN 39984	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ dom 𝐼) → (𝐼‘((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = (((ocA‘𝐾)‘𝑊)‘(𝐼‘𝑌)))
123	122	adantrl 714	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = (((ocA‘𝐾)‘𝑊)‘(𝐼‘𝑌)))
124	121, 123	ineq12d 4212	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((𝐼‘((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∩ (𝐼‘((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) = ((((ocA‘𝐾)‘𝑊)‘(𝐼‘𝑋)) ∩ (((ocA‘𝐾)‘𝑊)‘(𝐼‘𝑌))))
125	119, 124	eqtrd 2772	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) = ((((ocA‘𝐾)‘𝑊)‘(𝐼‘𝑋)) ∩ (((ocA‘𝐾)‘𝑊)‘(𝐼‘𝑌))))
126	125	fveq2d 6892	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((ocA‘𝐾)‘𝑊)‘(𝐼‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))) = (((ocA‘𝐾)‘𝑊)‘((((ocA‘𝐾)‘𝑊)‘(𝐼‘𝑋)) ∩ (((ocA‘𝐾)‘𝑊)‘(𝐼‘𝑌)))))
127	109, 126	eqtr4d 2775	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((𝐼‘𝑋)𝐽(𝐼‘𝑌)) = (((ocA‘𝐾)‘𝑊)‘(𝐼‘(((((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))))
128	45, 97, 127	3eqtr4d 2782	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 ∨ 𝑌)) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5147 dom cdm 5675 ran crn 5676 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 lecple 17200 occoc 17201 joincjn 18260 meetcmee 18261 Latclat 18380 OPcops 38030 OLcol 38032 OMLcoml 38033 HLchlt 38208 LHypclh 38843 LTrncltrn 38960 DIsoAcdia 39887 ocAcocaN 39978 vAcdjaN 39990

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-riotaBAD 37811

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-undef 8254 df-map 8818 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-oposet 38034 df-cmtN 38035 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-llines 38357 df-lplanes 38358 df-lvols 38359 df-lines 38360 df-psubsp 38362 df-pmap 38363 df-padd 38655 df-lhyp 38847 df-laut 38848 df-ldil 38963 df-ltrn 38964 df-trl 39018 df-disoa 39888 df-docaN 39979 df-djaN 39991

This theorem is referenced by: (None)

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