djajN - Mathbox for Norm Megill

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Theorem djajN 36426

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 34650	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2	1	ad2antrr 762	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝐾 ∈ Lat)
3		hlop 34649	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
4	3	ad2antrr 762	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝐾 ∈ OP)
5		eqid 2622	. . . . . . . . . 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 36326	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ (Base‘𝐾))
9	8	adantrr 753	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑋 ∈ (Base‘𝐾))
10		eqid 2622	. . . . . . . . 9 ⊢ (oc‘𝐾) = (oc‘𝐾)
11	5, 10	opoccl 34481	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
12	4, 9, 11	syl2anc 693	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
13	5, 6	lhpbase 35284	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
14	13	ad2antlr 763	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑊 ∈ (Base‘𝐾))
15	5, 10	opoccl 34481	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
16	4, 14, 15	syl2anc 693	. . . . . . 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 17051	. . . . . . 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 1326	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
20		eqid 2622	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
21	5, 20	latmcl 17052	. . . . . 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 1326	. . . . 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 36326	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ dom 𝐼) → 𝑌 ∈ (Base‘𝐾))
24	23	adantrl 752	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑌 ∈ (Base‘𝐾))
25	5, 10	opoccl 34481	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑌) ∈ (Base‘𝐾))
26	4, 24, 25	syl2anc 693	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑌) ∈ (Base‘𝐾))
27	5, 17	latjcl 17051	. . . . . . 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 1326	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
29	5, 20	latmcl 17052	. . . . . 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 1326	. . . . 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 17052	. . . . 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 1326	. . . 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 2622	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
34	5, 33, 20	latmle2 17077	. . . . . 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 1326	. . . . 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 17077	. . . . . 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 1326	. . . . 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 17058	. . . 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 36325	. . . . 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 957	. . 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 2622	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
43		eqid 2622	. . . 4 ⊢ ((ocA‘𝐾)‘𝑊) = ((ocA‘𝐾)‘𝑊)
44	17, 20, 10, 6, 42, 7, 43	diaocN 36414	. . 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 487	. 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 34644	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)
47	46	ad2antrr 762	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝐾 ∈ OML)
48	5, 17	latjcl 17051	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∨ 𝑌) ∈ (Base‘𝐾))
49	2, 9, 24, 48	syl3anc 1326	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋 ∨ 𝑌) ∈ (Base‘𝐾))
50	33, 6, 7	diadmleN 36327	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋(le‘𝐾)𝑊)
51	50	adantrr 753	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑋(le‘𝐾)𝑊)
52	33, 6, 7	diadmleN 36327	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ dom 𝐼) → 𝑌(le‘𝐾)𝑊)
53	52	adantrl 752	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑌(le‘𝐾)𝑊)
54	5, 33, 17	latjle12 17062	. . . . . . 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 1328	. . . . . 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 956	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋 ∨ 𝑌)(le‘𝐾)𝑊)
57	5, 33, 17, 20, 10	omlspjN 34548	. . . . 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 1331	. . . 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 17074	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑊) ∨ ((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
60	2, 16, 59	syl2anc 693	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘𝑊) ∨ ((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
61	60	oveq2d 6666	. . . . . 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 17095	. . . . . . . 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 1328	. . . . . . 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 34648	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
65	64	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝐾 ∈ OL)
66	5, 17, 20, 10	oldmm2 34505	. . . . . . . . . 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 1326	. . . . . . . . 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 34508	. . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . 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 17079	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋(meet‘𝐾)𝑊) = 𝑋))
71	2, 9, 14, 70	syl3anc 1326	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋(meet‘𝐾)𝑊) = 𝑋))
72	51, 71	mpbid 222	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋(meet‘𝐾)𝑊) = 𝑋)
73	72	fveq2d 6195	. . . . . . . . . . . . . . 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 34504	. . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . 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 2658	. . . . . . . . . . . . . 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 17079	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑌(le‘𝐾)𝑊 ↔ (𝑌(meet‘𝐾)𝑊) = 𝑌))
78	2, 24, 14, 77	syl3anc 1326	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑌(le‘𝐾)𝑊 ↔ (𝑌(meet‘𝐾)𝑊) = 𝑌))
79	53, 78	mpbid 222	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑌(meet‘𝐾)𝑊) = 𝑌)
80	79	fveq2d 6195	. . . . . . . . . . . . . . 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 34504	. . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . 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 2658	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑌) = (((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)))
84	76, 83	oveq12d 6668	. . . . . . . . . . . . 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 2656	. . . . . . . . . . . 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 6665	. . . . . . . . . . 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 34522	. . . . . . . . . . . 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 1328	. . . . . . . . . . 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 2656	. . . . . . . . . 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 6195	. . . . . . . . 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 2658	. . . . . . . 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 6665	. . . . . . 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 2658	. . . . . 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 2658	. . . . 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 6665	. . . 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 2658	. . 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 6195	. 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 473	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
99	6, 7	diaclN 36339	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ ran 𝐼)
100	99	adantrr 753	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘𝑋) ∈ ran 𝐼)
101	6, 42, 7	diaelrnN 36334	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑋) ∈ ran 𝐼) → (𝐼‘𝑋) ⊆ ((LTrn‘𝐾)‘𝑊))
102	100, 101	syldan 487	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘𝑋) ⊆ ((LTrn‘𝐾)‘𝑊))
103	6, 7	diaclN 36339	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ dom 𝐼) → (𝐼‘𝑌) ∈ ran 𝐼)
104	103	adantrl 752	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘𝑌) ∈ ran 𝐼)
105	6, 42, 7	diaelrnN 36334	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑌) ∈ ran 𝐼) → (𝐼‘𝑌) ⊆ ((LTrn‘𝐾)‘𝑊))
106	104, 105	syldan 487	. . . 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 36424	. . . 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 1324	. . 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 17077	. . . . . . . 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 1326	. . . . . . 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 36325	. . . . . . . 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 957	. . . . . 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 36325	. . . . . . . 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 957	. . . . . 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 36345	. . . . . 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 1324	. . . . 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 36414	. . . . . . 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 753	. . . . . 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 36414	. . . . . . 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 752	. . . . . 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 3815	. . . . 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 2656	. . . 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 6195	. . 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 2659	. 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 2666	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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 class class class wbr 4653 dom cdm 5114 ran crn 5115 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 lecple 15948 occoc 15949 joincjn 16944 meetcmee 16945 Latclat 17045 OPcops 34459 OLcol 34461 OMLcoml 34462 HLchlt 34637 LHypclh 35270 LTrncltrn 35387 DIsoAcdia 36317 ocAcocaN 36408 vAcdjaN 36420

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-riotaBAD 34239

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-undef 7399 df-map 7859 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-oposet 34463 df-cmtN 34464 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-disoa 36318 df-docaN 36409 df-djaN 36421

This theorem is referenced by: (None)

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