djajN - Mathbox for Norm Megill

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Theorem djajN 38888

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 37114	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2	1	ad2antrr 726	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝐾 ∈ Lat)
3		hlop 37113	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
4	3	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝐾 ∈ OP)
5		eqid 2737	. . . . . . . . . 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 38788	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ (Base‘𝐾))
9	8	adantrr 717	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑋 ∈ (Base‘𝐾))
10		eqid 2737	. . . . . . . . 9 ⊢ (oc‘𝐾) = (oc‘𝐾)
11	5, 10	opoccl 36945	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
12	4, 9, 11	syl2anc 587	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
13	5, 6	lhpbase 37749	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
14	13	ad2antlr 727	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑊 ∈ (Base‘𝐾))
15	5, 10	opoccl 36945	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
16	4, 14, 15	syl2anc 587	. . . . . . 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 17945	. . . . . . 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 1373	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘𝑋) ∨ ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
20		eqid 2737	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
21	5, 20	latmcl 17946	. . . . . 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 1373	. . . . 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 38788	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ dom 𝐼) → 𝑌 ∈ (Base‘𝐾))
24	23	adantrl 716	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑌 ∈ (Base‘𝐾))
25	5, 10	opoccl 36945	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑌) ∈ (Base‘𝐾))
26	4, 24, 25	syl2anc 587	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑌) ∈ (Base‘𝐾))
27	5, 17	latjcl 17945	. . . . . . 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 1373	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
29	5, 20	latmcl 17946	. . . . . 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 1373	. . . . 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 17946	. . . . 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 1373	. . . 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 2737	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
34	5, 33, 20	latmle2 17971	. . . . . 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 1373	. . . . 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 17971	. . . . . 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 1373	. . . . 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 17952	. . . 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 38787	. . . . 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 484	. . . 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 713	. . 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 2737	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
43		eqid 2737	. . . 4 ⊢ ((ocA‘𝐾)‘𝑊) = ((ocA‘𝐾)‘𝑊)
44	17, 20, 10, 6, 42, 7, 43	diaocN 38876	. . 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 594	. 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 37108	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OML)
47	46	ad2antrr 726	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝐾 ∈ OML)
48	5, 17	latjcl 17945	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∨ 𝑌) ∈ (Base‘𝐾))
49	2, 9, 24, 48	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋 ∨ 𝑌) ∈ (Base‘𝐾))
50	33, 6, 7	diadmleN 38789	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋(le‘𝐾)𝑊)
51	50	adantrr 717	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑋(le‘𝐾)𝑊)
52	33, 6, 7	diadmleN 38789	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ dom 𝐼) → 𝑌(le‘𝐾)𝑊)
53	52	adantrl 716	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑌(le‘𝐾)𝑊)
54	5, 33, 17	latjle12 17956	. . . . . . 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 1374	. . . . . 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 712	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋 ∨ 𝑌)(le‘𝐾)𝑊)
57	5, 33, 17, 20, 10	omlspjN 37012	. . . . 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 1377	. . . 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 17968	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑊) ∨ ((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
60	2, 16, 59	syl2anc 587	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘𝑊) ∨ ((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
61	60	oveq2d 7229	. . . . . 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 17989	. . . . . . . 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 1374	. . . . . . 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 37112	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
65	64	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝐾 ∈ OL)
66	5, 17, 20, 10	oldmm2 36969	. . . . . . . . . 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 1373	. . . . . . . . 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 36972	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 17973	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋(meet‘𝐾)𝑊) = 𝑋))
71	2, 9, 14, 70	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋(meet‘𝐾)𝑊) = 𝑋))
72	51, 71	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋(meet‘𝐾)𝑊) = 𝑋)
73	72	fveq2d 6721	. . . . . . . . . . . . . . 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 36968	. . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . 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 17973	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑌(le‘𝐾)𝑊 ↔ (𝑌(meet‘𝐾)𝑊) = 𝑌))
78	2, 24, 14, 77	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑌(le‘𝐾)𝑊 ↔ (𝑌(meet‘𝐾)𝑊) = 𝑌))
79	53, 78	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑌(meet‘𝐾)𝑊) = 𝑌)
80	79	fveq2d 6721	. . . . . . . . . . . . . . 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 36968	. . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑌) = (((oc‘𝐾)‘𝑌) ∨ ((oc‘𝐾)‘𝑊)))
84	76, 83	oveq12d 7231	. . . . . . . . . . . . 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 2777	. . . . . . . . . . . 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 7228	. . . . . . . . . . 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 36986	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 2777	. . . . . . . . . 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 6721	. . . . . . . . 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 2779	. . . . . . . 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 7228	. . . . . . 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 2779	. . . . . 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 2779	. . . . 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 7228	. . . 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 2779	. . 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 6721	. 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 486	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
99	6, 7	diaclN 38801	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ ran 𝐼)
100	99	adantrr 717	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘𝑋) ∈ ran 𝐼)
101	6, 42, 7	diaelrnN 38796	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑋) ∈ ran 𝐼) → (𝐼‘𝑋) ⊆ ((LTrn‘𝐾)‘𝑊))
102	100, 101	syldan 594	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘𝑋) ⊆ ((LTrn‘𝐾)‘𝑊))
103	6, 7	diaclN 38801	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ dom 𝐼) → (𝐼‘𝑌) ∈ ran 𝐼)
104	103	adantrl 716	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘𝑌) ∈ ran 𝐼)
105	6, 42, 7	diaelrnN 38796	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑌) ∈ ran 𝐼) → (𝐼‘𝑌) ⊆ ((LTrn‘𝐾)‘𝑊))
106	104, 105	syldan 594	. . . 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 38886	. . . 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 837	. . 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 17971	. . . . . . . 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 1373	. . . . . . 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 38787	. . . . . . . 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 484	. . . . . . 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 713	. . . . . 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 38787	. . . . . . . 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 484	. . . . . . 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 713	. . . . . 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 38807	. . . . . 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 837	. . . . 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 38876	. . . . . . 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 717	. . . . . 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 38876	. . . . . . 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 716	. . . . . 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 4128	. . . . 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 2777	. . . 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 6721	. . 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 2780	. 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 2787	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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∩ cin 3865 ⊆ wss 3866 class class class wbr 5053 dom cdm 5551 ran crn 5552 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 lecple 16809 occoc 16810 joincjn 17818 meetcmee 17819 Latclat 17937 OPcops 36923 OLcol 36925 OMLcoml 36926 HLchlt 37101 LHypclh 37735 LTrncltrn 37852 DIsoAcdia 38779 ocAcocaN 38870 vAcdjaN 38882

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-riotaBAD 36704

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-undef 8015 df-map 8510 df-proset 17802 df-poset 17820 df-plt 17836 df-lub 17852 df-glb 17853 df-join 17854 df-meet 17855 df-p0 17931 df-p1 17932 df-lat 17938 df-clat 18005 df-oposet 36927 df-cmtN 36928 df-ol 36929 df-oml 36930 df-covers 37017 df-ats 37018 df-atl 37049 df-cvlat 37073 df-hlat 37102 df-llines 37249 df-lplanes 37250 df-lvols 37251 df-lines 37252 df-psubsp 37254 df-pmap 37255 df-padd 37547 df-lhyp 37739 df-laut 37740 df-ldil 37855 df-ltrn 37856 df-trl 37910 df-disoa 38780 df-docaN 38871 df-djaN 38883

This theorem is referenced by: (None)

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