djajN - Mathbox for Norm Megill

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Theorem djajN 41139

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 39364	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2	1	ad2antrr 726	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝐾 ∈ Lat)
3		hlop 39363	. . . . . . . . 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 41039	. . . . . . . . 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 39195	. . . . . . . 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 40000	. . . . . . . . 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 39195	. . . . . . . 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 18484	. . . . . . 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 18485	. . . . . 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 41039	. . . . . . . . 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 39195	. . . . . . . 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 18484	. . . . . . 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 18485	. . . . . 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 18485	. . . . 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 18510	. . . . . 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 18510	. . . . . 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 18491	. . . 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 41038	. . . . 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 480	. . . 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 41127	. . 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 39358	. . . . . 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 18484	. . . . . 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 41040	. . . . . . 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 41040	. . . . . . 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 18495	. . . . . . 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 39262	. . . . 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 18507	. . . . . . . 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 7447	. . . . . 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 18528	. . . . . . . 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 39362	. . . . . . . . . . 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 39219	. . . . . . . . . 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 39222	. . . . . . . . . . . . . 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 18512	. . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋(meet‘𝐾)𝑊) = 𝑋)
73	72	fveq2d 6910	. . . . . . . . . . . . . . 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 39218	. . . . . . . . . . . . . . . 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 18512	. . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑌(meet‘𝐾)𝑊) = 𝑌)
80	79	fveq2d 6910	. . . . . . . . . . . . . . 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 39218	. . . . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . 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 7446	. . . . . . . . . . 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 39236	. . . . . . . . . . . 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 6910	. . . . . . . . 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 7446	. . . . . . 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 7446	. . . 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 6910	. 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 482	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
99	6, 7	diaclN 41052	. . . . . 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 41047	. . . . 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 41052	. . . . . 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 41047	. . . . 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 41137	. . . 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 18510	. . . . . . . 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 41038	. . . . . . . 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 480	. . . . . . 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 41038	. . . . . . . 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 480	. . . . . . 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 41058	. . . . . 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 41127	. . . . . . 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 41127	. . . . . . 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 4221	. . . . 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 6910	. . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 dom cdm 5685 ran crn 5686 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 occoc 17305 joincjn 18357 meetcmee 18358 Latclat 18476 OPcops 39173 OLcol 39175 OMLcoml 39176 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 DIsoAcdia 41030 ocAcocaN 41121 vAcdjaN 41133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-riotaBAD 38954

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-undef 8298 df-map 8868 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-oposet 39177 df-cmtN 39178 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-disoa 41031 df-docaN 41122 df-djaN 41134

This theorem is referenced by: (None)

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