Theorem cmtcomlemN 36378

Description: Lemma for cmtcomN 36379. (cmcmlem 29362 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cmtcom.b	⊢ 𝐵 = (Base‘𝐾)
cmtcom.c	⊢ 𝐶 = (cm‘𝐾)

Assertion

Ref	Expression
cmtcomlemN	⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → 𝑌𝐶𝑋))

Proof of Theorem cmtcomlemN

Step	Hyp	Ref	Expression
1		omllat 36372	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
2	1	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
3		omlop 36371	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
4		cmtcom.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝐾)
5		eqid 2821	. . . . . . . . . . . . . 14 ⊢ (oc‘𝐾) = (oc‘𝐾)
6	4, 5	opoccl 36324	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
7	3, 6	sylan 582	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
8	7	3adant3 1128	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
9		simp3 1134	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
10		eqid 2821	. . . . . . . . . . . 12 ⊢ (le‘𝐾) = (le‘𝐾)
11		eqid 2821	. . . . . . . . . . . 12 ⊢ (join‘𝐾) = (join‘𝐾)
12	4, 10, 11	latlej2 17665	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌(le‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))
13	2, 8, 9, 12	syl3anc 1367	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌(le‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))
14	4, 11	latjcl 17655	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵)
15	2, 8, 9, 14	syl3anc 1367	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵)
16		eqid 2821	. . . . . . . . . . . 12 ⊢ (meet‘𝐾) = (meet‘𝐾)
17	4, 10, 16	latleeqm2 17684	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵) → (𝑌(le‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ↔ ((((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)(meet‘𝐾)𝑌) = 𝑌))
18	2, 9, 15, 17	syl3anc 1367	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌(le‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ↔ ((((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)(meet‘𝐾)𝑌) = 𝑌))
19	13, 18	mpbid 234	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)(meet‘𝐾)𝑌) = 𝑌)
20	19	oveq2d 7166	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)((((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)(meet‘𝐾)𝑌)) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)𝑌))
21		omlol 36370	. . . . . . . . . 10 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
22	21	3ad2ant1 1129	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OL)
23	3	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
24	4, 5	opoccl 36324	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
25	23, 9, 24	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
26	4, 11	latjcl 17655	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵)
27	2, 8, 25, 26	syl3anc 1367	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵)
28	4, 16	latmassOLD 36359	. . . . . . . . 9 ⊢ ((𝐾 ∈ OL ∧ ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵 ∧ (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))(meet‘𝐾)𝑌) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)((((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)(meet‘𝐾)𝑌)))
29	22, 27, 15, 9, 28	syl13anc 1368	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))(meet‘𝐾)𝑌) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)((((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)(meet‘𝐾)𝑌)))
30	4, 11, 16, 5	oldmm1 36347	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌)) = (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))
31	21, 30	syl3an1 1159	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌)) = (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))
32	31	oveq1d 7165	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)𝑌))
33	20, 29, 32	3eqtr4rd 2867	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌) = (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))(meet‘𝐾)𝑌))
34	33	adantr 483	. . . . . 6 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → (((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌) = (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))(meet‘𝐾)𝑌))
35	4, 11, 16, 5	oldmj4 36354	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) = (𝑋(meet‘𝐾)𝑌))
36	21, 35	syl3an1 1159	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) = (𝑋(meet‘𝐾)𝑌))
37	4, 11, 16, 5	oldmj2 36352	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)) = (𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))
38	21, 37	syl3an1 1159	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)) = (𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))
39	36, 38	oveq12d 7168	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))(join‘𝐾)((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))) = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌))))
40	39	eqeq2d 2832	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = (((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))(join‘𝐾)((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))) ↔ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))))
41	40	biimpar 480	. . . . . . . . 9 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → 𝑋 = (((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))(join‘𝐾)((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))))
42	41	fveq2d 6668	. . . . . . . 8 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → ((oc‘𝐾)‘𝑋) = ((oc‘𝐾)‘(((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))(join‘𝐾)((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)))))
43	4, 11, 16, 5	oldmj4 36354	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OL ∧ (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵 ∧ (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))(join‘𝐾)((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)))) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)))
44	22, 27, 15, 43	syl3anc 1367	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))(join‘𝐾)((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)))) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)))
45	44	adantr 483	. . . . . . . 8 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → ((oc‘𝐾)‘(((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))(join‘𝐾)((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)))) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)))
46	42, 45	eqtr2d 2857	. . . . . . 7 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)) = ((oc‘𝐾)‘𝑋))
47	46	oveq1d 7165	. . . . . 6 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))(meet‘𝐾)𝑌) = (((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌))
48	34, 47	eqtrd 2856	. . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → (((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌) = (((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌))
49	48	oveq2d 7166	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌)) = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌)))
50		simp1 1132	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OML)
51	4, 16	latmcl 17656	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
52	1, 51	syl3an1 1159	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
53	50, 52, 9	3jca 1124	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐾 ∈ OML ∧ (𝑋(meet‘𝐾)𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
54	4, 10, 16	latmle2 17681	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌)(le‘𝐾)𝑌)
55	1, 54	syl3an1 1159	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌)(le‘𝐾)𝑌)
56	4, 10, 11, 16, 5	omllaw2N 36374	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ (𝑋(meet‘𝐾)𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(meet‘𝐾)𝑌)(le‘𝐾)𝑌 → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌)) = 𝑌))
57	53, 55, 56	sylc 65	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌)) = 𝑌)
58	57	adantr 483	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌)) = 𝑌)
59	4, 16	latmcom 17679	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)𝑋))
60	1, 59	syl3an1 1159	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)𝑋))
61	4, 16	latmcom 17679	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))
62	2, 8, 9, 61	syl3anc 1367	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))
63	60, 62	oveq12d 7168	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌)) = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋))))
64	63	adantr 483	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌)) = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋))))
65	49, 58, 64	3eqtr3d 2864	. . 3 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → 𝑌 = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋))))
66	65	ex 415	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌))) → 𝑌 = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))))
67		cmtcom.c	. . 3 ⊢ 𝐶 = (cm‘𝐾)
68	4, 11, 16, 5, 67	cmtvalN 36341	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))))
69	4, 11, 16, 5, 67	cmtvalN 36341	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌𝐶𝑋 ↔ 𝑌 = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))))
70	69	3com23 1122	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌𝐶𝑋 ↔ 𝑌 = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))))
71	66, 68, 70	3imtr4d 296	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → 𝑌𝐶𝑋))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   class class class wbr 5058  ‘cfv 6349  (class class class)co 7150  Basecbs 16477  lecple 16566  occoc 16567  joincjn 17548  meetcmee 17549  Latclat 17649  OPcops 36302  cmccmtN 36303  OLcol 36304  OMLcoml 36305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17532  df-poset 17550  df-lub 17578  df-glb 17579  df-join 17580  df-meet 17581  df-lat 17650  df-oposet 36306  df-cmtN 36307  df-ol 36308  df-oml 36309

This theorem is referenced by:  cmtcomN  36379