Theorem cmtcomlemN 33553

Description: Lemma for cmtcomN 33554. (cmcmlem 27834 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 33547	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
2	1	3ad2ant1 1075	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
3		omlop 33546	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
4		cmtcom.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝐾)
5		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (oc‘𝐾) = (oc‘𝐾)
6	4, 5	opoccl 33499	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
7	3, 6	sylan 487	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
8	7	3adant3 1074	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
9		simp3 1056	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
10		eqid 2610	. . . . . . . . . . . 12 ⊢ (le‘𝐾) = (le‘𝐾)
11		eqid 2610	. . . . . . . . . . . 12 ⊢ (join‘𝐾) = (join‘𝐾)
12	4, 10, 11	latlej2 16884	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌(le‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))
13	2, 8, 9, 12	syl3anc 1318	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌(le‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))
14	4, 11	latjcl 16874	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵)
15	2, 8, 9, 14	syl3anc 1318	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵)
16		eqid 2610	. . . . . . . . . . . 12 ⊢ (meet‘𝐾) = (meet‘𝐾)
17	4, 10, 16	latleeqm2 16903	. . . . . . . . . . 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 1318	. . . . . . . . . 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 221	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)(meet‘𝐾)𝑌) = 𝑌)
20	19	oveq2d 6565	. . . . . . . 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 33545	. . . . . . . . . 10 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
22	21	3ad2ant1 1075	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OL)
23	3	3ad2ant1 1075	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
24	4, 5	opoccl 33499	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
25	23, 9, 24	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
26	4, 11	latjcl 16874	. . . . . . . . . 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 1318	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵)
28	4, 16	latmassOLD 33534	. . . . . . . . 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 1320	. . . . . . . 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 33522	. . . . . . . . . 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 1351	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌)) = (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))
32	31	oveq1d 6564	. . . . . . . 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 2655	. . . . . . 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 480	. . . . . 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 33529	. . . . . . . . . . . . 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 1351	. . . . . . . . . . . 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 33527	. . . . . . . . . . . . 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 1351	. . . . . . . . . . . 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 6567	. . . . . . . . . . 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 2620	. . . . . . . . . 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 501	. . . . . . . . 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 6107	. . . . . . . 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 33529	. . . . . . . . . 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 1318	. . . . . . . . 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 480	. . . . . . . 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 2645	. . . . . . 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 6564	. . . . . 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 2644	. . . . 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 6565	. . . 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 1054	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OML)
51	4, 16	latmcl 16875	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
52	1, 51	syl3an1 1351	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
53	50, 52, 9	3jca 1235	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐾 ∈ OML ∧ (𝑋(meet‘𝐾)𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
54	4, 10, 16	latmle2 16900	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌)(le‘𝐾)𝑌)
55	1, 54	syl3an1 1351	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌)(le‘𝐾)𝑌)
56	4, 10, 11, 16, 5	omllaw2N 33549	. . . . . 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 63	. . . . 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 480	. . . 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 16898	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)𝑋))
60	1, 59	syl3an1 1351	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)𝑋))
61	4, 16	latmcom 16898	. . . . . . 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 1318	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))
63	60, 62	oveq12d 6567	. . . . 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 480	. . . 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 2652	. . 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 449	. 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 33516	. 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 33516	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌𝐶𝑋 ↔ 𝑌 = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))))
70	69	3com23 1263	. 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 282	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → 𝑌𝐶𝑋))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  occoc 15776  joincjn 16767  meetcmee 16768  Latclat 16868  OPcops 33477  cmccmtN 33478  OLcol 33479  OMLcoml 33480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-lat 16869  df-oposet 33481  df-cmtN 33482  df-ol 33483  df-oml 33484

This theorem is referenced by:  cmtcomN  33554