Theorem cmtcomlemN 40285

Description: Lemma for cmtcomN 40286. (cmcmlem 32186 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 40279	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
2	1	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
3		omlop 40278	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
4		cmtcom.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝐾)
5		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (oc‘𝐾) = (oc‘𝐾)
6	4, 5	opoccl 40231	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
7	3, 6	sylan 592	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
8	7	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
9		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
10		eqid 2761	. . . . . . . . . . . 12 ⊢ (le‘𝐾) = (le‘𝐾)
11		eqid 2761	. . . . . . . . . . . 12 ⊢ (join‘𝐾) = (join‘𝐾)
12	4, 10, 11	latlej2 18616	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌(le‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))
13	2, 8, 9, 12	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌(le‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))
14	4, 11	latjcl 18606	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵)
15	2, 8, 9, 14	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵)
16		eqid 2761	. . . . . . . . . . . 12 ⊢ (meet‘𝐾) = (meet‘𝐾)
17	4, 10, 16	latleeqm2 18635	. . . . . . . . . . 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 1398	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)(meet‘𝐾)𝑌) = 𝑌)
20	19	oveq2d 7434	. . . . . . . 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 40277	. . . . . . . . . 10 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
22	21	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OL)
23	3	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
24	4, 5	opoccl 40231	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
25	23, 9, 24	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
26	4, 11	latjcl 18606	. . . . . . . . . 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 1398	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵)
28	4, 16	latmassOLD 40266	. . . . . . . . 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 1399	. . . . . . . 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 40254	. . . . . . . . . 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 1181	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌)) = (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))
32	31	oveq1d 7433	. . . . . . . 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 2807	. . . . . . 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 486	. . . . . 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 40261	. . . . . . . . . . . . 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 1181	. . . . . . . . . . . 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 40259	. . . . . . . . . . . . 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 1181	. . . . . . . . . . . 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 7436	. . . . . . . . . . 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 2772	. . . . . . . . . 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 483	. . . . . . . . 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 6887	. . . . . . . 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 40261	. . . . . . . . . 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 1398	. . . . . . . . 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 486	. . . . . . . 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 2797	. . . . . . 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 7433	. . . . . 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 2796	. . . . 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 7434	. . . 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 1154	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OML)
51	4, 16	latmcl 18607	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
52	1, 51	syl3an1 1181	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
53	50, 52, 9	3jca 1146	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐾 ∈ OML ∧ (𝑋(meet‘𝐾)𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
54	4, 10, 16	latmle2 18632	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌)(le‘𝐾)𝑌)
55	1, 54	syl3an1 1181	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌)(le‘𝐾)𝑌)
56	4, 10, 11, 16, 5	omllaw2N 40281	. . . . . 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 66	. . . . 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 486	. . . 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 18630	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)𝑋))
60	1, 59	syl3an1 1181	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)𝑋))
61	4, 16	latmcom 18630	. . . . . . 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 1398	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))
63	60, 62	oveq12d 7436	. . . . 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 486	. . . 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 2804	. . 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 418	. 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 40248	. 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 40248	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌𝐶𝑋 ↔ 𝑌 = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))))
70	69	3com23 1144	. 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 297	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → 𝑌𝐶𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   class class class wbr 5103  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  lecple 17428  occoc 17429  joincjn 18478  meetcmee 18479  Latclat 18598  OPcops 40209  cmccmtN 40210  OLcol 40211  OMLcoml 40212

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-proset 18461  df-poset 18480  df-lub 18511  df-glb 18512  df-join 18513  df-meet 18514  df-lat 18599  df-oposet 40213  df-cmtN 40214  df-ol 40215  df-oml 40216

This theorem is used by:  cmtcomN  40286