Theorem cmtcomlemN 39907

Description: Lemma for cmtcomN 39908. (cmcmlem 31880 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 39901	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
2	1	3ad2ant1 1149	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
3		omlop 39900	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
4		cmtcom.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝐾)
5		eqid 2769	. . . . . . . . . . . . . 14 ⊢ (oc‘𝐾) = (oc‘𝐾)
6	4, 5	opoccl 39853	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
7	3, 6	sylan 591	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
8	7	3adant3 1148	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
9		simp3 1154	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
10		eqid 2769	. . . . . . . . . . . 12 ⊢ (le‘𝐾) = (le‘𝐾)
11		eqid 2769	. . . . . . . . . . . 12 ⊢ (join‘𝐾) = (join‘𝐾)
12	4, 10, 11	latlej2 18501	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌(le‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))
13	2, 8, 9, 12	syl3anc 1396	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌(le‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))
14	4, 11	latjcl 18491	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵)
15	2, 8, 9, 14	syl3anc 1396	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵)
16		eqid 2769	. . . . . . . . . . . 12 ⊢ (meet‘𝐾) = (meet‘𝐾)
17	4, 10, 16	latleeqm2 18520	. . . . . . . . . . 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 1396	. . . . . . . . . 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 7424	. . . . . . . 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 39899	. . . . . . . . . 10 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
22	21	3ad2ant1 1149	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OL)
23	3	3ad2ant1 1149	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
24	4, 5	opoccl 39853	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
25	23, 9, 24	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
26	4, 11	latjcl 18491	. . . . . . . . . 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 1396	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵)
28	4, 16	latmassOLD 39888	. . . . . . . . 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 1397	. . . . . . . 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 39876	. . . . . . . . . 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 1179	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌)) = (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))
32	31	oveq1d 7423	. . . . . . . 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 2815	. . . . . . 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 485	. . . . . 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 39883	. . . . . . . . . . . . 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 1179	. . . . . . . . . . . 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 39881	. . . . . . . . . . . . 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 1179	. . . . . . . . . . . 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 7426	. . . . . . . . . . 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 2780	. . . . . . . . . 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 482	. . . . . . . . 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 6883	. . . . . . . 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 39883	. . . . . . . . . 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 1396	. . . . . . . . 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 485	. . . . . . . 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 2805	. . . . . . 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 7423	. . . . . 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 2804	. . . . 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 7424	. . . 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 1152	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OML)
51	4, 16	latmcl 18492	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
52	1, 51	syl3an1 1179	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
53	50, 52, 9	3jca 1144	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐾 ∈ OML ∧ (𝑋(meet‘𝐾)𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
54	4, 10, 16	latmle2 18517	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌)(le‘𝐾)𝑌)
55	1, 54	syl3an1 1179	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌)(le‘𝐾)𝑌)
56	4, 10, 11, 16, 5	omllaw2N 39903	. . . . . 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 485	. . . 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 18515	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)𝑋))
60	1, 59	syl3an1 1179	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)𝑋))
61	4, 16	latmcom 18515	. . . . . . 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 1396	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))
63	60, 62	oveq12d 7426	. . . . 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 485	. . . 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 2812	. . 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 417	. 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 39870	. 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 39870	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌𝐶𝑋 ↔ 𝑌 = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))))
70	69	3com23 1142	. 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))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   class class class wbr 5110  ‘cfv 6534  (class class class)co 7408  Basecbs 17265  lecple 17313  occoc 17314  joincjn 18363  meetcmee 18364  Latclat 18483  OPcops 39831  cmccmtN 39832  OLcol 39833  OMLcoml 39834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-proset 18346  df-poset 18365  df-lub 18396  df-glb 18397  df-join 18398  df-meet 18399  df-lat 18484  df-oposet 39835  df-cmtN 39836  df-ol 39837  df-oml 39838

This theorem is referenced by:  cmtcomN  39908