cmtbr3N - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cmtbr3N		Structured version Visualization version GIF version

Theorem cmtbr3N 37789

Description: Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 30613 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
cmtbr2.b	⊢ 𝐵 = (Base‘𝐾)
cmtbr2.j	⊢ ∨ = (join‘𝐾)
cmtbr2.m	⊢ ∧ = (meet‘𝐾)
cmtbr2.o	⊢ ⊥ = (oc‘𝐾)
cmtbr2.c	⊢ 𝐶 = (cm‘𝐾)

**Assertion**
Ref	Expression
cmtbr3N	⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))

**Proof of Theorem cmtbr3N**
Step	Hyp	Ref	Expression
1		cmtbr2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		cmtbr2.c	. . . . 5 ⊢ 𝐶 = (cm‘𝐾)
3	1, 2	cmtcomN 37784	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑌𝐶𝑋))
4		cmtbr2.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
5		cmtbr2.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
6		cmtbr2.o	. . . . . 6 ⊢ ⊥ = (oc‘𝐾)
7	1, 4, 5, 6, 2	cmtbr2N 37788	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌𝐶𝑋 ↔ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
8	7	3com23 1126	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌𝐶𝑋 ↔ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
9	3, 8	bitrd 278	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
10		oveq2 7370	. . . . . 6 ⊢ (𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) → (𝑋 ∧ 𝑌) = (𝑋 ∧ ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
11	10	adantl 482	. . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))) → (𝑋 ∧ 𝑌) = (𝑋 ∧ ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
12		omlol 37775	. . . . . . . . 9 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
13	12	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OL)
14		simp2 1137	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
15		omllat 37777	. . . . . . . . . 10 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
16	15	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
17		simp3 1138	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
18	1, 4	latjcl 18342	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 ∨ 𝑋) ∈ 𝐵)
19	16, 17, 14, 18	syl3anc 1371	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ 𝑋) ∈ 𝐵)
20		omlop 37776	. . . . . . . . . . 11 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
21	20	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
22	1, 6	opoccl 37729	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
23	21, 14, 22	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
24	1, 4	latjcl 18342	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ( ⊥ ‘𝑋) ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) ∈ 𝐵)
25	16, 17, 23, 24	syl3anc 1371	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) ∈ 𝐵)
26	1, 5	latmassOLD 37764	. . . . . . . 8 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ (𝑌 ∨ 𝑋) ∈ 𝐵 ∧ (𝑌 ∨ ( ⊥ ‘𝑋)) ∈ 𝐵)) → ((𝑋 ∧ (𝑌 ∨ 𝑋)) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) = (𝑋 ∧ ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
27	13, 14, 19, 25, 26	syl13anc 1372	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (𝑌 ∨ 𝑋)) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) = (𝑋 ∧ ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
28	1, 4	latjcom 18350	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 ∨ 𝑋) = (𝑋 ∨ 𝑌))
29	16, 17, 14, 28	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ 𝑋) = (𝑋 ∨ 𝑌))
30	29	oveq2d 7378	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑌 ∨ 𝑋)) = (𝑋 ∧ (𝑋 ∨ 𝑌)))
31	1, 4, 5	latabs2 18379	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋)
32	15, 31	syl3an1 1163	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋)
33	30, 32	eqtrd 2771	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑌 ∨ 𝑋)) = 𝑋)
34	1, 4	latjcom 18350	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ( ⊥ ‘𝑋) ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) = (( ⊥ ‘𝑋) ∨ 𝑌))
35	16, 17, 23, 34	syl3anc 1371	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) = (( ⊥ ‘𝑋) ∨ 𝑌))
36	33, 35	oveq12d 7380	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (𝑌 ∨ 𝑋)) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) = (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
37	27, 36	eqtr3d 2773	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))) = (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
38	37	adantr 481	. . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))) → (𝑋 ∧ ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))) = (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
39	11, 38	eqtr2d 2772	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌))
40	39	ex 413	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
41	9, 40	sylbid 239	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
42		simp1 1136	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OML)
43	1, 6	opoccl 37729	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
44	21, 17, 43	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
45	1, 5	latmcl 18343	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵)
46	16, 14, 44, 45	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵)
47	42, 46, 14	3jca 1128	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐾 ∈ OML ∧ (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
48		eqid 2731	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
49	1, 48, 5	latmle1 18367	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌))(le‘𝐾)𝑋)
50	16, 14, 44, 49	syl3anc 1371	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌))(le‘𝐾)𝑋)
51	1, 48, 4, 5, 6	omllaw2N 37779	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑋 ∧ ( ⊥ ‘𝑌))(le‘𝐾)𝑋 → ((𝑋 ∧ ( ⊥ ‘𝑌)) ∨ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋)) = 𝑋))
52	47, 50, 51	sylc 65	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ ( ⊥ ‘𝑌)) ∨ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋)) = 𝑋)
53	1, 6	opoccl 37729	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∈ 𝐵)
54	21, 46, 53	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∈ 𝐵)
55	1, 5	latmcl 18343	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∈ 𝐵)
56	16, 54, 14, 55	syl3anc 1371	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∈ 𝐵)
57	1, 4	latjcom 18350	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵 ∧ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∈ 𝐵) → ((𝑋 ∧ ( ⊥ ‘𝑌)) ∨ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋)) = ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
58	16, 46, 56, 57	syl3anc 1371	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ ( ⊥ ‘𝑌)) ∨ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋)) = ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
59	52, 58	eqtr3d 2773	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 = ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
60	59	adantr 481	. . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)) → 𝑋 = ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
61	1, 4, 5, 6	oldmm3N 37754	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∨ 𝑌))
62	12, 61	syl3an1 1163	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∨ 𝑌))
63	62	oveq2d 7378	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌)))) = (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
64	1, 5	latmcom 18366	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌)))) = (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋))
65	16, 14, 54, 64	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌)))) = (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋))
66	63, 65	eqtr3d 2773	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋))
67	66	eqeq1d 2733	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) ↔ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) = (𝑋 ∧ 𝑌)))
68		oveq1 7369	. . . . . . 7 ⊢ ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) = (𝑋 ∧ 𝑌) → ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
69	67, 68	syl6bi 252	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
70	69	imp 407	. . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)) → ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
71	60, 70	eqtrd 2771	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)) → 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
72	71	ex 413	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
73	1, 4, 5, 6, 2	cmtvalN 37746	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
74	72, 73	sylibrd 258	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → 𝑋𝐶𝑌))
75	41, 74	impbid 211	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5110 ‘cfv 6501 (class class class)co 7362 Basecbs 17094 lecple 17154 occoc 17155 joincjn 18214 meetcmee 18215 Latclat 18334 OPcops 37707 cmccmtN 37708 OLcol 37709 OMLcoml 37710

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-proset 18198 df-poset 18216 df-lub 18249 df-glb 18250 df-join 18251 df-meet 18252 df-lat 18335 df-oposet 37711 df-cmtN 37712 df-ol 37713 df-oml 37714

This theorem is referenced by: cmtbr4N 37790 omlfh1N 37793

W3C validator