cmtbr3N - Mathbox for Norm Megill

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Theorem cmtbr3N 39210

Description: Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 31640 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 39205	. . . 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 39209	. . . . 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 279	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
10		oveq2 7456	. . . . . 6 ⊢ (𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) → (𝑋 ∧ 𝑌) = (𝑋 ∧ ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
11	10	adantl 481	. . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))) → (𝑋 ∧ 𝑌) = (𝑋 ∧ ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
12		omlol 39196	. . . . . . . . 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 39198	. . . . . . . . . 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 18509	. . . . . . . . 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 39197	. . . . . . . . . . 11 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
21	20	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
22	1, 6	opoccl 39150	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
23	21, 14, 22	syl2anc 583	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
24	1, 4	latjcl 18509	. . . . . . . . 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 39185	. . . . . . . 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 18517	. . . . . . . . . . 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 7464	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑌 ∨ 𝑋)) = (𝑋 ∧ (𝑋 ∨ 𝑌)))
31	1, 4, 5	latabs2 18546	. . . . . . . . . 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 2780	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑌 ∨ 𝑋)) = 𝑋)
34	1, 4	latjcom 18517	. . . . . . . . 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 7466	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (𝑌 ∨ 𝑋)) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) = (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
37	27, 36	eqtr3d 2782	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))) = (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
38	37	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))) → (𝑋 ∧ ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))) = (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
39	11, 38	eqtr2d 2781	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌))
40	39	ex 412	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
41	9, 40	sylbid 240	. 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 39150	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
44	21, 17, 43	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
45	1, 5	latmcl 18510	. . . . . . . . . 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 2740	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
49	1, 48, 5	latmle1 18534	. . . . . . . . 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 39200	. . . . . . . 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 39150	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∈ 𝐵)
54	21, 46, 53	syl2anc 583	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∈ 𝐵)
55	1, 5	latmcl 18510	. . . . . . . . 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 18517	. . . . . . . 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 2782	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 = ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
60	59	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)) → 𝑋 = ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
61	1, 4, 5, 6	oldmm3N 39175	. . . . . . . . . . 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 7464	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌)))) = (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
64	1, 5	latmcom 18533	. . . . . . . . . 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 2782	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋))
67	66	eqeq1d 2742	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) ↔ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) = (𝑋 ∧ 𝑌)))
68		oveq1 7455	. . . . . . 7 ⊢ ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) = (𝑋 ∧ 𝑌) → ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
69	67, 68	biimtrdi 253	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
70	69	imp 406	. . . . 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 2780	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)) → 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
72	71	ex 412	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
73	1, 4, 5, 6, 2	cmtvalN 39167	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
74	72, 73	sylibrd 259	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → 𝑋𝐶𝑌))
75	41, 74	impbid 212	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 lecple 17318 occoc 17319 joincjn 18381 meetcmee 18382 Latclat 18501 OPcops 39128 cmccmtN 39129 OLcol 39130 OMLcoml 39131

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-proset 18365 df-poset 18383 df-lub 18416 df-glb 18417 df-join 18418 df-meet 18419 df-lat 18502 df-oposet 39132 df-cmtN 39133 df-ol 39134 df-oml 39135

This theorem is referenced by: cmtbr4N 39211 omlfh1N 39214

W3C validator