cmtbr3N - Mathbox for Norm Megill

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Theorem cmtbr3N 39252

Description: Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 31571 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 39247	. . . 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 39251	. . . . 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 7361	. . . . . 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 39238	. . . . . . . . 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 39240	. . . . . . . . . 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 18364	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 ∨ 𝑋) ∈ 𝐵)
19	16, 17, 14, 18	syl3anc 1373	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ 𝑋) ∈ 𝐵)
20		omlop 39239	. . . . . . . . . . 11 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
21	20	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
22	1, 6	opoccl 39192	. . . . . . . . . 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 18364	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ( ⊥ ‘𝑋) ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) ∈ 𝐵)
25	16, 17, 23, 24	syl3anc 1373	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) ∈ 𝐵)
26	1, 5	latmassOLD 39227	. . . . . . . 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 1374	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (𝑌 ∨ 𝑋)) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) = (𝑋 ∧ ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
28	1, 4	latjcom 18372	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 ∨ 𝑋) = (𝑋 ∨ 𝑌))
29	16, 17, 14, 28	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ 𝑋) = (𝑋 ∨ 𝑌))
30	29	oveq2d 7369	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑌 ∨ 𝑋)) = (𝑋 ∧ (𝑋 ∨ 𝑌)))
31	1, 4, 5	latabs2 18401	. . . . . . . . . 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 2764	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑌 ∨ 𝑋)) = 𝑋)
34	1, 4	latjcom 18372	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ( ⊥ ‘𝑋) ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) = (( ⊥ ‘𝑋) ∨ 𝑌))
35	16, 17, 23, 34	syl3anc 1373	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) = (( ⊥ ‘𝑋) ∨ 𝑌))
36	33, 35	oveq12d 7371	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (𝑌 ∨ 𝑋)) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) = (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
37	27, 36	eqtr3d 2766	. . . . . 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 2765	. . . 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 39192	. . . . . . . . . . 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 18365	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵)
46	16, 14, 44, 45	syl3anc 1373	. . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
49	1, 48, 5	latmle1 18389	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌))(le‘𝐾)𝑋)
50	16, 14, 44, 49	syl3anc 1373	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌))(le‘𝐾)𝑋)
51	1, 48, 4, 5, 6	omllaw2N 39242	. . . . . . . 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 39192	. . . . . . . . . 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 18365	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∈ 𝐵)
56	16, 54, 14, 55	syl3anc 1373	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∈ 𝐵)
57	1, 4	latjcom 18372	. . . . . . . 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 1373	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ ( ⊥ ‘𝑌)) ∨ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋)) = ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
59	52, 58	eqtr3d 2766	. . . . . 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 39217	. . . . . . . . . . 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 7369	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌)))) = (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
64	1, 5	latmcom 18388	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌)))) = (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋))
65	16, 14, 54, 64	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌)))) = (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋))
66	63, 65	eqtr3d 2766	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋))
67	66	eqeq1d 2731	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) ↔ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) = (𝑋 ∧ 𝑌)))
68		oveq1 7360	. . . . . . 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 2764	. . . 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 39209	. . 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 1086 = wceq 1540 ∈ wcel 2109 class class class wbr 5095 ‘cfv 6486 (class class class)co 7353 Basecbs 17139 lecple 17187 occoc 17188 joincjn 18236 meetcmee 18237 Latclat 18356 OPcops 39170 cmccmtN 39171 OLcol 39172 OMLcoml 39173

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-proset 18219 df-poset 18238 df-lub 18269 df-glb 18270 df-join 18271 df-meet 18272 df-lat 18357 df-oposet 39174 df-cmtN 39175 df-ol 39176 df-oml 39177

This theorem is referenced by: cmtbr4N 39253 omlfh1N 39256

W3C validator