cmtbr3N - Mathbox for Norm Megill

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Theorem cmtbr3N 33559

Description: Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 27851 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 33554	. . . 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 33558	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌𝐶𝑋 ↔ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
8	7	3com23 1263	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌𝐶𝑋 ↔ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
9	3, 8	bitrd 267	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
10		oveq2 6557	. . . . . 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 33545	. . . . . . . . 9 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
13	12	3ad2ant1 1075	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OL)
14		simp2 1055	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
15		omllat 33547	. . . . . . . . . 10 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
16	15	3ad2ant1 1075	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
17		simp3 1056	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
18	1, 4	latjcl 16874	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 ∨ 𝑋) ∈ 𝐵)
19	16, 17, 14, 18	syl3anc 1318	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ 𝑋) ∈ 𝐵)
20		omlop 33546	. . . . . . . . . . 11 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
21	20	3ad2ant1 1075	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
22	1, 6	opoccl 33499	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
23	21, 14, 22	syl2anc 691	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
24	1, 4	latjcl 16874	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ( ⊥ ‘𝑋) ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) ∈ 𝐵)
25	16, 17, 23, 24	syl3anc 1318	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) ∈ 𝐵)
26	1, 5	latmassOLD 33534	. . . . . . . 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 1320	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (𝑌 ∨ 𝑋)) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) = (𝑋 ∧ ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
28	1, 4	latjcom 16882	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 ∨ 𝑋) = (𝑋 ∨ 𝑌))
29	16, 17, 14, 28	syl3anc 1318	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ 𝑋) = (𝑋 ∨ 𝑌))
30	29	oveq2d 6565	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑌 ∨ 𝑋)) = (𝑋 ∧ (𝑋 ∨ 𝑌)))
31	1, 4, 5	latabs2 16911	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋)
32	15, 31	syl3an1 1351	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋)
33	30, 32	eqtrd 2644	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑌 ∨ 𝑋)) = 𝑋)
34	1, 4	latjcom 16882	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ( ⊥ ‘𝑋) ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) = (( ⊥ ‘𝑋) ∨ 𝑌))
35	16, 17, 23, 34	syl3anc 1318	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) = (( ⊥ ‘𝑋) ∨ 𝑌))
36	33, 35	oveq12d 6567	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (𝑌 ∨ 𝑋)) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) = (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
37	27, 36	eqtr3d 2646	. . . . . 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 2645	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌))
40	39	ex 449	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
41	9, 40	sylbid 229	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
42		simp1 1054	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OML)
43	1, 6	opoccl 33499	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
44	21, 17, 43	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
45	1, 5	latmcl 16875	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵)
46	16, 14, 44, 45	syl3anc 1318	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵)
47	42, 46, 14	3jca 1235	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐾 ∈ OML ∧ (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
48		eqid 2610	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
49	1, 48, 5	latmle1 16899	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌))(le‘𝐾)𝑋)
50	16, 14, 44, 49	syl3anc 1318	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌))(le‘𝐾)𝑋)
51	1, 48, 4, 5, 6	omllaw2N 33549	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑋 ∧ ( ⊥ ‘𝑌))(le‘𝐾)𝑋 → ((𝑋 ∧ ( ⊥ ‘𝑌)) ∨ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋)) = 𝑋))
52	47, 50, 51	sylc 63	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ ( ⊥ ‘𝑌)) ∨ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋)) = 𝑋)
53	1, 6	opoccl 33499	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∈ 𝐵)
54	21, 46, 53	syl2anc 691	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∈ 𝐵)
55	1, 5	latmcl 16875	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∈ 𝐵)
56	16, 54, 14, 55	syl3anc 1318	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∈ 𝐵)
57	1, 4	latjcom 16882	. . . . . . . 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 1318	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ ( ⊥ ‘𝑌)) ∨ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋)) = ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
59	52, 58	eqtr3d 2646	. . . . . 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 33524	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∨ 𝑌))
62	12, 61	syl3an1 1351	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∨ 𝑌))
63	62	oveq2d 6565	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌)))) = (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
64	1, 5	latmcom 16898	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌)))) = (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋))
65	16, 14, 54, 64	syl3anc 1318	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌)))) = (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋))
66	63, 65	eqtr3d 2646	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋))
67	66	eqeq1d 2612	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) ↔ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) = (𝑋 ∧ 𝑌)))
68		oveq1 6556	. . . . . . 7 ⊢ ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) = (𝑋 ∧ 𝑌) → ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
69	67, 68	syl6bi 242	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
70	69	imp 444	. . . . 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 2644	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)) → 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
72	71	ex 449	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
73	1, 4, 5, 6, 2	cmtvalN 33516	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
74	72, 73	sylibrd 248	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → 𝑋𝐶𝑌))
75	41, 74	impbid 201	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 occoc 15776 joincjn 16767 meetcmee 16768 Latclat 16868 OPcops 33477 cmccmtN 33478 OLcol 33479 OMLcoml 33480

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-preset 16751 df-poset 16769 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-lat 16869 df-oposet 33481 df-cmtN 33482 df-ol 33483 df-oml 33484

This theorem is referenced by: cmtbr4N 33560 omlfh1N 33563

W3C validator