cmtbr3N - Mathbox for Norm Megill

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Theorem cmtbr3N 37195

Description: Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 29871 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 37190	. . . 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 37194	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌𝐶𝑋 ↔ 𝑌 = ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
8	7	3com23 1124	. . . 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 7263	. . . . . 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 37181	. . . . . . . . 9 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
13	12	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OL)
14		simp2 1135	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
15		omllat 37183	. . . . . . . . . 10 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
16	15	3ad2ant1 1131	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
17		simp3 1136	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
18	1, 4	latjcl 18072	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 ∨ 𝑋) ∈ 𝐵)
19	16, 17, 14, 18	syl3anc 1369	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ 𝑋) ∈ 𝐵)
20		omlop 37182	. . . . . . . . . . 11 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
21	20	3ad2ant1 1131	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
22	1, 6	opoccl 37135	. . . . . . . . . 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 18072	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ( ⊥ ‘𝑋) ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) ∈ 𝐵)
25	16, 17, 23, 24	syl3anc 1369	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) ∈ 𝐵)
26	1, 5	latmassOLD 37170	. . . . . . . 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 1370	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (𝑌 ∨ 𝑋)) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) = (𝑋 ∧ ((𝑌 ∨ 𝑋) ∧ (𝑌 ∨ ( ⊥ ‘𝑋)))))
28	1, 4	latjcom 18080	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 ∨ 𝑋) = (𝑋 ∨ 𝑌))
29	16, 17, 14, 28	syl3anc 1369	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ 𝑋) = (𝑋 ∨ 𝑌))
30	29	oveq2d 7271	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑌 ∨ 𝑋)) = (𝑋 ∧ (𝑋 ∨ 𝑌)))
31	1, 4, 5	latabs2 18109	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋)
32	15, 31	syl3an1 1161	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋)
33	30, 32	eqtrd 2778	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑌 ∨ 𝑋)) = 𝑋)
34	1, 4	latjcom 18080	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ( ⊥ ‘𝑋) ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) = (( ⊥ ‘𝑋) ∨ 𝑌))
35	16, 17, 23, 34	syl3anc 1369	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ∨ ( ⊥ ‘𝑋)) = (( ⊥ ‘𝑋) ∨ 𝑌))
36	33, 35	oveq12d 7273	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (𝑌 ∨ 𝑋)) ∧ (𝑌 ∨ ( ⊥ ‘𝑋))) = (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
37	27, 36	eqtr3d 2780	. . . . . 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 2779	. . . 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 239	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
42		simp1 1134	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OML)
43	1, 6	opoccl 37135	. . . . . . . . . . 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 18073	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵)
46	16, 14, 44, 45	syl3anc 1369	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵)
47	42, 46, 14	3jca 1126	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐾 ∈ OML ∧ (𝑋 ∧ ( ⊥ ‘𝑌)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
48		eqid 2738	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
49	1, 48, 5	latmle1 18097	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌))(le‘𝐾)𝑋)
50	16, 14, 44, 49	syl3anc 1369	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑌))(le‘𝐾)𝑋)
51	1, 48, 4, 5, 6	omllaw2N 37185	. . . . . . . 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 37135	. . . . . . . . . 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 18073	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∈ 𝐵)
56	16, 54, 14, 55	syl3anc 1369	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∈ 𝐵)
57	1, 4	latjcom 18080	. . . . . . . 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 1369	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ ( ⊥ ‘𝑌)) ∨ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋)) = ((( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
59	52, 58	eqtr3d 2780	. . . . . 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 37160	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∨ 𝑌))
62	12, 61	syl3an1 1161	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∨ 𝑌))
63	62	oveq2d 7271	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌)))) = (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
64	1, 5	latmcom 18096	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌)))) = (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋))
65	16, 14, 54, 64	syl3anc 1369	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌)))) = (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋))
66	63, 65	eqtr3d 2780	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋))
67	66	eqeq1d 2740	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) ↔ (( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) ∧ 𝑋) = (𝑋 ∧ 𝑌)))
68		oveq1 7262	. . . . . . 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 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 2778	. . . 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 37152	. . 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 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 lecple 16895 occoc 16896 joincjn 17944 meetcmee 17945 Latclat 18064 OPcops 37113 cmccmtN 37114 OLcol 37115 OMLcoml 37116

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-proset 17928 df-poset 17946 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-lat 18065 df-oposet 37117 df-cmtN 37118 df-ol 37119 df-oml 37120

This theorem is referenced by: cmtbr4N 37196 omlfh1N 37199

W3C validator