Theorem cmtbr4N 39257

Description: Alternate definition for the commutes relation. (cmbr4i 31621 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cmtbr4.b	⊢ 𝐵 = (Base‘𝐾)
cmtbr4.l	⊢ ≤ = (le‘𝐾)
cmtbr4.j	⊢ ∨ = (join‘𝐾)
cmtbr4.m	⊢ ∧ = (meet‘𝐾)
cmtbr4.o	⊢ ⊥ = (oc‘𝐾)
cmtbr4.c	⊢ 𝐶 = (cm‘𝐾)

Assertion

Ref	Expression
cmtbr4N	⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌))

Proof of Theorem cmtbr4N

Step	Hyp	Ref	Expression
1		cmtbr4.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		cmtbr4.j	. . 3 ⊢ ∨ = (join‘𝐾)
3		cmtbr4.m	. . 3 ⊢ ∧ = (meet‘𝐾)
4		cmtbr4.o	. . 3 ⊢ ⊥ = (oc‘𝐾)
5		cmtbr4.c	. . 3 ⊢ 𝐶 = (cm‘𝐾)
6	1, 2, 3, 4, 5	cmtbr3N 39256	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
7		omllat 39244	. . . . 5 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
8		cmtbr4.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
9	1, 8, 3	latmle2 18511	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌)
10	7, 9	syl3an1 1163	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌)
11		breq1 5145	. . . 4 ⊢ ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌 ↔ (𝑋 ∧ 𝑌) ≤ 𝑌))
12	10, 11	syl5ibrcom 247	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌))
13	7	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
14		simp2 1137	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
15		omlop 39243	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
16	15	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
17	1, 4	opoccl 39196	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
18	16, 14, 17	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
19		simp3 1138	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
20	1, 2	latjcl 18485	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵)
21	13, 18, 19, 20	syl3anc 1372	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵)
22	1, 8, 3	latmle1 18510	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋)
23	13, 14, 21, 22	syl3anc 1372	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋)
24	23	anim1i 615	. . . . . . 7 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋 ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌))
25	24	ex 412	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌 → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋 ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌)))
26	1, 3	latmcl 18486	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ∈ 𝐵)
27	13, 14, 21, 26	syl3anc 1372	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ∈ 𝐵)
28	1, 8, 3	latlem12 18512	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋 ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌) ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ (𝑋 ∧ 𝑌)))
29	13, 27, 14, 19, 28	syl13anc 1373	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋 ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌) ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ (𝑋 ∧ 𝑌)))
30	25, 29	sylibd 239	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌 → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ (𝑋 ∧ 𝑌)))
31	1, 8, 2	latlej2 18495	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (( ⊥ ‘𝑋) ∨ 𝑌))
32	13, 18, 19, 31	syl3anc 1372	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (( ⊥ ‘𝑋) ∨ 𝑌))
33	1, 8, 3	latmlem2 18516	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑌 ∈ 𝐵 ∧ (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (𝑌 ≤ (( ⊥ ‘𝑋) ∨ 𝑌) → (𝑋 ∧ 𝑌) ≤ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌))))
34	13, 19, 21, 14, 33	syl13anc 1373	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ≤ (( ⊥ ‘𝑋) ∨ 𝑌) → (𝑋 ∧ 𝑌) ≤ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌))))
35	32, 34	mpd 15	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))
36	30, 35	jctird 526	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌 → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)))))
37	1, 3	latmcl 18486	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
38	7, 37	syl3an1 1163	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
39	1, 8	latasymb 18488	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ∈ 𝐵) → (((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌))) ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
40	13, 27, 38, 39	syl3anc 1372	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ (𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑌) ≤ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌))) ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
41	36, 40	sylibd 239	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌 → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
42	12, 41	impbid 212	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌))
43	6, 42	bitrd 279	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   class class class wbr 5142  ‘cfv 6560  (class class class)co 7432  Basecbs 17248  lecple 17305  occoc 17306  joincjn 18358  meetcmee 18359  Latclat 18477  OPcops 39174  cmccmtN 39175  OMLcoml 39177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-proset 18341  df-poset 18360  df-lub 18392  df-glb 18393  df-join 18394  df-meet 18395  df-lat 18478  df-oposet 39178  df-cmtN 39179  df-ol 39180  df-oml 39181

This theorem is referenced by:  lecmtN  39258