Theorem cmtbr4N 37006

Description: Alternate definition for the commutes relation. (cmbr4i 29682 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 37005	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
7		omllat 36993	. . . . 5 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
8		cmtbr4.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
9	1, 8, 3	latmle2 17971	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌)
10	7, 9	syl3an1 1165	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌)
11		breq1 5056	. . . 4 ⊢ ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌 ↔ (𝑋 ∧ 𝑌) ≤ 𝑌))
12	10, 11	syl5ibrcom 250	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌))
13	7	3ad2ant1 1135	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
14		simp2 1139	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
15		omlop 36992	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
16	15	3ad2ant1 1135	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
17	1, 4	opoccl 36945	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
18	16, 14, 17	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
19		simp3 1140	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
20	1, 2	latjcl 17945	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵)
21	13, 18, 19, 20	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵)
22	1, 8, 3	latmle1 17970	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋)
23	13, 14, 21, 22	syl3anc 1373	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋)
24	23	anim1i 618	. . . . . . 7 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋 ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌))
25	24	ex 416	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌 → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋 ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌)))
26	1, 3	latmcl 17946	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ∈ 𝐵)
27	13, 14, 21, 26	syl3anc 1373	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ∈ 𝐵)
28	1, 8, 3	latlem12 17972	. . . . . . 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 1374	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋 ∧ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌) ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ (𝑋 ∧ 𝑌)))
30	25, 29	sylibd 242	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌 → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ (𝑋 ∧ 𝑌)))
31	1, 8, 2	latlej2 17955	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (( ⊥ ‘𝑋) ∨ 𝑌))
32	13, 18, 19, 31	syl3anc 1373	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (( ⊥ ‘𝑋) ∨ 𝑌))
33	1, 8, 3	latmlem2 17976	. . . . . . 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 1374	. . . . . 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 530	. . . 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 17946	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
38	7, 37	syl3an1 1165	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
39	1, 8	latasymb 17948	. . . . 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 1373	. . . 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 242	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌 → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
42	12, 41	impbid 215	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌) ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌))
43	6, 42	bitrd 282	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   class class class wbr 5053  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  lecple 16809  occoc 16810  joincjn 17818  meetcmee 17819  Latclat 17937  OPcops 36923  cmccmtN 36924  OMLcoml 36926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-proset 17802  df-poset 17820  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-lat 17938  df-oposet 36927  df-cmtN 36928  df-ol 36929  df-oml 36930

This theorem is referenced by:  lecmtN  37007