Theorem cmtbr4N 39631

Description: Alternate definition for the commutes relation. (cmbr4i 31689 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 39630	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
7		omllat 39618	. . . . 5 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
8		cmtbr4.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
9	1, 8, 3	latmle2 18400	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌)
10	7, 9	syl3an1 1164	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌)
11		breq1 5103	. . . 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 1134	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
14		simp2 1138	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
15		omlop 39617	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
16	15	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
17	1, 4	opoccl 39570	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
18	16, 14, 17	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
19		simp3 1139	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
20	1, 2	latjcl 18374	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵)
21	13, 18, 19, 20	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵)
22	1, 8, 3	latmle1 18399	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋)
23	13, 14, 21, 22	syl3anc 1374	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋)
24	23	anim1i 616	. . . . . . 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 18375	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ∈ 𝐵)
27	13, 14, 21, 26	syl3anc 1374	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ∈ 𝐵)
28	1, 8, 3	latlem12 18401	. . . . . . 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 1375	. . . . . 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 18384	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (( ⊥ ‘𝑋) ∨ 𝑌))
32	13, 18, 19, 31	syl3anc 1374	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (( ⊥ ‘𝑋) ∨ 𝑌))
33	1, 8, 3	latmlem2 18405	. . . . . . 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 1375	. . . . . 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 18375	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
38	7, 37	syl3an1 1164	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
39	1, 8	latasymb 18377	. . . . 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 1374	. . . 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 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  lecple 17196  occoc 17197  joincjn 18246  meetcmee 18247  Latclat 18366  OPcops 39548  cmccmtN 39549  OMLcoml 39551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-proset 18229  df-poset 18248  df-lub 18279  df-glb 18280  df-join 18281  df-meet 18282  df-lat 18367  df-oposet 39552  df-cmtN 39553  df-ol 39554  df-oml 39555

This theorem is referenced by:  lecmtN  39632