Theorem cmtbr4N 39211

Description: Alternate definition for the commutes relation. (cmbr4i 31633 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 39210	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌)))
7		omllat 39198	. . . . 5 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
8		cmtbr4.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
9	1, 8, 3	latmle2 18535	. . . . 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 5169	. . . 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 39197	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
16	15	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
17	1, 4	opoccl 39150	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
18	16, 14, 17	syl2anc 583	. . . . . . . . . 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 18509	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵)
21	13, 18, 19, 20	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵)
22	1, 8, 3	latmle1 18534	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋)
23	13, 14, 21, 22	syl3anc 1371	. . . . . . . 8 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑋)
24	23	anim1i 614	. . . . . . 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 18510	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (( ⊥ ‘𝑋) ∨ 𝑌) ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ∈ 𝐵)
27	13, 14, 21, 26	syl3anc 1371	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ∈ 𝐵)
28	1, 8, 3	latlem12 18536	. . . . . . 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 1372	. . . . . 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 18519	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (( ⊥ ‘𝑋) ∨ 𝑌))
32	13, 18, 19, 31	syl3anc 1371	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (( ⊥ ‘𝑋) ∨ 𝑌))
33	1, 8, 3	latmlem2 18540	. . . . . . 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 1372	. . . . . 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 18510	. . . . . 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 18512	. . . . 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 1371	. . . 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 1537   ∈ wcel 2108   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  lecple 17318  occoc 17319  joincjn 18381  meetcmee 18382  Latclat 18501  OPcops 39128  cmccmtN 39129  OMLcoml 39131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-proset 18365  df-poset 18383  df-lub 18416  df-glb 18417  df-join 18418  df-meet 18419  df-lat 18502  df-oposet 39132  df-cmtN 39133  df-ol 39134  df-oml 39135

This theorem is referenced by:  lecmtN  39212