Theorem latmassOLD 39230

Description: Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 4228 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
olmass.b	⊢ 𝐵 = (Base‘𝐾)
olmass.m	⊢ ∧ = (meet‘𝐾)

Assertion

Ref	Expression
latmassOLD	⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = (𝑋 ∧ (𝑌 ∧ 𝑍)))

Proof of Theorem latmassOLD

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ OL)
2		ollat 39214	. . . . . 6 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)
3	2	adantr 480	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat)
4		olop 39215	. . . . . . 7 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
5	4	adantr 480	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ OP)
6		simpr1 1195	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
7		olmass.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
8		eqid 2737	. . . . . . 7 ⊢ (oc‘𝐾) = (oc‘𝐾)
9	7, 8	opoccl 39195	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
10	5, 6, 9	syl2anc 584	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
11		simpr2 1196	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
12	7, 8	opoccl 39195	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
13	5, 11, 12	syl2anc 584	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
14		eqid 2737	. . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾)
15	7, 14	latjcl 18484	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵)
16	3, 10, 13, 15	syl3anc 1373	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵)
17		simpr3 1197	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
18		olmass.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
19	7, 14, 18, 8	oldmj3 39224	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(join‘𝐾)((oc‘𝐾)‘𝑍))) = (((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) ∧ 𝑍))
20	1, 16, 17, 19	syl3anc 1373	. . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(join‘𝐾)((oc‘𝐾)‘𝑍))) = (((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) ∧ 𝑍))
21	7, 8	opoccl 39195	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑍 ∈ 𝐵) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
22	5, 17, 21	syl2anc 584	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
23	7, 14	latjass 18528	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵)) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(join‘𝐾)((oc‘𝐾)‘𝑍)) = (((oc‘𝐾)‘𝑋)(join‘𝐾)(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍))))
24	3, 10, 13, 22, 23	syl13anc 1374	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(join‘𝐾)((oc‘𝐾)‘𝑍)) = (((oc‘𝐾)‘𝑋)(join‘𝐾)(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍))))
25	24	fveq2d 6910	. . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(join‘𝐾)((oc‘𝐾)‘𝑍))) = ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍)))))
26	7, 14, 18, 8	oldmj4 39225	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) = (𝑋 ∧ 𝑌))
27	26	3adant3r3 1185	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) = (𝑋 ∧ 𝑌))
28	27	oveq1d 7446	. . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) ∧ 𝑍) = ((𝑋 ∧ 𝑌) ∧ 𝑍))
29	20, 25, 28	3eqtr3rd 2786	. 2 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍)))))
30	7, 14	latjcl 18484	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → (((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍)) ∈ 𝐵)
31	3, 13, 22, 30	syl3anc 1373	. . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍)) ∈ 𝐵)
32	7, 14, 18, 8	oldmj2 39223	. . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍)) ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍)))) = (𝑋 ∧ ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍)))))
33	1, 6, 31, 32	syl3anc 1373	. 2 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍)))) = (𝑋 ∧ ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍)))))
34	7, 14, 18, 8	oldmj4 39225	. . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍))) = (𝑌 ∧ 𝑍))
35	34	3adant3r1 1183	. . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍))) = (𝑌 ∧ 𝑍))
36	35	oveq2d 7447	. 2 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍)))) = (𝑋 ∧ (𝑌 ∧ 𝑍)))
37	29, 33, 36	3eqtrd 2781	1 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = (𝑋 ∧ (𝑌 ∧ 𝑍)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  occoc 17305  joincjn 18357  meetcmee 18358  Latclat 18476  OPcops 39173  OLcol 39175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-lat 18477  df-oposet 39177  df-ol 39179

This theorem is referenced by:  latm12  39231  latm32  39232  latmrot  39233  latm4  39234  cmtcomlemN  39249  cmtbr3N  39255  omlfh1N  39259  dalawlem2  39874  dalawlem7  39879  dalawlem11  39883  dalawlem12  39884  lhp2at0  40034  cdleme20d  40314  cdleme23b  40352  cdlemh2  40818  dia2dimlem2  41067  dihmeetbclemN  41306