Theorem latmassOLD 39215

Description: Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 4187 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 39199	. . . . . 6 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)
3	2	adantr 480	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat)
4		olop 39200	. . . . . . 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 2729	. . . . . . 7 ⊢ (oc‘𝐾) = (oc‘𝐾)
9	7, 8	opoccl 39180	. . . . . 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 39180	. . . . . 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 2729	. . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾)
15	7, 14	latjcl 18380	. . . . 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 39209	. . . 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 39180	. . . . . 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 18424	. . . . 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 6844	. . 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 39210	. . . . 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 7384	. . 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 2773	. 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 18380	. . . 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 39208	. . 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 39210	. . . 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 7385	. 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 2768	1 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = (𝑋 ∧ (𝑌 ∧ 𝑍)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  occoc 17204  joincjn 18252  meetcmee 18253  Latclat 18372  OPcops 39158  OLcol 39160

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-proset 18235  df-poset 18254  df-lub 18285  df-glb 18286  df-join 18287  df-meet 18288  df-lat 18373  df-oposet 39162  df-ol 39164

This theorem is referenced by:  latm12  39216  latm32  39217  latmrot  39218  latm4  39219  cmtcomlemN  39234  cmtbr3N  39240  omlfh1N  39244  dalawlem2  39859  dalawlem7  39864  dalawlem11  39868  dalawlem12  39869  lhp2at0  40019  cdleme20d  40299  cdleme23b  40337  cdlemh2  40803  dia2dimlem2  41052  dihmeetbclemN  41291