Theorem latmassOLD 37455

Description: Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 4163 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 483	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ OL)
2		ollat 37439	. . . . . 6 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)
3	2	adantr 481	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat)
4		olop 37440	. . . . . . 7 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
5	4	adantr 481	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ OP)
6		simpr1 1193	. . . . . 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 37420	. . . . . 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 1194	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
12	7, 8	opoccl 37420	. . . . . 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 18224	. . . . 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 1370	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵)
17		simpr3 1195	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
18		olmass.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
19	7, 14, 18, 8	oldmj3 37449	. . . 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 1370	. . 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 37420	. . . . . 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 18268	. . . . 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 1371	. . . 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 6813	. . 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 37450	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) = (𝑋 ∧ 𝑌))
27	26	3adant3r3 1183	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) = (𝑋 ∧ 𝑌))
28	27	oveq1d 7328	. . 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 18224	. . . 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 1370	. . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍)) ∈ 𝐵)
32	7, 14, 18, 8	oldmj2 37448	. . 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 1370	. 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 37450	. . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍))) = (𝑌 ∧ 𝑍))
35	34	3adant3r1 1181	. . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍))) = (𝑌 ∧ 𝑍))
36	35	oveq2d 7329	. 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 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ‘cfv 6463  (class class class)co 7313  Basecbs 16979  occoc 17037  joincjn 18096  meetcmee 18097  Latclat 18216  OPcops 37398  OLcol 37400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-riota 7270  df-ov 7316  df-oprab 7317  df-proset 18080  df-poset 18098  df-lub 18131  df-glb 18132  df-join 18133  df-meet 18134  df-lat 18217  df-oposet 37402  df-ol 37404

This theorem is referenced by:  latm12  37456  latm32  37457  latmrot  37458  latm4  37459  cmtcomlemN  37474  cmtbr3N  37480  omlfh1N  37484  dalawlem2  38098  dalawlem7  38103  dalawlem11  38107  dalawlem12  38108  lhp2at0  38258  cdleme20d  38538  cdleme23b  38576  cdlemh2  39042  dia2dimlem2  39291  dihmeetbclemN  39530