Theorem latmassOLD 38927

Description: Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 4221 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 481	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ OL)
2		ollat 38911	. . . . . 6 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)
3	2	adantr 479	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat)
4		olop 38912	. . . . . . 7 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
5	4	adantr 479	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ OP)
6		simpr1 1191	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
7		olmass.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
8		eqid 2726	. . . . . . 7 ⊢ (oc‘𝐾) = (oc‘𝐾)
9	7, 8	opoccl 38892	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
10	5, 6, 9	syl2anc 582	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
11		simpr2 1192	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
12	7, 8	opoccl 38892	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
13	5, 11, 12	syl2anc 582	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
14		eqid 2726	. . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾)
15	7, 14	latjcl 18464	. . . . 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 1368	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵)
17		simpr3 1193	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
18		olmass.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
19	7, 14, 18, 8	oldmj3 38921	. . . 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 1368	. . 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 38892	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑍 ∈ 𝐵) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
22	5, 17, 21	syl2anc 582	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
23	7, 14	latjass 18508	. . . . 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 1369	. . . 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 6905	. . 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 38922	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) = (𝑋 ∧ 𝑌))
27	26	3adant3r3 1181	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) = (𝑋 ∧ 𝑌))
28	27	oveq1d 7439	. . 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 2775	. 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 18464	. . . 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 1368	. . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍)) ∈ 𝐵)
32	7, 14, 18, 8	oldmj2 38920	. . 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 1368	. 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 38922	. . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍))) = (𝑌 ∧ 𝑍))
35	34	3adant3r1 1179	. . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍))) = (𝑌 ∧ 𝑍))
36	35	oveq2d 7440	. 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 2770	1 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = (𝑋 ∧ (𝑌 ∧ 𝑍)))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  ‘cfv 6554  (class class class)co 7424  Basecbs 17213  occoc 17274  joincjn 18336  meetcmee 18337  Latclat 18456  OPcops 38870  OLcol 38872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-proset 18320  df-poset 18338  df-lub 18371  df-glb 18372  df-join 18373  df-meet 18374  df-lat 18457  df-oposet 38874  df-ol 38876

This theorem is referenced by:  latm12  38928  latm32  38929  latmrot  38930  latm4  38931  cmtcomlemN  38946  cmtbr3N  38952  omlfh1N  38956  dalawlem2  39571  dalawlem7  39576  dalawlem11  39580  dalawlem12  39581  lhp2at0  39731  cdleme20d  40011  cdleme23b  40049  cdlemh2  40515  dia2dimlem2  40764  dihmeetbclemN  41003