Theorem latmassOLD 40003

Description: Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 4180 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 487	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ OL)
2		ollat 39987	. . . . . 6 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)
3	2	adantr 485	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat)
4		olop 39988	. . . . . . 7 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
5	4	adantr 485	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ OP)
6		simpr1 1213	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
7		olmass.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
8		eqid 2763	. . . . . . 7 ⊢ (oc‘𝐾) = (oc‘𝐾)
9	7, 8	opoccl 39968	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
10	5, 6, 9	syl2anc 595	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
11		simpr2 1214	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
12	7, 8	opoccl 39968	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
13	5, 11, 12	syl2anc 595	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
14		eqid 2763	. . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾)
15	7, 14	latjcl 18490	. . . . 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 1398	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵)
17		simpr3 1215	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
18		olmass.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
19	7, 14, 18, 8	oldmj3 39997	. . . 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 1398	. . 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 39968	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑍 ∈ 𝐵) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
22	5, 17, 21	syl2anc 595	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
23	7, 14	latjass 18534	. . . . 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 1399	. . . 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 6885	. . 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 39998	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) = (𝑋 ∧ 𝑌))
27	26	3adant3r3 1203	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) = (𝑋 ∧ 𝑌))
28	27	oveq1d 7425	. . 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 2807	. 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 18490	. . . 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 1398	. . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍)) ∈ 𝐵)
32	7, 14, 18, 8	oldmj2 39996	. . 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 1398	. 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 39998	. . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍))) = (𝑌 ∧ 𝑍))
35	34	3adant3r1 1201	. . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌)(join‘𝐾)((oc‘𝐾)‘𝑍))) = (𝑌 ∧ 𝑍))
36	35	oveq2d 7426	. 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 2802	1 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = (𝑋 ∧ (𝑌 ∧ 𝑍)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  occoc 17313  joincjn 18362  meetcmee 18363  Latclat 18482  OPcops 39946  OLcol 39948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-proset 18345  df-poset 18364  df-lub 18395  df-glb 18396  df-join 18397  df-meet 18398  df-lat 18483  df-oposet 39950  df-ol 39952

This theorem is referenced by:  latm12  40004  latm32  40005  latmrot  40006  latm4  40007  cmtcomlemN  40022  cmtbr3N  40028  omlfh1N  40032  dalawlem2  40646  dalawlem7  40651  dalawlem11  40655  dalawlem12  40656  lhp2at0  40806  cdleme20d  41086  cdleme23b  41124  cdlemh2  41590  dia2dimlem2  41839  dihmeetbclemN  42078