Theorem cm2j 11196

Description: A lattice element that commutes with two others also commutes with their join. Theorem 4.2 of [Beran] p. 49.

Assertion

Ref	Expression
cm2j	|- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> A C_H (B vH C))

Proof of Theorem cm2j

Step	Hyp	Ref	Expression
1		cmcm 11190	. . . . . . . . . . 11 |- ((A e. CH /\ B e. CH) -> (A C_H B <-> B C_H A))
2		cmbr 11160	. . . . . . . . . . . 12 |- ((B e. CH /\ A e. CH) -> (B C_H A <-> B = ((B i^i A) vH (B i^i (_|_` A)))))
3	2	ancoms 484	. . . . . . . . . . 11 |- ((A e. CH /\ B e. CH) -> (B C_H A <-> B = ((B i^i A) vH (B i^i (_|_` A)))))
4	1, 3	bitrd 587	. . . . . . . . . 10 |- ((A e. CH /\ B e. CH) -> (A C_H B <-> B = ((B i^i A) vH (B i^i (_|_` A)))))
5	4	biimpa 460	. . . . . . . . 9 |- (((A e. CH /\ B e. CH) /\ A C_H B) -> B = ((B i^i A) vH (B i^i (_|_` A))))
6		incom 2787	. . . . . . . . . 10 |- (B i^i A) = (A i^i B)
7		incom 2787	. . . . . . . . . 10 |- (B i^i (_|_` A)) = ((_|_` A) i^i B)
8	6, 7	opreq12i 4894	. . . . . . . . 9 |- ((B i^i A) vH (B i^i (_|_` A))) = ((A i^i B) vH ((_|_` A) i^i B))
9	5, 8	syl6eq 1944	. . . . . . . 8 |- (((A e. CH /\ B e. CH) /\ A C_H B) -> B = ((A i^i B) vH ((_|_` A) i^i B)))
10	9	3adantl3 1034	. . . . . . 7 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ A C_H B) -> B = ((A i^i B) vH ((_|_` A) i^i B)))
11	10	adantrr 431	. . . . . 6 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> B = ((A i^i B) vH ((_|_` A) i^i B)))
12		cmcm 11190	. . . . . . . . . . 11 |- ((A e. CH /\ C e. CH) -> (A C_H C <-> C C_H A))
13		cmbr 11160	. . . . . . . . . . . 12 |- ((C e. CH /\ A e. CH) -> (C C_H A <-> C = ((C i^i A) vH (C i^i (_|_` A)))))
14	13	ancoms 484	. . . . . . . . . . 11 |- ((A e. CH /\ C e. CH) -> (C C_H A <-> C = ((C i^i A) vH (C i^i (_|_` A)))))
15	12, 14	bitrd 587	. . . . . . . . . 10 |- ((A e. CH /\ C e. CH) -> (A C_H C <-> C = ((C i^i A) vH (C i^i (_|_` A)))))
16	15	biimpa 460	. . . . . . . . 9 |- (((A e. CH /\ C e. CH) /\ A C_H C) -> C = ((C i^i A) vH (C i^i (_|_` A))))
17		incom 2787	. . . . . . . . . 10 |- (C i^i A) = (A i^i C)
18		incom 2787	. . . . . . . . . 10 |- (C i^i (_|_` A)) = ((_|_` A) i^i C)
19	17, 18	opreq12i 4894	. . . . . . . . 9 |- ((C i^i A) vH (C i^i (_|_` A))) = ((A i^i C) vH ((_|_` A) i^i C))
20	16, 19	syl6eq 1944	. . . . . . . 8 |- (((A e. CH /\ C e. CH) /\ A C_H C) -> C = ((A i^i C) vH ((_|_` A) i^i C)))
21	20	3adantl2 1033	. . . . . . 7 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ A C_H C) -> C = ((A i^i C) vH ((_|_` A) i^i C)))
22	21	adantrl 430	. . . . . 6 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> C = ((A i^i C) vH ((_|_` A) i^i C)))
23	11, 22	opreq12d 4900	. . . . 5 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> (B vH C) = (((A i^i B) vH ((_|_` A) i^i B)) vH ((A i^i C) vH ((_|_` A) i^i C))))
24		chj4 11091	. . . . . . . 8 |- ((((A i^i B) e. CH /\ ((_|_` A) i^i B) e. CH) /\ ((A i^i C) e. CH /\ ((_|_` A) i^i C) e. CH)) -> (((A i^i B) vH ((_|_` A) i^i B)) vH ((A i^i C) vH ((_|_` A) i^i C))) = (((A i^i B) vH (A i^i C)) vH (((_|_` A) i^i B) vH ((_|_` A) i^i C))))
25		chincl 11055	. . . . . . . . 9 |- ((A e. CH /\ B e. CH) -> (A i^i B) e. CH)
26		chincl 11055	. . . . . . . . . 10 |- (((_|_` A) e. CH /\ B e. CH) -> ((_|_` A) i^i B) e. CH)
27		choccl 10817	. . . . . . . . . 10 |- (A e. CH -> (_|_` A) e. CH)
28	26, 27	sylan 497	. . . . . . . . 9 |- ((A e. CH /\ B e. CH) -> ((_|_` A) i^i B) e. CH)
29	25, 28	jca 310	. . . . . . . 8 |- ((A e. CH /\ B e. CH) -> ((A i^i B) e. CH /\ ((_|_` A) i^i B) e. CH))
30		chincl 11055	. . . . . . . . 9 |- ((A e. CH /\ C e. CH) -> (A i^i C) e. CH)
31		chincl 11055	. . . . . . . . . 10 |- (((_|_` A) e. CH /\ C e. CH) -> ((_|_` A) i^i C) e. CH)
32	31, 27	sylan 497	. . . . . . . . 9 |- ((A e. CH /\ C e. CH) -> ((_|_` A) i^i C) e. CH)
33	30, 32	jca 310	. . . . . . . 8 |- ((A e. CH /\ C e. CH) -> ((A i^i C) e. CH /\ ((_|_` A) i^i C) e. CH))
34	24, 29, 33	syl2an 503	. . . . . . 7 |- (((A e. CH /\ B e. CH) /\ (A e. CH /\ C e. CH)) -> (((A i^i B) vH ((_|_` A) i^i B)) vH ((A i^i C) vH ((_|_` A) i^i C))) = (((A i^i B) vH (A i^i C)) vH (((_|_` A) i^i B) vH ((_|_` A) i^i C))))
35	34	3impdi 1152	. . . . . 6 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (((A i^i B) vH ((_|_` A) i^i B)) vH ((A i^i C) vH ((_|_` A) i^i C))) = (((A i^i B) vH (A i^i C)) vH (((_|_` A) i^i B) vH ((_|_` A) i^i C))))
36	35	adantr 425	. . . . 5 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> (((A i^i B) vH ((_|_` A) i^i B)) vH ((A i^i C) vH ((_|_` A) i^i C))) = (((A i^i B) vH (A i^i C)) vH (((_|_` A) i^i B) vH ((_|_` A) i^i C))))
37		fh1 11194	. . . . . . 7 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> (A i^i (B vH C)) = ((A i^i B) vH (A i^i C)))
38		incom 2787	. . . . . . 7 |- (A i^i (B vH C)) = ((B vH C) i^i A)
39	37, 38	syl5reqr 1943	. . . . . 6 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> ((A i^i B) vH (A i^i C)) = ((B vH C) i^i A))
40		id 73	. . . . . . . . . 10 |- (B e. CH -> B e. CH)
41		id 73	. . . . . . . . . 10 |- (C e. CH -> C e. CH)
42	27, 40, 41	3anim123i 1053	. . . . . . . . 9 |- ((A e. CH /\ B e. CH /\ C e. CH) -> ((_|_` A) e. CH /\ B e. CH /\ C e. CH))
43	42	adantr 425	. . . . . . . 8 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> ((_|_` A) e. CH /\ B e. CH /\ C e. CH))
44		cmcm3 11191	. . . . . . . . . . 11 |- ((A e. CH /\ B e. CH) -> (A C_H B <-> (_|_` A) C_H B))
45	44	3adant3 896	. . . . . . . . . 10 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A C_H B <-> (_|_` A) C_H B))
46		cmcm3 11191	. . . . . . . . . . 11 |- ((A e. CH /\ C e. CH) -> (A C_H C <-> (_|_` A) C_H C))
47	46	3adant2 895	. . . . . . . . . 10 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A C_H C <-> (_|_` A) C_H C))
48	45, 47	anbi12d 690	. . . . . . . . 9 |- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A C_H B /\ A C_H C) <-> ((_|_` A) C_H B /\ (_|_` A) C_H C)))
49	48	biimpa 460	. . . . . . . 8 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> ((_|_` A) C_H B /\ (_|_` A) C_H C))
50		fh1 11194	. . . . . . . 8 |- ((((_|_` A) e. CH /\ B e. CH /\ C e. CH) /\ ((_|_` A) C_H B /\ (_|_` A) C_H C)) -> ((_|_` A) i^i (B vH C)) = (((_|_` A) i^i B) vH ((_|_` A) i^i C)))
51	43, 49, 50	syl11anc 524	. . . . . . 7 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> ((_|_` A) i^i (B vH C)) = (((_|_` A) i^i B) vH ((_|_` A) i^i C)))
52		incom 2787	. . . . . . 7 |- ((_|_` A) i^i (B vH C)) = ((B vH C) i^i (_|_` A))
53	51, 52	syl5reqr 1943	. . . . . 6 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> (((_|_` A) i^i B) vH ((_|_` A) i^i C)) = ((B vH C) i^i (_|_` A)))
54	39, 53	opreq12d 4900	. . . . 5 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> (((A i^i B) vH (A i^i C)) vH (((_|_` A) i^i B) vH ((_|_` A) i^i C))) = (((B vH C) i^i A) vH ((B vH C) i^i (_|_` A))))
55	23, 36, 54	3eqtrd 1929	. . . 4 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> (B vH C) = (((B vH C) i^i A) vH ((B vH C) i^i (_|_` A))))
56	55	ex 402	. . 3 |- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A C_H B /\ A C_H C) -> (B vH C) = (((B vH C) i^i A) vH ((B vH C) i^i (_|_` A)))))
57		cmcm 11190	. . . . . 6 |- ((A e. CH /\ (B vH C) e. CH) -> (A C_H (B vH C) <-> (B vH C) C_H A))
58		cmbr 11160	. . . . . . 7 |- (((B vH C) e. CH /\ A e. CH) -> ((B vH C) C_H A <-> (B vH C) = (((B vH C) i^i A) vH ((B vH C) i^i (_|_` A)))))
59	58	ancoms 484	. . . . . 6 |- ((A e. CH /\ (B vH C) e. CH) -> ((B vH C) C_H A <-> (B vH C) = (((B vH C) i^i A) vH ((B vH C) i^i (_|_` A)))))
60	57, 59	bitrd 587	. . . . 5 |- ((A e. CH /\ (B vH C) e. CH) -> (A C_H (B vH C) <-> (B vH C) = (((B vH C) i^i A) vH ((B vH C) i^i (_|_` A)))))
61		chjcl 10962	. . . . 5 |- ((B e. CH /\ C e. CH) -> (B vH C) e. CH)
62	60, 61	sylan2 500	. . . 4 |- ((A e. CH /\ (B e. CH /\ C e. CH)) -> (A C_H (B vH C) <-> (B vH C) = (((B vH C) i^i A) vH ((B vH C) i^i (_|_` A)))))
63	62	3impb 1063	. . 3 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A C_H (B vH C) <-> (B vH C) = (((B vH C) i^i A) vH ((B vH C) i^i (_|_` A)))))
64	56, 63	sylibrd 221	. 2 |- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A C_H B /\ A C_H C) -> A C_H (B vH C)))
65	64	imp 377	1 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> A C_H (B vH C))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   i^i cin 2592   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CHcch 10430  _|_cort 10431   vH chj 10434   C_H ccm 10437

This theorem is referenced by:  cm2ji 11201

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906  ax-hilex 10501  ax-hfvadd 10502  ax-hvcom 10503  ax-hvass 10504  ax-hv0cl 10505  ax-hvaddid 10506  ax-hfvmul 10507  ax-hvmulid 10508  ax-hvmulass 10509  ax-hvdistr1 10510  ax-hvdistr2 10511  ax-hvmul0 10512  ax-hfi 10579  ax-his1 10582  ax-his2 10583  ax-his3 10584  ax-his4 10585  ax-hcompl 10704

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-q 7436  df-fl 7463  df-ioo 7528  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240  df-top 8861  df-bases 8863  df-topgen 8864  df-cld 8939  df-ntr 8940  df-cls 8941  df-cn 9030  df-cnp 9031  df-haus 9059  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-vs 9550  df-nm 9551  df-ims 9552  df-ip 9689  df-ph 9813  df-hnorm 10469  df-hvsub 10472  df-hlim 10473  df-hcau 10474  df-sh 10709  df-ch 10725  df-oc 10757  df-ch0 10758  df-shsum 10906  df-chj 10908  df-cm 11159

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