Theorem fh2 11195

Description: Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Second of two parts. Theorem 5 of [Kalmbach] p. 25.

Assertion

Ref	Expression
fh2	|- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> (A i^i (B vH C)) = ((A i^i B) vH (A i^i C)))

Proof of Theorem fh2

Step	Hyp	Ref	Expression
1		chjcl 10962	. . . . . . . 8 |- (((A i^i B) e. CH /\ (A i^i C) e. CH) -> ((A i^i B) vH (A i^i C)) e. CH)
2		chincl 11055	. . . . . . . 8 |- ((A e. CH /\ B e. CH) -> (A i^i B) e. CH)
3		chincl 11055	. . . . . . . 8 |- ((A e. CH /\ C e. CH) -> (A i^i C) e. CH)
4	1, 2, 3	syl2an 503	. . . . . . 7 |- (((A e. CH /\ B e. CH) /\ (A e. CH /\ C e. CH)) -> ((A i^i B) vH (A i^i C)) e. CH)
5	4	anandis 570	. . . . . 6 |- ((A e. CH /\ (B e. CH /\ C e. CH)) -> ((A i^i B) vH (A i^i C)) e. CH)
6		chincl 11055	. . . . . . . 8 |- ((A e. CH /\ (B vH C) e. CH) -> (A i^i (B vH C)) e. CH)
7		chjcl 10962	. . . . . . . 8 |- ((B e. CH /\ C e. CH) -> (B vH C) e. CH)
8	6, 7	sylan2 500	. . . . . . 7 |- ((A e. CH /\ (B e. CH /\ C e. CH)) -> (A i^i (B vH C)) e. CH)
9		chsh 10729	. . . . . . 7 |- ((A i^i (B vH C)) e. CH -> (A i^i (B vH C)) e. SH)
10	8, 9	syl 12	. . . . . 6 |- ((A e. CH /\ (B e. CH /\ C e. CH)) -> (A i^i (B vH C)) e. SH)
11	5, 10	jca 310	. . . . 5 |- ((A e. CH /\ (B e. CH /\ C e. CH)) -> (((A i^i B) vH (A i^i C)) e. CH /\ (A i^i (B vH C)) e. SH))
12	11	3impb 1063	. . . 4 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (((A i^i B) vH (A i^i C)) e. CH /\ (A i^i (B vH C)) e. SH))
13	12	adantr 425	. . 3 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> (((A i^i B) vH (A i^i C)) e. CH /\ (A i^i (B vH C)) e. SH))
14		ledi 11096	. . . 4 |- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A i^i B) vH (A i^i C)) C_ (A i^i (B vH C)))
15	14	adantr 425	. . 3 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> ((A i^i B) vH (A i^i C)) C_ (A i^i (B vH C)))
16		chdmj1 11085	. . . . . . . . . . 11 |- (((A i^i B) e. CH /\ (A i^i C) e. CH) -> (_|_` ((A i^i B) vH (A i^i C))) = ((_|_` (A i^i B)) i^i (_|_` (A i^i C))))
17	16, 2, 3	syl2an 503	. . . . . . . . . 10 |- (((A e. CH /\ B e. CH) /\ (A e. CH /\ C e. CH)) -> (_|_` ((A i^i B) vH (A i^i C))) = ((_|_` (A i^i B)) i^i (_|_` (A i^i C))))
18		chdmm1 11081	. . . . . . . . . . . 12 |- ((A e. CH /\ B e. CH) -> (_|_` (A i^i B)) = ((_|_` A) vH (_|_` B)))
19	18	adantr 425	. . . . . . . . . . 11 |- (((A e. CH /\ B e. CH) /\ (A e. CH /\ C e. CH)) -> (_|_` (A i^i B)) = ((_|_` A) vH (_|_` B)))
20	19	ineq1d 2795	. . . . . . . . . 10 |- (((A e. CH /\ B e. CH) /\ (A e. CH /\ C e. CH)) -> ((_|_` (A i^i B)) i^i (_|_` (A i^i C))) = (((_|_` A) vH (_|_` B)) i^i (_|_` (A i^i C))))
21	17, 20	eqtrd 1925	. . . . . . . . 9 |- (((A e. CH /\ B e. CH) /\ (A e. CH /\ C e. CH)) -> (_|_` ((A i^i B) vH (A i^i C))) = (((_|_` A) vH (_|_` B)) i^i (_|_` (A i^i C))))
22	21	3impdi 1152	. . . . . . . 8 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (_|_` ((A i^i B) vH (A i^i C))) = (((_|_` A) vH (_|_` B)) i^i (_|_` (A i^i C))))
23	22	ineq2d 2796	. . . . . . 7 |- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A i^i (B vH C)) i^i (_|_` ((A i^i B) vH (A i^i C)))) = ((A i^i (B vH C)) i^i (((_|_` A) vH (_|_` B)) i^i (_|_` (A i^i C)))))
24	23	adantr 425	. . . . . 6 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> ((A i^i (B vH C)) i^i (_|_` ((A i^i B) vH (A i^i C)))) = ((A i^i (B vH C)) i^i (((_|_` A) vH (_|_` B)) i^i (_|_` (A i^i C)))))
25		cmcm2 11192	. . . . . . . . . . . . 13 |- ((A e. CH /\ B e. CH) -> (A C_H B <-> A C_H (_|_` B)))
26		cmcm 11190	. . . . . . . . . . . . 13 |- ((A e. CH /\ B e. CH) -> (A C_H B <-> B C_H A))
27		cmbr3 11184	. . . . . . . . . . . . . 14 |- ((A e. CH /\ (_|_` B) e. CH) -> (A C_H (_|_` B) <-> (A i^i ((_|_` A) vH (_|_` B))) = (A i^i (_|_` B))))
28		choccl 10817	. . . . . . . . . . . . . 14 |- (B e. CH -> (_|_` B) e. CH)
29	27, 28	sylan2 500	. . . . . . . . . . . . 13 |- ((A e. CH /\ B e. CH) -> (A C_H (_|_` B) <-> (A i^i ((_|_` A) vH (_|_` B))) = (A i^i (_|_` B))))
30	25, 26, 29	3bitr3d 607	. . . . . . . . . . . 12 |- ((A e. CH /\ B e. CH) -> (B C_H A <-> (A i^i ((_|_` A) vH (_|_` B))) = (A i^i (_|_` B))))
31	30	biimpa 460	. . . . . . . . . . 11 |- (((A e. CH /\ B e. CH) /\ B C_H A) -> (A i^i ((_|_` A) vH (_|_` B))) = (A i^i (_|_` B)))
32		incom 2787	. . . . . . . . . . 11 |- (A i^i (_|_` B)) = ((_|_` B) i^i A)
33	31, 32	syl6eq 1944	. . . . . . . . . 10 |- (((A e. CH /\ B e. CH) /\ B C_H A) -> (A i^i ((_|_` A) vH (_|_` B))) = ((_|_` B) i^i A))
34	33	3adantl3 1034	. . . . . . . . 9 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ B C_H A) -> (A i^i ((_|_` A) vH (_|_` B))) = ((_|_` B) i^i A))
35	34	adantrr 431	. . . . . . . 8 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> (A i^i ((_|_` A) vH (_|_` B))) = ((_|_` B) i^i A))
36	35	ineq1d 2795	. . . . . . 7 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> ((A i^i ((_|_` A) vH (_|_` B))) i^i ((B vH C) i^i (_|_` (A i^i C)))) = (((_|_` B) i^i A) i^i ((B vH C) i^i (_|_` (A i^i C)))))
37		in4 2808	. . . . . . 7 |- ((A i^i (B vH C)) i^i (((_|_` A) vH (_|_` B)) i^i (_|_` (A i^i C)))) = ((A i^i ((_|_` A) vH (_|_` B))) i^i ((B vH C) i^i (_|_` (A i^i C))))
38	36, 37	syl5eq 1940	. . . . . 6 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> ((A i^i (B vH C)) i^i (((_|_` A) vH (_|_` B)) i^i (_|_` (A i^i C)))) = (((_|_` B) i^i A) i^i ((B vH C) i^i (_|_` (A i^i C)))))
39	24, 38	eqtrd 1925	. . . . 5 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> ((A i^i (B vH C)) i^i (_|_` ((A i^i B) vH (A i^i C)))) = (((_|_` B) i^i A) i^i ((B vH C) i^i (_|_` (A i^i C)))))
40		in4 2808	. . . . 5 |- (((_|_` B) i^i A) i^i ((B vH C) i^i (_|_` (A i^i C)))) = (((_|_` B) i^i (B vH C)) i^i (A i^i (_|_` (A i^i C))))
41	39, 40	syl6eq 1944	. . . 4 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> ((A i^i (B vH C)) i^i (_|_` ((A i^i B) vH (A i^i C)))) = (((_|_` B) i^i (B vH C)) i^i (A i^i (_|_` (A i^i C)))))
42		ococ 10881	. . . . . . . . . 10 |- (B e. CH -> (_|_` (_|_` B)) = B)
43	42	opreq1d 4897	. . . . . . . . 9 |- (B e. CH -> ((_|_` (_|_` B)) vH C) = (B vH C))
44	43	ineq2d 2796	. . . . . . . 8 |- (B e. CH -> ((_|_` B) i^i ((_|_` (_|_` B)) vH C)) = ((_|_` B) i^i (B vH C)))
45	44	3ad2ant2 898	. . . . . . 7 |- ((A e. CH /\ B e. CH /\ C e. CH) -> ((_|_` B) i^i ((_|_` (_|_` B)) vH C)) = ((_|_` B) i^i (B vH C)))
46	45	adantr 425	. . . . . 6 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> ((_|_` B) i^i ((_|_` (_|_` B)) vH C)) = ((_|_` B) i^i (B vH C)))
47		cmcm3 11191	. . . . . . . . . 10 |- ((B e. CH /\ C e. CH) -> (B C_H C <-> (_|_` B) C_H C))
48		cmbr3 11184	. . . . . . . . . . 11 |- (((_|_` B) e. CH /\ C e. CH) -> ((_|_` B) C_H C <-> ((_|_` B) i^i ((_|_` (_|_` B)) vH C)) = ((_|_` B) i^i C)))
49	48, 28	sylan 497	. . . . . . . . . 10 |- ((B e. CH /\ C e. CH) -> ((_|_` B) C_H C <-> ((_|_` B) i^i ((_|_` (_|_` B)) vH C)) = ((_|_` B) i^i C)))
50	47, 49	bitrd 587	. . . . . . . . 9 |- ((B e. CH /\ C e. CH) -> (B C_H C <-> ((_|_` B) i^i ((_|_` (_|_` B)) vH C)) = ((_|_` B) i^i C)))
51	50	biimpa 460	. . . . . . . 8 |- (((B e. CH /\ C e. CH) /\ B C_H C) -> ((_|_` B) i^i ((_|_` (_|_` B)) vH C)) = ((_|_` B) i^i C))
52	51	3adantl1 1032	. . . . . . 7 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ B C_H C) -> ((_|_` B) i^i ((_|_` (_|_` B)) vH C)) = ((_|_` B) i^i C))
53	52	adantrl 430	. . . . . 6 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> ((_|_` B) i^i ((_|_` (_|_` B)) vH C)) = ((_|_` B) i^i C))
54	46, 53	eqtr3d 1927	. . . . 5 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> ((_|_` B) i^i (B vH C)) = ((_|_` B) i^i C))
55	54	ineq1d 2795	. . . 4 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> (((_|_` B) i^i (B vH C)) i^i (A i^i (_|_` (A i^i C)))) = (((_|_` B) i^i C) i^i (A i^i (_|_` (A i^i C)))))
56		chocin 11051	. . . . . . . . . . 11 |- ((A i^i C) e. CH -> ((A i^i C) i^i (_|_` (A i^i C))) = 0H)
57	3, 56	syl 12	. . . . . . . . . 10 |- ((A e. CH /\ C e. CH) -> ((A i^i C) i^i (_|_` (A i^i C))) = 0H)
58		in12 2805	. . . . . . . . . . 11 |- (C i^i (A i^i (_|_` (A i^i C)))) = (A i^i (C i^i (_|_` (A i^i C))))
59		inass 2804	. . . . . . . . . . 11 |- ((A i^i C) i^i (_|_` (A i^i C))) = (A i^i (C i^i (_|_` (A i^i C))))
60	58, 59	eqtr4i 1911	. . . . . . . . . 10 |- (C i^i (A i^i (_|_` (A i^i C)))) = ((A i^i C) i^i (_|_` (A i^i C)))
61	57, 60	syl5eq 1940	. . . . . . . . 9 |- ((A e. CH /\ C e. CH) -> (C i^i (A i^i (_|_` (A i^i C)))) = 0H)
62	61	ineq2d 2796	. . . . . . . 8 |- ((A e. CH /\ C e. CH) -> ((_|_` B) i^i (C i^i (A i^i (_|_` (A i^i C))))) = ((_|_` B) i^i 0H))
63		inass 2804	. . . . . . . 8 |- (((_|_` B) i^i C) i^i (A i^i (_|_` (A i^i C)))) = ((_|_` B) i^i (C i^i (A i^i (_|_` (A i^i C)))))
64	62, 63	syl5eq 1940	. . . . . . 7 |- ((A e. CH /\ C e. CH) -> (((_|_` B) i^i C) i^i (A i^i (_|_` (A i^i C)))) = ((_|_` B) i^i 0H))
65	64	3adant2 895	. . . . . 6 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (((_|_` B) i^i C) i^i (A i^i (_|_` (A i^i C)))) = ((_|_` B) i^i 0H))
66		chm0 11047	. . . . . . . 8 |- ((_|_` B) e. CH -> ((_|_` B) i^i 0H) = 0H)
67	28, 66	syl 12	. . . . . . 7 |- (B e. CH -> ((_|_` B) i^i 0H) = 0H)
68	67	3ad2ant2 898	. . . . . 6 |- ((A e. CH /\ B e. CH /\ C e. CH) -> ((_|_` B) i^i 0H) = 0H)
69	65, 68	eqtrd 1925	. . . . 5 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (((_|_` B) i^i C) i^i (A i^i (_|_` (A i^i C)))) = 0H)
70	69	adantr 425	. . . 4 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> (((_|_` B) i^i C) i^i (A i^i (_|_` (A i^i C)))) = 0H)
71	41, 55, 70	3eqtrd 1929	. . 3 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> ((A i^i (B vH C)) i^i (_|_` ((A i^i B) vH (A i^i C)))) = 0H)
72		pjoml 10902	. . 3 |- (((((A i^i B) vH (A i^i C)) e. CH /\ (A i^i (B vH C)) e. SH) /\ (((A i^i B) vH (A i^i C)) C_ (A i^i (B vH C)) /\ ((A i^i (B vH C)) i^i (_|_` ((A i^i B) vH (A i^i C)))) = 0H)) -> ((A i^i B) vH (A i^i C)) = (A i^i (B vH C)))
73	13, 15, 71, 72	syl12anc 1098	. 2 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> ((A i^i B) vH (A i^i C)) = (A i^i (B vH C)))
74	73	eqcomd 1889	1 |- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> (A i^i (B vH C)) = ((A i^i B) vH (A i^i C)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   i^i cin 2592   C_ wss 2593   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  SHcsh 10429  CHcch 10430  _|_cort 10431   vH chj 10434  0Hc0h 10436   C_H ccm 10437

This theorem is referenced by:  fh2i 11198  atordi 11956  irredlem2 11963

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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