Theorem irredi 11966

Description: The Hilbert lattice is irreducible: any element that commutes with all elements must be zero or one. Theorem 14.8.4 of [BeltramettiCassinelli] p. 166.

Hypotheses

Ref	Expression
irred.1	|- A e. CH
irred.2	|- (x e. CH -> A C_H x)

Assertion

Ref	Expression
irredi	|- (A = 0H \/ A = ~H)

Distinct variable group:   x,A

Proof of Theorem irredi

Step	Hyp	Ref	Expression
1		eqid 1884	. . 3 |- 0H = 0H
2		ioran 331	. . . . . 6 |- (-. (A = 0H \/ (_|_` A) = 0H) <-> (-. A = 0H /\ -. (_|_` A) = 0H))
3		df-ne 2019	. . . . . . 7 |- (A =/= 0H <-> -. A = 0H)
4		df-ne 2019	. . . . . . 7 |- ((_|_` A) =/= 0H <-> -. (_|_` A) = 0H)
5	3, 4	anbi12i 540	. . . . . 6 |- ((A =/= 0H /\ (_|_` A) =/= 0H) <-> (-. A = 0H /\ -. (_|_` A) = 0H))
6	2, 5	bitr4i 193	. . . . 5 |- (-. (A = 0H \/ (_|_` A) = 0H) <-> (A =/= 0H /\ (_|_` A) =/= 0H))
7		irred.1	. . . . . . . . 9 |- A e. CH
8	7	hatomici 11931	. . . . . . . 8 |- (A =/= 0H -> E.p e. Atoms p C_ A)
9	7	choccli 10818	. . . . . . . . 9 |- (_|_` A) e. CH
10	9	hatomici 11931	. . . . . . . 8 |- ((_|_` A) =/= 0H -> E.q e. Atoms q C_ (_|_` A))
11	8, 10	anim12i 360	. . . . . . 7 |- ((A =/= 0H /\ (_|_` A) =/= 0H) -> (E.p e. Atoms p C_ A /\ E.q e. Atoms q C_ (_|_` A)))
12		reeanv 2249	. . . . . . 7 |- (E.p e. Atoms E.q e. Atoms (p C_ A /\ q C_ (_|_` A)) <-> (E.p e. Atoms p C_ A /\ E.q e. Atoms q C_ (_|_` A)))
13	11, 12	sylibr 217	. . . . . 6 |- ((A =/= 0H /\ (_|_` A) =/= 0H) -> E.p e. Atoms E.q e. Atoms (p C_ A /\ q C_ (_|_` A)))
14		simpll 448	. . . . . . . . . . 11 |- (((p e. Atoms /\ p C_ A) /\ (q e. Atoms /\ q C_ (_|_` A))) -> p e. Atoms)
15		simprl 450	. . . . . . . . . . 11 |- (((p e. Atoms /\ p C_ A) /\ (q e. Atoms /\ q C_ (_|_` A))) -> q e. Atoms)
16		sstr 2625	. . . . . . . . . . . . . . 15 |- ((q C_ (_|_` A) /\ (_|_` A) C_ (_|_` p)) -> q C_ (_|_` p))
17		atelch 11916	. . . . . . . . . . . . . . . . 17 |- (p e. Atoms -> p e. CH)
18		chsscon3 11056	. . . . . . . . . . . . . . . . . 18 |- ((p e. CH /\ A e. CH) -> (p C_ A <-> (_|_` A) C_ (_|_` p)))
19	7, 18	mpan2 760	. . . . . . . . . . . . . . . . 17 |- (p e. CH -> (p C_ A <-> (_|_` A) C_ (_|_` p)))
20	17, 19	syl 12	. . . . . . . . . . . . . . . 16 |- (p e. Atoms -> (p C_ A <-> (_|_` A) C_ (_|_` p)))
21	20	biimpa 460	. . . . . . . . . . . . . . 15 |- ((p e. Atoms /\ p C_ A) -> (_|_` A) C_ (_|_` p))
22	16, 21	sylan2 500	. . . . . . . . . . . . . 14 |- ((q C_ (_|_` A) /\ (p e. Atoms /\ p C_ A)) -> q C_ (_|_` p))
23	22	ancoms 484	. . . . . . . . . . . . 13 |- (((p e. Atoms /\ p C_ A) /\ q C_ (_|_` A)) -> q C_ (_|_` p))
24		atn0 11917	. . . . . . . . . . . . . . . 16 |- (p e. Atoms -> p =/= 0H)
25	24	adantr 425	. . . . . . . . . . . . . . 15 |- ((p e. Atoms /\ q C_ (_|_` p)) -> p =/= 0H)
26		sseq1 2637	. . . . . . . . . . . . . . . . . . . . 21 |- (p = q -> (p C_ (_|_` p) <-> q C_ (_|_` p)))
27	26	bicomd 580	. . . . . . . . . . . . . . . . . . . 20 |- (p = q -> (q C_ (_|_` p) <-> p C_ (_|_` p)))
28		chssoc 11052	. . . . . . . . . . . . . . . . . . . . 21 |- (p e. CH -> (p C_ (_|_` p) <-> p = 0H))
29	17, 28	syl 12	. . . . . . . . . . . . . . . . . . . 20 |- (p e. Atoms -> (p C_ (_|_` p) <-> p = 0H))
30	27, 29	sylan9bbr 600	. . . . . . . . . . . . . . . . . . 19 |- ((p e. Atoms /\ p = q) -> (q C_ (_|_` p) <-> p = 0H))
31	30	biimpa 460	. . . . . . . . . . . . . . . . . 18 |- (((p e. Atoms /\ p = q) /\ q C_ (_|_` p)) -> p = 0H)
32	31	an1rs 547	. . . . . . . . . . . . . . . . 17 |- (((p e. Atoms /\ q C_ (_|_` p)) /\ p = q) -> p = 0H)
33	32	ex 402	. . . . . . . . . . . . . . . 16 |- ((p e. Atoms /\ q C_ (_|_` p)) -> (p = q -> p = 0H))
34	33	necon3d 2041	. . . . . . . . . . . . . . 15 |- ((p e. Atoms /\ q C_ (_|_` p)) -> (p =/= 0H -> p =/= q))
35	25, 34	mpd 29	. . . . . . . . . . . . . 14 |- ((p e. Atoms /\ q C_ (_|_` p)) -> p =/= q)
36	35	adantlr 429	. . . . . . . . . . . . 13 |- (((p e. Atoms /\ p C_ A) /\ q C_ (_|_` p)) -> p =/= q)
37	23, 36	syldan 516	. . . . . . . . . . . 12 |- (((p e. Atoms /\ p C_ A) /\ q C_ (_|_` A)) -> p =/= q)
38	37	adantrl 430	. . . . . . . . . . 11 |- (((p e. Atoms /\ p C_ A) /\ (q e. Atoms /\ q C_ (_|_` A))) -> p =/= q)
39		superpos 11926	. . . . . . . . . . 11 |- ((p e. Atoms /\ q e. Atoms /\ p =/= q) -> E.r e. Atoms (r =/= p /\ r =/= q /\ r C_ (p vH q)))
40	14, 15, 38, 39	syl111anc 1100	. . . . . . . . . 10 |- (((p e. Atoms /\ p C_ A) /\ (q e. Atoms /\ q C_ (_|_` A))) -> E.r e. Atoms (r =/= p /\ r =/= q /\ r C_ (p vH q)))
41		irred.2	. . . . . . . . . . . . . . . . . 18 |- (x e. CH -> A C_H x)
42	7, 41	irredlem4 11965	. . . . . . . . . . . . . . . . 17 |- ((((p e. Atoms /\ p C_ A) /\ (q e. Atoms /\ q C_ (_|_` A))) /\ (r e. Atoms /\ r C_ (p vH q))) -> (r = p \/ r = q))
43	42	anassrs 489	. . . . . . . . . . . . . . . 16 |- (((((p e. Atoms /\ p C_ A) /\ (q e. Atoms /\ q C_ (_|_` A))) /\ r e. Atoms) /\ r C_ (p vH q)) -> (r = p \/ r = q))
44	43	pm2.24d 120	. . . . . . . . . . . . . . 15 |- (((((p e. Atoms /\ p C_ A) /\ (q e. Atoms /\ q C_ (_|_` A))) /\ r e. Atoms) /\ r C_ (p vH q)) -> (-. (r = p \/ r = q) -> -. 0H = 0H))
45	44	ex 402	. . . . . . . . . . . . . 14 |- ((((p e. Atoms /\ p C_ A) /\ (q e. Atoms /\ q C_ (_|_` A))) /\ r e. Atoms) -> (r C_ (p vH q) -> (-. (r = p \/ r = q) -> -. 0H = 0H)))
46	45	com23 36	. . . . . . . . . . . . 13 |- ((((p e. Atoms /\ p C_ A) /\ (q e. Atoms /\ q C_ (_|_` A))) /\ r e. Atoms) -> (-. (r = p \/ r = q) -> (r C_ (p vH q) -> -. 0H = 0H)))
47	46	imp3a 388	. . . . . . . . . . . 12 |- ((((p e. Atoms /\ p C_ A) /\ (q e. Atoms /\ q C_ (_|_` A))) /\ r e. Atoms) -> ((-. (r = p \/ r = q) /\ r C_ (p vH q)) -> -. 0H = 0H))
48		df-3an 860	. . . . . . . . . . . . 13 |- ((r =/= p /\ r =/= q /\ r C_ (p vH q)) <-> ((r =/= p /\ r =/= q) /\ r C_ (p vH q)))
49		neanior 2097	. . . . . . . . . . . . . 14 |- ((r =/= p /\ r =/= q) <-> -. (r = p \/ r = q))
50	49	anbi1i 539	. . . . . . . . . . . . 13 |- (((r =/= p /\ r =/= q) /\ r C_ (p vH q)) <-> (-. (r = p \/ r = q) /\ r C_ (p vH q)))
51	48, 50	bitri 190	. . . . . . . . . . . 12 |- ((r =/= p /\ r =/= q /\ r C_ (p vH q)) <-> (-. (r = p \/ r = q) /\ r C_ (p vH q)))
52	47, 51	syl5ib 223	. . . . . . . . . . 11 |- ((((p e. Atoms /\ p C_ A) /\ (q e. Atoms /\ q C_ (_|_` A))) /\ r e. Atoms) -> ((r =/= p /\ r =/= q /\ r C_ (p vH q)) -> -. 0H = 0H))
53	52	r19.23adva 2216	. . . . . . . . . 10 |- (((p e. Atoms /\ p C_ A) /\ (q e. Atoms /\ q C_ (_|_` A))) -> (E.r e. Atoms (r =/= p /\ r =/= q /\ r C_ (p vH q)) -> -. 0H = 0H))
54	40, 53	mpd 29	. . . . . . . . 9 |- (((p e. Atoms /\ p C_ A) /\ (q e. Atoms /\ q C_ (_|_` A))) -> -. 0H = 0H)
55	54	an4s 566	. . . . . . . 8 |- (((p e. Atoms /\ q e. Atoms) /\ (p C_ A /\ q C_ (_|_` A))) -> -. 0H = 0H)
56	55	ex 402	. . . . . . 7 |- ((p e. Atoms /\ q e. Atoms) -> ((p C_ A /\ q C_ (_|_` A)) -> -. 0H = 0H))
57	56	r19.23aivv 2217	. . . . . 6 |- (E.p e. Atoms E.q e. Atoms (p C_ A /\ q C_ (_|_` A)) -> -. 0H = 0H)
58	13, 57	syl 12	. . . . 5 |- ((A =/= 0H /\ (_|_` A) =/= 0H) -> -. 0H = 0H)
59	6, 58	sylbi 216	. . . 4 |- (-. (A = 0H \/ (_|_` A) = 0H) -> -. 0H = 0H)
60	59	con4i 90	. . 3 |- (0H = 0H -> (A = 0H \/ (_|_` A) = 0H))
61	1, 60	ax-mp 7	. 2 |- (A = 0H \/ (_|_` A) = 0H)
62		fveq2 4681	. . . 4 |- ((_|_` A) = 0H -> (_|_` (_|_` A)) = (_|_` 0H))
63	7	ococi 10880	. . . 4 |- (_|_` (_|_` A)) = A
64		choc0 10923	. . . 4 |- (_|_` 0H) = ~H
65	62, 63, 64	3eqtr3g 1952	. . 3 |- ((_|_` A) = 0H -> A = ~H)
66	65	orim2i 365	. 2 |- ((A = 0H \/ (_|_` A) = 0H) -> (A = 0H \/ A = ~H))
67	61, 66	ax-mp 7	1 |- (A = 0H \/ A = ~H)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106   C_ wss 2593   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  ~Hchil 10420  CHcch 10430  _|_cort 10431   vH chj 10434  0Hc0h 10436   C_H ccm 10437  Atomscat 10465

This theorem is referenced by:  irred 11967

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-pj 10870  df-shsum 10906  df-span 10907  df-chj 10908  df-chsup 10909  df-cm 11159  df-cv 11851  df-at 11910

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