Theorem atom1d 11925

Description: The 1-dimensional subspaces of Hilbert space are its atoms. Part of Remark 10.3.5 of [BeltramettiCassinelli] p. 107.

Assertion

Ref	Expression
atom1d	|- (A e. Atoms <-> E.x e. ~H (x =/= 0h /\ A = (span` {x})))

Distinct variable group:   x,A

Proof of Theorem atom1d

Step	Hyp	Ref	Expression
1		elat2 11912	. . . 4 |- (A e. Atoms <-> (A e. CH /\ (A =/= 0H /\ A.y e. CH (y C_ A -> (y = A \/ y = 0H)))))
2		chne0 11050	. . . . . 6 |- (A e. CH -> (A =/= 0H <-> E.x e. A x =/= 0h))
3		ax-17 1317	. . . . . . 7 |- (A e. CH -> A.x A e. CH)
4		ax-17 1317	. . . . . . . 8 |- (A.y e. CH (y C_ A -> (y = A \/ y = 0H)) -> A.xA.y e. CH (y C_ A -> (y = A \/ y = 0H)))
5		hbre1 2150	. . . . . . . 8 |- (E.x e. ~H (x =/= 0h /\ A = (_|_` (_|_` {x}))) -> A.xE.x e. ~H (x =/= 0h /\ A = (_|_` (_|_` {x}))))
6	4, 5	hbim 1354	. . . . . . 7 |- ((A.y e. CH (y C_ A -> (y = A \/ y = 0H)) -> E.x e. ~H (x =/= 0h /\ A = (_|_` (_|_` {x})))) -> A.x(A.y e. CH (y C_ A -> (y = A \/ y = 0H)) -> E.x e. ~H (x =/= 0h /\ A = (_|_` (_|_` {x})))))
7		chel 10733	. . . . . . . . . . 11 |- ((A e. CH /\ x e. A) -> x e. ~H)
8	7	adantrr 431	. . . . . . . . . 10 |- ((A e. CH /\ (x e. A /\ x =/= 0h)) -> x e. ~H)
9	8	adantrr 431	. . . . . . . . 9 |- ((A e. CH /\ ((x e. A /\ x =/= 0h) /\ A.y e. CH (y C_ A -> (y = A \/ y = 0H)))) -> x e. ~H)
10		simprlr 457	. . . . . . . . 9 |- ((A e. CH /\ ((x e. A /\ x =/= 0h) /\ A.y e. CH (y C_ A -> (y = A \/ y = 0H)))) -> x =/= 0h)
11		h1dn0 11108	. . . . . . . . . . . . . 14 |- ((x e. ~H /\ x =/= 0h) -> (_|_` (_|_` {x})) =/= 0H)
12	11, 7	sylan 497	. . . . . . . . . . . . 13 |- (((A e. CH /\ x e. A) /\ x =/= 0h) -> (_|_` (_|_` {x})) =/= 0H)
13	12	anasss 488	. . . . . . . . . . . 12 |- ((A e. CH /\ (x e. A /\ x =/= 0h)) -> (_|_` (_|_` {x})) =/= 0H)
14	13	adantrr 431	. . . . . . . . . . 11 |- ((A e. CH /\ ((x e. A /\ x =/= 0h) /\ A.y e. CH (y C_ A -> (y = A \/ y = 0H)))) -> (_|_` (_|_` {x})) =/= 0H)
15		ch1dle 11924	. . . . . . . . . . . . . . . 16 |- ((A e. CH /\ x e. A) -> (_|_` (_|_` {x})) C_ A)
16		snssi 3129	. . . . . . . . . . . . . . . . . 18 |- (x e. ~H -> {x} C_ ~H)
17		occl 10815	. . . . . . . . . . . . . . . . . 18 |- ({x} C_ ~H -> (_|_` {x}) e. CH)
18	7, 16, 17	3syl 24	. . . . . . . . . . . . . . . . 17 |- ((A e. CH /\ x e. A) -> (_|_` {x}) e. CH)
19		choccl 10817	. . . . . . . . . . . . . . . . 17 |- ((_|_` {x}) e. CH -> (_|_` (_|_` {x})) e. CH)
20		sseq1 2637	. . . . . . . . . . . . . . . . . . 19 |- (y = (_|_` (_|_` {x})) -> (y C_ A <-> (_|_` (_|_` {x})) C_ A))
21		eqeq1 1890	. . . . . . . . . . . . . . . . . . . 20 |- (y = (_|_` (_|_` {x})) -> (y = A <-> (_|_` (_|_` {x})) = A))
22		eqeq1 1890	. . . . . . . . . . . . . . . . . . . 20 |- (y = (_|_` (_|_` {x})) -> (y = 0H <-> (_|_` (_|_` {x})) = 0H))
23	21, 22	orbi12d 689	. . . . . . . . . . . . . . . . . . 19 |- (y = (_|_` (_|_` {x})) -> ((y = A \/ y = 0H) <-> ((_|_` (_|_` {x})) = A \/ (_|_` (_|_` {x})) = 0H)))
24	20, 23	imbi12d 688	. . . . . . . . . . . . . . . . . 18 |- (y = (_|_` (_|_` {x})) -> ((y C_ A -> (y = A \/ y = 0H)) <-> ((_|_` (_|_` {x})) C_ A -> ((_|_` (_|_` {x})) = A \/ (_|_` (_|_` {x})) = 0H))))
25	24	rcla4v 2376	. . . . . . . . . . . . . . . . 17 |- ((_|_` (_|_` {x})) e. CH -> (A.y e. CH (y C_ A -> (y = A \/ y = 0H)) -> ((_|_` (_|_` {x})) C_ A -> ((_|_` (_|_` {x})) = A \/ (_|_` (_|_` {x})) = 0H))))
26	18, 19, 25	3syl 24	. . . . . . . . . . . . . . . 16 |- ((A e. CH /\ x e. A) -> (A.y e. CH (y C_ A -> (y = A \/ y = 0H)) -> ((_|_` (_|_` {x})) C_ A -> ((_|_` (_|_` {x})) = A \/ (_|_` (_|_` {x})) = 0H))))
27	15, 26	mpid 58	. . . . . . . . . . . . . . 15 |- ((A e. CH /\ x e. A) -> (A.y e. CH (y C_ A -> (y = A \/ y = 0H)) -> ((_|_` (_|_` {x})) = A \/ (_|_` (_|_` {x})) = 0H)))
28	27	impr 422	. . . . . . . . . . . . . 14 |- ((A e. CH /\ (x e. A /\ A.y e. CH (y C_ A -> (y = A \/ y = 0H)))) -> ((_|_` (_|_` {x})) = A \/ (_|_` (_|_` {x})) = 0H))
29	28	adantrlr 437	. . . . . . . . . . . . 13 |- ((A e. CH /\ ((x e. A /\ x =/= 0h) /\ A.y e. CH (y C_ A -> (y = A \/ y = 0H)))) -> ((_|_` (_|_` {x})) = A \/ (_|_` (_|_` {x})) = 0H))
30	29	ord 249	. . . . . . . . . . . 12 |- ((A e. CH /\ ((x e. A /\ x =/= 0h) /\ A.y e. CH (y C_ A -> (y = A \/ y = 0H)))) -> (-. (_|_` (_|_` {x})) = A -> (_|_` (_|_` {x})) = 0H))
31		nne 2021	. . . . . . . . . . . 12 |- (-. (_|_` (_|_` {x})) =/= 0H <-> (_|_` (_|_` {x})) = 0H)
32	30, 31	syl6ibr 230	. . . . . . . . . . 11 |- ((A e. CH /\ ((x e. A /\ x =/= 0h) /\ A.y e. CH (y C_ A -> (y = A \/ y = 0H)))) -> (-. (_|_` (_|_` {x})) = A -> -. (_|_` (_|_` {x})) =/= 0H))
33	14, 32	mt4d 130	. . . . . . . . . 10 |- ((A e. CH /\ ((x e. A /\ x =/= 0h) /\ A.y e. CH (y C_ A -> (y = A \/ y = 0H)))) -> (_|_` (_|_` {x})) = A)
34	33	eqcomd 1889	. . . . . . . . 9 |- ((A e. CH /\ ((x e. A /\ x =/= 0h) /\ A.y e. CH (y C_ A -> (y = A \/ y = 0H)))) -> A = (_|_` (_|_` {x})))
35		ra4e 2156	. . . . . . . . 9 |- ((x e. ~H /\ (x =/= 0h /\ A = (_|_` (_|_` {x})))) -> E.x e. ~H (x =/= 0h /\ A = (_|_` (_|_` {x}))))
36	9, 10, 34, 35	syl12anc 1098	. . . . . . . 8 |- ((A e. CH /\ ((x e. A /\ x =/= 0h) /\ A.y e. CH (y C_ A -> (y = A \/ y = 0H)))) -> E.x e. ~H (x =/= 0h /\ A = (_|_` (_|_` {x}))))
37	36	exp44 416	. . . . . . 7 |- (A e. CH -> (x e. A -> (x =/= 0h -> (A.y e. CH (y C_ A -> (y = A \/ y = 0H)) -> E.x e. ~H (x =/= 0h /\ A = (_|_` (_|_` {x})))))))
38	3, 6, 37	r19.23ad 2213	. . . . . 6 |- (A e. CH -> (E.x e. A x =/= 0h -> (A.y e. CH (y C_ A -> (y = A \/ y = 0H)) -> E.x e. ~H (x =/= 0h /\ A = (_|_` (_|_` {x}))))))
39	2, 38	sylbid 220	. . . . 5 |- (A e. CH -> (A =/= 0H -> (A.y e. CH (y C_ A -> (y = A \/ y = 0H)) -> E.x e. ~H (x =/= 0h /\ A = (_|_` (_|_` {x}))))))
40	39	imp32 390	. . . 4 |- ((A e. CH /\ (A =/= 0H /\ A.y e. CH (y C_ A -> (y = A \/ y = 0H)))) -> E.x e. ~H (x =/= 0h /\ A = (_|_` (_|_` {x}))))
41	1, 40	sylbi 216	. . 3 |- (A e. Atoms -> E.x e. ~H (x =/= 0h /\ A = (_|_` (_|_` {x}))))
42		eleq1 1957	. . . . . . . 8 |- (A = (_|_` (_|_` {x})) -> (A e. Atoms <-> (_|_` (_|_` {x})) e. Atoms))
43		h1da 11921	. . . . . . . 8 |- ((x e. ~H /\ x =/= 0h) -> (_|_` (_|_` {x})) e. Atoms)
44	42, 43	syl5bir 227	. . . . . . 7 |- (A = (_|_` (_|_` {x})) -> ((x e. ~H /\ x =/= 0h) -> A e. Atoms))
45	44	exp3a 405	. . . . . 6 |- (A = (_|_` (_|_` {x})) -> (x e. ~H -> (x =/= 0h -> A e. Atoms)))
46	45	com3l 38	. . . . 5 |- (x e. ~H -> (x =/= 0h -> (A = (_|_` (_|_` {x})) -> A e. Atoms)))
47	46	imp3a 388	. . . 4 |- (x e. ~H -> ((x =/= 0h /\ A = (_|_` (_|_` {x}))) -> A e. Atoms))
48	47	r19.23aiv 2211	. . 3 |- (E.x e. ~H (x =/= 0h /\ A = (_|_` (_|_` {x}))) -> A e. Atoms)
49	41, 48	impbii 174	. 2 |- (A e. Atoms <-> E.x e. ~H (x =/= 0h /\ A = (_|_` (_|_` {x}))))
50		spansn 11115	. . . . 5 |- (x e. ~H -> (span` {x}) = (_|_` (_|_` {x})))
51	50	eqeq2d 1895	. . . 4 |- (x e. ~H -> (A = (span` {x}) <-> A = (_|_` (_|_` {x}))))
52	51	anbi2d 678	. . 3 |- (x e. ~H -> ((x =/= 0h /\ A = (span` {x})) <-> (x =/= 0h /\ A = (_|_` (_|_` {x})))))
53	52	rexbiia 2134	. 2 |- (E.x e. ~H (x =/= 0h /\ A = (span` {x})) <-> E.x e. ~H (x =/= 0h /\ A = (_|_` (_|_` {x}))))
54	49, 53	bitr4i 193	1 |- (A e. Atoms <-> E.x e. ~H (x =/= 0h /\ A = (span` {x})))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   C_ wss 2593  {csn 3044  ` cfv 3998  ~Hchil 10420  0hc0v 10423  CHcch 10430  _|_cort 10431  spancspn 10433  0Hc0h 10436  Atomscat 10465

This theorem is referenced by:  superpos 11926  chcv1 11927  chjatom 11929

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-span 10907  df-cv 11851  df-at 11910

Copyright terms: Public domain