Theorem hmopidmpj 10080

Description: An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Halmos seems to omit the proof that H is a closed subspace, which is not trivial as hmopidmch 10079 shows.)

Hypotheses

Ref	Expression
hmopidmch.1	|- H = {x e. H~ | (T` x) = x}
hmopidmch.2	|- (T e. HrmOp /\ (T o. T) = T)

Assertion

Ref	Expression
hmopidmpj	|- T = (proj` H)

Distinct variable group:   x,T

Proof of Theorem hmopidmpj

Step	Hyp	Ref	Expression
1		hmopidmch.2	. . . . . . 7 |- (T e. HrmOp /\ (T o. T) = T)
2	1	pm3.26i 320	. . . . . 6 |- T e. HrmOp
3		hmoplint 9866	. . . . . 6 |- (T e. HrmOp -> T e. LinOp)
4	2, 3	ax-mp 7	. . . . 5 |- T e. LinOp
5	4	lnopf 9893	. . . 4 |- T:H~-->H~
6		ffn 3627	. . . 4 |- (T:H~-->H~ -> T Fn H~)
7	5, 6	ax-mp 7	. . 3 |- T Fn H~
8		hmopidmch.1	. . . . 5 |- H = {x e. H~ | (T` x) = x}
9	8, 1	hmopidmch 10079	. . . 4 |- H e. CH
10	9	pjfn 9646	. . 3 |- (proj` H) Fn H~
11		eqfnfv 3797	. . 3 |- ((T Fn H~ /\ (proj` H) Fn H~) -> (T = (proj` H) <-> (H~ = H~ /\ A.y e. H~ (T` y) = ((proj` H)` y))))
12	7, 10, 11	mp2an 697	. 2 |- (T = (proj` H) <-> (H~ = H~ /\ A.y e. H~ (T` y) = ((proj` H)` y)))
13		eqid 1475	. 2 |- H~ = H~
14	9	pjv 9650	. . . . 5 |- (((T` y) e. H /\ (y -h (T` y)) e. (_|_` H)) -> ((proj` H)` ((T` y) +h (y -h (T` y)))) = (T` y))
15	5	ffvelrni 3815	. . . . . . 7 |- (y e. H~ -> (T` y) e. H~)
16	5, 5	hoco 9690	. . . . . . . 8 |- (y e. H~ -> ((T o. T)` y) = (T` (T` y)))
17	1	pm3.27i 324	. . . . . . . . 9 |- (T o. T) = T
18	17	fveq1i 3725	. . . . . . . 8 |- ((T o. T)` y) = (T` y)
19	16, 18	syl5reqr 1522	. . . . . . 7 |- (y e. H~ -> (T` (T` y)) = (T` y))
20	15, 19	jca 288	. . . . . 6 |- (y e. H~ -> ((T` y) e. H~ /\ (T` (T` y)) = (T` y)))
21		fveq2 3724	. . . . . . . 8 |- (x = (T` y) -> (T` x) = (T` (T` y)))
22		id 59	. . . . . . . 8 |- (x = (T` y) -> x = (T` y))
23	21, 22	eqeq12d 1489	. . . . . . 7 |- (x = (T` y) -> ((T` x) = x <-> (T` (T` y)) = (T` y)))
24	23, 8	elrab2 1907	. . . . . 6 |- ((T` y) e. H <-> ((T` y) e. H~ /\ (T` (T` y)) = (T` y)))
25	20, 24	sylibr 200	. . . . 5 |- (y e. H~ -> (T` y) e. H)
26		hvsubclt 8887	. . . . . . . 8 |- ((y e. H~ /\ (T` y) e. H~) -> (y -h (T` y)) e. H~)
27	15, 26	mpdan 704	. . . . . . 7 |- (y e. H~ -> (y -h (T` y)) e. H~)
28		his2subt 8958	. . . . . . . . . 10 |- ((y e. H~ /\ (T` y) e. H~ /\ z e. H~) -> ((y -h (T` y)) .ih z) = ((y .ih z) - ((T` y) .ih z)))
29		pm3.26 319	. . . . . . . . . 10 |- ((y e. H~ /\ z e. H) -> y e. H~)
30	15	adantr 389	. . . . . . . . . 10 |- ((y e. H~ /\ z e. H) -> (T` y) e. H~)
31		ssrab2 2131	. . . . . . . . . . . . 13 |- {x e. H~ | (T` x) = x} (_ H~
32	8, 31	eqsstr 2091	. . . . . . . . . . . 12 |- H (_ H~
33	32	sseli 2065	. . . . . . . . . . 11 |- (z e. H -> z e. H~)
34	33	adantl 388	. . . . . . . . . 10 |- ((y e. H~ /\ z e. H) -> z e. H~)
35	28, 29, 30, 34	syl3anc 858	. . . . . . . . 9 |- ((y e. H~ /\ z e. H) -> ((y -h (T` y)) .ih z) = ((y .ih z) - ((T` y) .ih z)))
36		hmopt 9846	. . . . . . . . . . . . 13 |- ((T e. HrmOp /\ y e. H~ /\ z e. H~) -> (y .ih (T` z)) = ((T` y) .ih z))
37	2, 36	mp3an1 903	. . . . . . . . . . . 12 |- ((y e. H~ /\ z e. H~) -> (y .ih (T` z)) = ((T` y) .ih z))
38	37, 33	sylan2 451	. . . . . . . . . . 11 |- ((y e. H~ /\ z e. H) -> (y .ih (T` z)) = ((T` y) .ih z))
39		fveq2 3724	. . . . . . . . . . . . . . . 16 |- (x = z -> (T` x) = (T` z))
40		id 59	. . . . . . . . . . . . . . . 16 |- (x = z -> x = z)
41	39, 40	eqeq12d 1489	. . . . . . . . . . . . . . 15 |- (x = z -> ((T` x) = x <-> (T` z) = z))
42	41, 8	elrab2 1907	. . . . . . . . . . . . . 14 |- (z e. H <-> (z e. H~ /\ (T` z) = z))
43	42	pm3.27bi 326	. . . . . . . . . . . . 13 |- (z e. H -> (T` z) = z)
44	43	opreq2d 3976	. . . . . . . . . . . 12 |- (z e. H -> (y .ih (T` z)) = (y .ih z))
45	44	adantl 388	. . . . . . . . . . 11 |- ((y e. H~ /\ z e. H) -> (y .ih (T` z)) = (y .ih z))
46	38, 45	eqtr3d 1509	. . . . . . . . . 10 |- ((y e. H~ /\ z e. H) -> ((T` y) .ih z) = (y .ih z))
47	46	opreq2d 3976	. . . . . . . . 9 |- ((y e. H~ /\ z e. H) -> ((y .ih z) - ((T` y) .ih z)) = ((y .ih z) - (y .ih z)))
48		hiclt 8947	. . . . . . . . . . 11 |- ((y e. H~ /\ z e. H~) -> (y .ih z) e. CC)
49	48, 33	sylan2 451	. . . . . . . . . 10 |- ((y e. H~ /\ z e. H) -> (y .ih z) e. CC)
50		subidt 5395	. . . . . . . . . 10 |- ((y .ih z) e. CC -> ((y .ih z) - (y .ih z)) = 0)
51	49, 50	syl 10	. . . . . . . . 9 |- ((y e. H~ /\ z e. H) -> ((y .ih z) - (y .ih z)) = 0)
52	35, 47, 51	3eqtrd 1511	. . . . . . . 8 |- ((y e. H~ /\ z e. H) -> ((y -h (T` y)) .ih z) = 0)
53	52	r19.21aiva 1714	. . . . . . 7 |- (y e. H~ -> A.z e. H ((y -h (T` y)) .ih z) = 0)
54	27, 53	jca 288	. . . . . 6 |- (y e. H~ -> ((y -h (T` y)) e. H~ /\ A.z e. H ((y -h (T` y)) .ih z) = 0))
55		ocelt 9154	. . . . . . 7 |- (H (_ H~ -> ((y -h (T` y)) e. (_|_` H) <-> ((y -h (T` y)) e. H~ /\ A.z e. H ((y -h (T` y)) .ih z) = 0)))
56	32, 55	ax-mp 7	. . . . . 6 |- ((y -h (T` y)) e. (_|_` H) <-> ((y -h (T` y)) e. H~ /\ A.z e. H ((y -h (T` y)) .ih z) = 0))
57	54, 56	sylibr 200	. . . . 5 |- (y e. H~ -> (y -h (T` y)) e. (_|_` H))
58	14, 25, 57	sylanc 471	. . . 4 |- (y e. H~ -> ((proj` H)` ((T` y) +h (y -h (T` y)))) = (T` y))
59		hvpncan3t 8911	. . . . . 6 |- (((T` y) e. H~ /\ y e. H~) -> ((T` y) +h (y -h (T` y))) = y)
60	15, 59	mpancom 705	. . . . 5 |- (y e. H~ -> ((T` y) +h (y -h (T` y))) = y)
61	60	fveq2d 3728	. . . 4 |- (y e. H~ -> ((proj` H)` ((T` y) +h (y -h (T` y)))) = ((proj` H)` y))
62	58, 61	eqtr3d 1509	. . 3 |- (y e. H~ -> (T` y) = ((proj` H)` y))
63	62	rgen 1698	. 2 |- A.y e. H~ (T` y) = ((proj` H)` y)
64	12, 13, 63	mpbir2an 730	1 |- T = (proj` H)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  {crab 1648   (_ wss 2047   o. ccom 3174   Fn wfn 3177  -->wf 3178  ` cfv 3182  (class class class)co 3963  CCcc 5232  0cc0 5234   - cmin 5292  H~chil 8788   +h cva 8789   -h cmv 8792   .ih csp 8793  _|_cort 8799  projcpj 8806  LinOpclo 8816  HrmOpcho 8819

This theorem is referenced by:  hmopidmpjt 10082

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625  ax-ac 4744  ax-hilex 8869  ax-hfvadd 8870  ax-hvcom 8871  ax-hvass 8872  ax-hv0cl 8873  ax-hvaddid 8874  ax-hfvmul 8875  ax-hvmulid 8876  ax-hvmulass 8877  ax-hvdistr1 8878  ax-hvdistr2 8879  ax-hvmul0 8880  ax-hfi 8946  ax-his1 8949  ax-his2 8950  ax-his3 8951  ax-his4 8952  ax-hcompl 9071

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-map 4324  df-en 4368  df-dom 4369  df-sdom 4370  df-sup 4574  df-r1 4643  df-rank 4644  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-n 5925  df-2 5970  df-3 5971  df-4 5972  df-n0 6100  df-z 6136  df-fl 6224  df-q 6256  df-seq1 6308  df-shft 6341  df-ioo 6361  df-uz 6418  df-fz 6468  df-seqz 6533  df-exp 6569  df-sqr 6670  df-re 6751  df-im 6752  df-cj 6753  df-abs 6754  df-clim 6975  df-sum 6980  df-top 7592  df-bases 7594  df-topgen 7595  df-cld 7663  df-ntr 7664  df-cls 7665  df-cn 7754  df-cnp 7755  df-haus 7782  df-met 7793  df-bl 7795  df-opn 7796  df-lm 7922  df-grp 8037  df-gid 8038  df-ginv 8039  df-gdiv 8040  df-abl 8100  df-vc 8165  df-nv 8211  df-va 8214  df-ba 8215  df-sm 8216  df-0v 8217  df-vs 8218  df-nm 8219  df-ims 8220  df-ip 8350  df-ph 8472  df-hnorm 8837  df-hvsub 8840  df-hlim 8841