Theorem riesz4 9996

Description: A continuous linear functional can be expressed as an inner product. Uniqueness part of Theorem 3.9 of [Beran] p. 104.

Hypotheses

Ref	Expression
nlelch.1	|- T e. LinFn
nlelch.2	|- T e. ConFn

Assertion

Ref	Expression
riesz4	|- E!w e. H~ A.v e. H~ (T` v) = (v .ih w)

Distinct variable group:   w,v,T

Proof of Theorem riesz4

Step	Hyp	Ref	Expression
1		opreq2 3969	. . . . 5 |- (w = u -> (v .ih w) = (v .ih u))
2	1	eqeq2d 1486	. . . 4 |- (w = u -> ((T` v) = (v .ih w) <-> (T` v) = (v .ih u)))
3	2	ralbidv 1663	. . 3 |- (w = u -> (A.v e. H~ (T` v) = (v .ih w) <-> A.v e. H~ (T` v) = (v .ih u)))
4	3	reu4 1934	. 2 |- (E!w e. H~ A.v e. H~ (T` v) = (v .ih w) <-> (E.w e. H~ A.v e. H~ (T` v) = (v .ih w) /\ A.w e. H~ A.u e. H~ ((A.v e. H~ (T` v) = (v .ih w) /\ A.v e. H~ (T` v) = (v .ih u)) -> w = u)))
5		nlelch.1	. . 3 |- T e. LinFn
6		nlelch.2	. . 3 |- T e. ConFn
7	5, 6	riesz3 9995	. 2 |- E.w e. H~ A.v e. H~ (T` v) = (v .ih w)
8		hvsubclt 8887	. . . . . 6 |- ((w e. H~ /\ u e. H~) -> (w -h u) e. H~)
9		opreq1 3968	. . . . . . . . 9 |- (v = (w -h u) -> (v .ih w) = ((w -h u) .ih w))
10		opreq1 3968	. . . . . . . . 9 |- (v = (w -h u) -> (v .ih u) = ((w -h u) .ih u))
11	9, 10	opreq12d 3978	. . . . . . . 8 |- (v = (w -h u) -> ((v .ih w) - (v .ih u)) = (((w -h u) .ih w) - ((w -h u) .ih u)))
12	11	eqeq1d 1483	. . . . . . 7 |- (v = (w -h u) -> (((v .ih w) - (v .ih u)) = 0 <-> (((w -h u) .ih w) - ((w -h u) .ih u)) = 0))
13	12	rcla4v 1873	. . . . . 6 |- ((w -h u) e. H~ -> (A.v e. H~ ((v .ih w) - (v .ih u)) = 0 -> (((w -h u) .ih w) - ((w -h u) .ih u)) = 0))
14	8, 13	syl 10	. . . . 5 |- ((w e. H~ /\ u e. H~) -> (A.v e. H~ ((v .ih w) - (v .ih u)) = 0 -> (((w -h u) .ih w) - ((w -h u) .ih u)) = 0))
15		normclt 8991	. . . . . . . . . 10 |- ((w -h u) e. H~ -> (normh` (w -h u)) e. RR)
16	15	recnd 5315	. . . . . . . . 9 |- ((w -h u) e. H~ -> (normh` (w -h u)) e. CC)
17		sqeq0t 6613	. . . . . . . . 9 |- ((normh` (w -h u)) e. CC -> (((normh` (w -h u))^2) = 0 <-> (normh` (w -h u)) = 0))
18	16, 17	syl 10	. . . . . . . 8 |- ((w -h u) e. H~ -> (((normh` (w -h u))^2) = 0 <-> (normh` (w -h u)) = 0))
19		norm-it 8996	. . . . . . . 8 |- ((w -h u) e. H~ -> ((normh` (w -h u)) = 0 <-> (w -h u) = 0h))
20	18, 19	bitrd 528	. . . . . . 7 |- ((w -h u) e. H~ -> (((normh` (w -h u))^2) = 0 <-> (w -h u) = 0h))
21	8, 20	syl 10	. . . . . 6 |- ((w e. H~ /\ u e. H~) -> (((normh` (w -h u))^2) = 0 <-> (w -h u) = 0h))
22		normsqt 9001	. . . . . . . . 9 |- ((w -h u) e. H~ -> ((normh` (w -h u))^2) = ((w -h u) .ih (w -h u)))
23	8, 22	syl 10	. . . . . . . 8 |- ((w e. H~ /\ u e. H~) -> ((normh` (w -h u))^2) = ((w -h u) .ih (w -h u)))
24		his2sub2t 8959	. . . . . . . . 9 |- (((w -h u) e. H~ /\ w e. H~ /\ u e. H~) -> ((w -h u) .ih (w -h u)) = (((w -h u) .ih w) - ((w -h u) .ih u)))
25		pm3.26 319	. . . . . . . . 9 |- ((w e. H~ /\ u e. H~) -> w e. H~)
26		pm3.27 323	. . . . . . . . 9 |- ((w e. H~ /\ u e. H~) -> u e. H~)
27	24, 8, 25, 26	syl3anc 858	. . . . . . . 8 |- ((w e. H~ /\ u e. H~) -> ((w -h u) .ih (w -h u)) = (((w -h u) .ih w) - ((w -h u) .ih u)))
28	23, 27	eqtrd 1507	. . . . . . 7 |- ((w e. H~ /\ u e. H~) -> ((normh` (w -h u))^2) = (((w -h u) .ih w) - ((w -h u) .ih u)))
29	28	eqeq1d 1483	. . . . . 6 |- ((w e. H~ /\ u e. H~) -> (((normh` (w -h u))^2) = 0 <-> (((w -h u) .ih w) - ((w -h u) .ih u)) = 0))
30		hvsubeq0t 8935	. . . . . 6 |- ((w e. H~ /\ u e. H~) -> ((w -h u) = 0h <-> w = u))
31	21, 29, 30	3bitr3d 548	. . . . 5 |- ((w e. H~ /\ u e. H~) -> ((((w -h u) .ih w) - ((w -h u) .ih u)) = 0 <-> w = u))
32	14, 31	sylibd 202	. . . 4 |- ((w e. H~ /\ u e. H~) -> (A.v e. H~ ((v .ih w) - (v .ih u)) = 0 -> w = u))
33		r19.26 1750	. . . . 5 |- (A.v e. H~ ((T` v) = (v .ih w) /\ (T` v) = (v .ih u)) <-> (A.v e. H~ (T` v) = (v .ih w) /\ A.v e. H~ (T` v) = (v .ih u)))
34		opreq12 3970	. . . . . . . 8 |- (((T` v) = (v .ih w) /\ (T` v) = (v .ih u)) -> ((T` v) - (T` v)) = ((v .ih w) - (v .ih u)))
35	34	adantl 388	. . . . . . 7 |- ((v e. H~ /\ ((T` v) = (v .ih w) /\ (T` v) = (v .ih u))) -> ((T` v) - (T` v)) = ((v .ih w) - (v .ih u)))
36	5	lnfnf 9970	. . . . . . . . . 10 |- T:H~-->CC
37	36	ffvelrni 3815	. . . . . . . . 9 |- (v e. H~ -> (T` v) e. CC)
38		subidt 5395	. . . . . . . . 9 |- ((T` v) e. CC -> ((T` v) - (T` v)) = 0)
39	37, 38	syl 10	. . . . . . . 8 |- (v e. H~ -> ((T` v) - (T` v)) = 0)
40	39	adantr 389	. . . . . . 7 |- ((v e. H~ /\ ((T` v) = (v .ih w) /\ (T` v) = (v .ih u))) -> ((T` v) - (T` v)) = 0)
41	35, 40	eqtr3d 1509	. . . . . 6 |- ((v e. H~ /\ ((T` v) = (v .ih w) /\ (T` v) = (v .ih u))) -> ((v .ih w) - (v .ih u)) = 0)
42	41	r19.20ia 1705	. . . . 5 |- (A.v e. H~ ((T` v) = (v .ih w) /\ (T` v) = (v .ih u)) -> A.v e. H~ ((v .ih w) - (v .ih u)) = 0)
43	33, 42	sylbir 201	. . . 4 |- ((A.v e. H~ (T` v) = (v .ih w) /\ A.v e. H~ (T` v) = (v .ih u)) -> A.v e. H~ ((v .ih w) - (v .ih u)) = 0)
44	32, 43	syl5 21	. . 3 |- ((w e. H~ /\ u e. H~) -> ((A.v e. H~ (T` v) = (v .ih w) /\ A.v e. H~ (T` v) = (v .ih u)) -> w = u))
45	44	rgen2a 1699	. 2 |- A.w e. H~ A.u e. H~ ((A.v e. H~ (T` v) = (v .ih w) /\ A.v e. H~ (T` v) = (v .ih u)) -> w = u)
46	4, 7, 45	mpbir2an 730	1 |- E!w e. H~ A.v e. H~ (T` v) = (v .ih w)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646  E!wreu 1647  ` cfv 3182  (class class class)co 3963  CCcc 5232  0cc0 5234   - cmin 5292  2c2 5961  ^cexp 6568  H~chil 8788  0hc0v 8791   -h cmv 8792   .ih csp 8793  normhcno 8794  ConFnccnf 8822  LinFnclf 8823

This theorem is referenced by:  riesz4t 9997

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 &