Theorem riesz1t 9998

Description: Part 1 of the Riesz representation theorem for bounded linear functionals. A linear functional is bounded iff its value can be expressed as an inner product. Part of Theorem 17.3 of [Halmos] p. 31. For part 2, see riesz2t 9999. For the continuous linear functional version, see riesz3 9995 and riesz4t 9997.

Assertion

Ref	Expression
riesz1t	|- (T e. LinFn -> ((normfn` T) e. RR <-> E.y e. H~ A.x e. H~ (T` x) = (x .ih y)))

Distinct variable group:   x,y,T

Proof of Theorem riesz1t

Step	Hyp	Ref	Expression
1		lnfncnbdt 9992	. 2 |- (T e. LinFn -> (T e. ConFn <-> (normfn` T) e. RR))
2		elin 2207	. . . . 5 |- (T e. (LinFn i^i ConFn) <-> (T e. LinFn /\ T e. ConFn))
3		fveq1 3723	. . . . . . . 8 |- (T = if(T e. (LinFn i^i ConFn), T, (H~ X. {0})) -> (T` x) = (if(T e. (LinFn i^i ConFn), T, (H~ X. {0}))` x))
4	3	eqeq1d 1483	. . . . . . 7 |- (T = if(T e. (LinFn i^i ConFn), T, (H~ X. {0})) -> ((T` x) = (x .ih y) <-> (if(T e. (LinFn i^i ConFn), T, (H~ X. {0}))` x) = (x .ih y)))
5	4	rexralbidv 1682	. . . . . 6 |- (T = if(T e. (LinFn i^i ConFn), T, (H~ X. {0})) -> (E.y e. H~ A.x e. H~ (T` x) = (x .ih y) <-> E.y e. H~ A.x e. H~ (if(T e. (LinFn i^i ConFn), T, (H~ X. {0}))` x) = (x .ih y)))
6		inss1 2230	. . . . . . . 8 |- (LinFn i^i ConFn) (_ LinFn
7		elin 2207	. . . . . . . . . 10 |- ((H~ X. {0}) e. (LinFn i^i ConFn) <-> ((H~ X. {0}) e. LinFn /\ (H~ X. {0}) e. ConFn))
8		0lnfn 9909	. . . . . . . . . 10 |- (H~ X. {0}) e. LinFn
9		0cnfn 9904	. . . . . . . . . 10 |- (H~ X. {0}) e. ConFn
10	7, 8, 9	mpbir2an 730	. . . . . . . . 9 |- (H~ X. {0}) e. (LinFn i^i ConFn)
11	10	elimel 2394	. . . . . . . 8 |- if(T e. (LinFn i^i ConFn), T, (H~ X. {0})) e. (LinFn i^i ConFn)
12	6, 11	sselii 2066	. . . . . . 7 |- if(T e. (LinFn i^i ConFn), T, (H~ X. {0})) e. LinFn
13		inss2 2231	. . . . . . . 8 |- (LinFn i^i ConFn) (_ ConFn
14	13, 11	sselii 2066	. . . . . . 7 |- if(T e. (LinFn i^i ConFn), T, (H~ X. {0})) e. ConFn
15	12, 14	riesz3 9995	. . . . . 6 |- E.y e. H~ A.x e. H~ (if(T e. (LinFn i^i ConFn), T, (H~ X. {0}))` x) = (x .ih y)
16	5, 15	dedth 2383	. . . . 5 |- (T e. (LinFn i^i ConFn) -> E.y e. H~ A.x e. H~ (T` x) = (x .ih y))
17	2, 16	sylbir 201	. . . 4 |- ((T e. LinFn /\ T e. ConFn) -> E.y e. H~ A.x e. H~ (T` x) = (x .ih y))
18	17	ex 373	. . 3 |- (T e. LinFn -> (T e. ConFn -> E.y e. H~ A.x e. H~ (T` x) = (x .ih y)))
19		fveq2 3724	. . . . . . . . . . . 12 |- ((T` x) = (x .ih y) -> (abs` (T` x)) = (abs` (x .ih y)))
20	19	adantl 388	. . . . . . . . . . 11 |- ((((T e. LinFn /\ x e. H~) /\ y e. H~) /\ (T` x) = (x .ih y)) -> (abs` (T` x)) = (abs` (x .ih y)))
21		bcst 9048	. . . . . . . . . . . . . 14 |- ((x e. H~ /\ y e. H~) -> (abs` (x .ih y)) <_ ((normh` x) x. (normh` y)))
22		axmulcom 5276	. . . . . . . . . . . . . . . 16 |- (((normh` x) e. CC /\ (normh` y) e. CC) -> ((normh` x) x. (normh` y)) = ((normh` y) x. (normh` x)))
23		recnt 5313	. . . . . . . . . . . . . . . 16 |- ((normh` x) e. RR -> (normh` x) e. CC)
24		recnt 5313	. . . . . . . . . . . . . . . 16 |- ((normh` y) e. RR -> (normh` y) e. CC)
25	22, 23, 24	syl2an 454	. . . . . . . . . . . . . . 15 |- (((normh` x) e. RR /\ (normh` y) e. RR) -> ((normh` x) x. (normh` y)) = ((normh` y) x. (normh` x)))
26		normclt 8991	. . . . . . . . . . . . . . 15 |- (x e. H~ -> (normh` x) e. RR)
27		normclt 8991	. . . . . . . . . . . . . . 15 |- (y e. H~ -> (normh` y) e. RR)
28	25, 26, 27	syl2an 454	. . . . . . . . . . . . . 14 |- ((x e. H~ /\ y e. H~) -> ((normh` x) x. (normh` y)) = ((normh` y) x. (normh` x)))
29	21, 28	breqtrd 2639	. . . . . . . . . . . . 13 |- ((x e. H~ /\ y e. H~) -> (abs` (x .ih y)) <_ ((normh` y) x. (normh` x)))
30	29	adantll 392	. . . . . . . . . . . 12 |- (((T e. LinFn /\ x e. H~) /\ y e. H~) -> (abs` (x .ih y)) <_ ((normh` y) x. (normh` x)))
31	30	adantr 389	. . . . . . . . . . 11 |- ((((T e. LinFn /\ x e. H~) /\ y e. H~) /\ (T` x) = (x .ih y)) -> (abs` (x .ih y)) <_ ((normh` y) x. (normh` x)))
32	20, 31	eqbrtrd 2635	. . . . . . . . . 10 |- ((((T e. LinFn /\ x e. H~) /\ y e. H~) /\ (T` x) = (x .ih y)) -> (abs` (T` x)) <_ ((normh` y) x. (normh` x)))
33	32	ex 373	. . . . . . . . 9 |- (((T e. LinFn /\ x e. H~) /\ y e. H~) -> ((T` x) = (x .ih y) -> (abs` (T` x)) <_ ((normh` y) x. (normh` x))))
34	33	an1rs 489	. . . . . . . 8 |- (((T e. LinFn /\ y e. H~) /\ x e. H~) -> ((T` x) = (x .ih y) -> (abs` (T` x)) <_ ((normh` y) x. (normh` x))))
35	34	r19.20dva 1709	. . . . . . 7 |- ((T e. LinFn /\ y e. H~) -> (A.x e. H~ (T` x) = (x .ih y) -> A.x e. H~ (abs` (T` x)) <_ ((normh` y) x. (normh` x))))
36	27	adantl 388	. . . . . . 7 |- ((T e. LinFn /\ y e. H~) -> (normh` y) e. RR)
37	35, 36	jctild 601	. . . . . 6 |- ((T e. LinFn /\ y e. H~) -> (A.x e. H~ (T` x) = (x .ih y) -> ((normh` y) e. RR /\ A.x e. H~ (abs` (T` x)) <_ ((normh` y) x. (normh` x)))))
38		opreq1 3968	. . . . . . . . 9 |- (z = (normh` y) -> (z x. (normh` x)) = ((normh` y) x. (normh` x)))
39	38	breq2d 2630	. . . . . . . 8 |- (z = (normh` y) -> ((abs` (T` x)) <_ (z x. (normh` x)) <-> (abs` (T` x)) <_ ((normh` y) x. (normh` x))))
40	39	ralbidv 1663	. . . . . . 7 |- (z = (normh` y) -> (A.x e. H~ (abs` (T` x)) <_ (z x. (normh` x)) <-> A.x e. H~ (abs` (T` x)) <_ ((normh` y) x. (normh` x))))
41	40	rcla4ev 1877	. . . . . 6 |- (((normh` y) e. RR /\ A.x e. H~ (abs` (T` x)) <_ ((normh` y) x. (normh` x))) -> E.z e. RR A.x e. H~ (abs` (T` x)) <_ (z x. (normh` x)))
42	37, 41	syl6 22	. . . . 5 |- ((T e. LinFn /\ y e. H~) -> (A.x e. H~ (T` x) = (x .ih y) -> E.z e. RR A.x e. H~ (abs` (T` x)) <_ (z x. (normh` x))))
43	42	r19.23adva 1747	. . . 4 |- (T e. LinFn -> (E.y e. H~ A.x e. H~ (T` x) = (x .ih y) -> E.z e. RR A.x e. H~ (abs` (T` x)) <_ (z x. (normh` x))))
44		lnfncont 9991	. . . 4 |- (T e. LinFn -> (T e. ConFn <-> E.z e. RR A.x e. H~ (abs` (T` x)) <_ (z x. (normh` x))))
45	43, 44	sylibrd 204	. . 3 |- (T e. LinFn -> (E.y e. H~ A.x e. H~ (T` x) = (x .ih y) -> T e. ConFn))
46	18, 45	impbid 516	. 2 |- (T e. LinFn -> (T e. ConFn <-> E.y e. H~ A.x e. H~ (T` x) = (x .ih y)))
47	1, 46	bitr3d 530	1 |- (T e. LinFn -> ((normfn` T) e. RR <-> E.y e. H~ A.x e. H~ (T` x) = (x .ih y)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646   i^i cin 2046  ifcif 2361  {csn 2409   class class class wbr 2619   X. cxp 3168  ` cfv 3182  (class class class)co 3963  CCcc 5232  RRcr 5233  0cc0 5234   x. cmul 5239   <_ cle 5295  abscabs 6750  H~chil 8788   .ih csp 8793  normhcno 8794  norm_fncnmf 8820  ConFnccnf 8822  LinFnclf 8823

This theorem is referenced by:  rnbra 10040

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