Theorem nmbdfnlbt 9979

Description: A lower bound for the norm of a bounded linear functional.

Assertion

Ref	Expression
nmbdfnlbt	|- ((T e. LinFn /\ (normfn` T) e. RR /\ A e. H~) -> (abs` (T` A)) <_ ((normfn` T) x. (normh` A)))

Proof of Theorem nmbdfnlbt

Step	Hyp	Ref	Expression
1		fveq1 3723	. . . . . 6 |- (T = if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) -> (T` A) = (if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))` A))
2	1	fveq2d 3728	. . . . 5 |- (T = if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) -> (abs` (T` A)) = (abs` (if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))` A)))
3		fveq2 3724	. . . . . 6 |- (T = if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) -> (normfn` T) = (normfn` if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))))
4	3	opreq1d 3975	. . . . 5 |- (T = if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) -> ((normfn` T) x. (normh` A)) = ((normfn` if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))) x. (normh` A)))
5	2, 4	breq12d 2631	. . . 4 |- (T = if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) -> ((abs` (T` A)) <_ ((normfn` T) x. (normh` A)) <-> (abs` (if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))` A)) <_ ((normfn` if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))) x. (normh` A))))
6	5	imbi2d 612	. . 3 |- (T = if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) -> ((A e. H~ -> (abs` (T` A)) <_ ((normfn` T) x. (normh` A))) <-> (A e. H~ -> (abs` (if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))` A)) <_ ((normfn` if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))) x. (normh` A)))))
7		eleq1 1534	. . . . . 6 |- (T = if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) -> (T e. LinFn <-> if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) e. LinFn))
8	3	eleq1d 1540	. . . . . 6 |- (T = if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) -> ((normfn` T) e. RR <-> (normfn` if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))) e. RR))
9	7, 8	anbi12d 628	. . . . 5 |- (T = if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) -> ((T e. LinFn /\ (normfn` T) e. RR) <-> (if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) e. LinFn /\ (normfn` if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))) e. RR)))
10		eleq1 1534	. . . . . 6 |- ((H~ X. {0}) = if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) -> ((H~ X. {0}) e. LinFn <-> if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) e. LinFn))
11		fveq2 3724	. . . . . . 7 |- ((H~ X. {0}) = if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) -> (normfn` (H~ X. {0})) = (normfn` if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))))
12	11	eleq1d 1540	. . . . . 6 |- ((H~ X. {0}) = if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) -> ((normfn` (H~ X. {0})) e. RR <-> (normfn` if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))) e. RR))
13	10, 12	anbi12d 628	. . . . 5 |- ((H~ X. {0}) = if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) -> (((H~ X. {0}) e. LinFn /\ (normfn` (H~ X. {0})) e. RR) <-> (if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) e. LinFn /\ (normfn` if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))) e. RR)))
14		0lnfn 9909	. . . . . 6 |- (H~ X. {0}) e. LinFn
15		nmfn0 9911	. . . . . . 7 |- (normfn` (H~ X. {0})) = 0
16		0re 5440	. . . . . . 7 |- 0 e. RR
17	15, 16	eqeltr 1544	. . . . . 6 |- (normfn` (H~ X. {0})) e. RR
18	14, 17	pm3.2i 285	. . . . 5 |- ((H~ X. {0}) e. LinFn /\ (normfn` (H~ X. {0})) e. RR)
19	9, 13, 18	elimhyp 2390	. . . 4 |- (if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0})) e. LinFn /\ (normfn` if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))) e. RR)
20	19	nmbdfnlb 9978	. . 3 |- (A e. H~ -> (abs` (if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))` A)) <_ ((normfn` if((T e. LinFn /\ (normfn` T) e. RR), T, (H~ X. {0}))) x. (normh` A)))
21	6, 20	dedth 2383	. 2 |- ((T e. LinFn /\ (normfn` T) e. RR) -> (A e. H~ -> (abs` (T` A)) <_ ((normfn` T) x. (normh` A))))
22	21	3impia 830	1 |- ((T e. LinFn /\ (normfn` T) e. RR /\ A e. H~) -> (abs` (T` A)) <_ ((normfn` T) x. (normh` A)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  ifcif 2361  {csn 2409   class class class wbr 2619   X. cxp 3168  ` cfv 3182  (class class class)co 3963  RRcr 5233  0cc0 5234   x. cmul 5239   <_ cle 5295  abscabs 6750  H~chil 8788  normhcno 8794  norm_fncnmf 8820  LinFnclf 8823

This theorem is referenced by:  lnfncnbdt 9992

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-inf2 4625  ax-hilex 8869  ax-hfvadd 8870  ax-hv0cl 8873  ax-hvaddid 8874  ax-hfvmul 8875  ax-hvmulid 8876  ax-hvmul0 8880  ax-hfi 8946  ax-his1 8949  ax-his3 8951  ax-his4 8952

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-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-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-n0 6100  df-z 6136  df-seq1 6308  df-exp 6569  df-sqr 6670  df-re 6751  df-im 6752  df-cj 6753  df-abs 6754  df-hnorm 8837  df-nmfn 9771  df-lnfn 9774

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