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Theorem nmbdfnlb 23558
Description: A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
nmbdfnlb |- ( ( T e. LinFn /\ ( normfn ` T ) e. RR /\ A e. ~H ) -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
Proof of Theorem nmbdfnlb
StepHypRef Expression
1 fveq1 5730 . . . . . 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 ) )
21fveq2d 5735 . . . . 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 5731 . . . . . 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 } ) ) ) )
43oveq1d 6099 . . . . 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 ) ) )
52, 4breq12d 4228 . . . 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 ) ) ) )
65imbi2d 309 . . 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 2498 . . . . . 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 ) )
83eleq1d 2504 . . . . . 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 ) )
97, 8anbi12d 693 . . . . 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 2498 . . . . . 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 5731 . . . . . . 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 } ) ) ) )
1211eleq1d 2504 . . . . . 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 ) )
1310, 12anbi12d 693 . . . . 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 23493 . . . . . 6 |- ( ~H X. { 0 } ) e. LinFn
15 nmfn0 23495 . . . . . . 7 |- ( normfn ` ( ~H X. { 0 } ) ) = 0
16 0re 9096 . . . . . . 7 |- 0 e. RR
1715, 16eqeltri 2508 . . . . . 6 |- ( normfn ` ( ~H X. { 0 } ) ) e. RR
1814, 17pm3.2i 443 . . . . 5 |- ( ( ~H X. { 0 } ) e. LinFn /\ ( normfn ` ( ~H X. { 0 } ) ) e. RR )
199, 13, 18elimhyp 3789 . . . 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 )
2019nmbdfnlbi 23557 . . 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 ) ) )
216, 20dedth 3782 . 2 |- ( ( T e. LinFn /\ ( normfn ` T ) e. RR ) -> ( A e. ~H -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) ) )
22213impia 1151 1 |- ( ( T e. LinFn /\ ( normfn ` T ) e. RR /\ A e. ~H ) -> ( abs ` ( T ` A ) ) <_ ( ( normfn ` T ) x. ( normh ` A ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 360   /\ w3a 937   = wceq 1653   e. wcel 1726   ifcif 3741   {csn 3816   class class class wbr 4215   X. cxp 4879   ` cfv 5457  (class class class)co 6084   RRcr 8994   0cc0 8995   x. cmul 9000   <_ cle 9126   abscabs 12044   ~Hchil 22427   normhcno 22431   normfncnmf 22459   LinFnclf 22462
This theorem is referenced by:  lnfncnbd  23565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-hilex 22507  ax-hfvadd 22508  ax-hv0cl 22511  ax-hvaddid 22512  ax-hfvmul 22513  ax-hvmulid 22514  ax-hvmul0 22518  ax-hfi 22586  ax-his1 22589  ax-his3 22591  ax-his4 22592
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-seq 11329  df-exp 11388  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-hnorm 22476  df-nmfn 23353  df-lnfn 23356
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