Theorem nmbdfnlb 30131

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 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))

Proof of Theorem nmbdfnlb

Step	Hyp	Ref	Expression
1		fveq1 6716	. . . . . 6 ⊢ (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (𝑇‘𝐴) = (if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴))
2	1	fveq2d 6721	. . . . 5 ⊢ (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (abs‘(𝑇‘𝐴)) = (abs‘(if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴)))
3		fveq2 6717	. . . . . 6 ⊢ (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (normfn‘𝑇) = (normfn‘if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))))
4	3	oveq1d 7228	. . . . 5 ⊢ (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((normfn‘𝑇) · (normℎ‘𝐴)) = ((normfn‘if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) · (normℎ‘𝐴)))
5	2, 4	breq12d 5066	. . . 4 ⊢ (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)) ↔ (abs‘(if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴)) ≤ ((normfn‘if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) · (normℎ‘𝐴))))
6	5	imbi2d 344	. . 3 ⊢ (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))) ↔ (𝐴 ∈ ℋ → (abs‘(if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴)) ≤ ((normfn‘if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) · (normℎ‘𝐴)))))
7		eleq1 2825	. . . . . 6 ⊢ (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (𝑇 ∈ LinFn ↔ if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn))
8	3	eleq1d 2822	. . . . . 6 ⊢ (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((normfn‘𝑇) ∈ ℝ ↔ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ))
9	7, 8	anbi12d 634	. . . . 5 ⊢ (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ) ↔ (if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn ∧ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ)))
10		eleq1 2825	. . . . . 6 ⊢ (( ℋ × {0}) = if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (( ℋ × {0}) ∈ LinFn ↔ if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn))
11		fveq2 6717	. . . . . . 7 ⊢ (( ℋ × {0}) = if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (normfn‘( ℋ × {0})) = (normfn‘if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))))
12	11	eleq1d 2822	. . . . . 6 ⊢ (( ℋ × {0}) = if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((normfn‘( ℋ × {0})) ∈ ℝ ↔ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ))
13	10, 12	anbi12d 634	. . . . 5 ⊢ (( ℋ × {0}) = if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((( ℋ × {0}) ∈ LinFn ∧ (normfn‘( ℋ × {0})) ∈ ℝ) ↔ (if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn ∧ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ)))
14		0lnfn 30066	. . . . . 6 ⊢ ( ℋ × {0}) ∈ LinFn
15		nmfn0 30068	. . . . . . 7 ⊢ (normfn‘( ℋ × {0})) = 0
16		0re 10835	. . . . . . 7 ⊢ 0 ∈ ℝ
17	15, 16	eqeltri 2834	. . . . . 6 ⊢ (normfn‘( ℋ × {0})) ∈ ℝ
18	14, 17	pm3.2i 474	. . . . 5 ⊢ (( ℋ × {0}) ∈ LinFn ∧ (normfn‘( ℋ × {0})) ∈ ℝ)
19	9, 13, 18	elimhyp 4504	. . . 4 ⊢ (if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn ∧ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ)
20	19	nmbdfnlbi 30130	. . 3 ⊢ (𝐴 ∈ ℋ → (abs‘(if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴)) ≤ ((normfn‘if((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) · (normℎ‘𝐴)))
21	6, 20	dedth 4497	. 2 ⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ) → (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))))
22	21	3impia 1119	1 ⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ifcif 4439  {csn 4541   class class class wbr 5053   × cxp 5549  ‘cfv 6380  (class class class)co 7213  ℝcr 10728  0cc0 10729   · cmul 10734   ≤ cle 10868  abscabs 14797   ℋchba 29000  norm_ℎcno 29004  norm_fncnmf 29032  LinFnclf 29035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807  ax-hilex 29080  ax-hfvadd 29081  ax-hv0cl 29084  ax-hvaddid 29085  ax-hfvmul 29086  ax-hvmulid 29087  ax-hvmul0 29091  ax-hfi 29160  ax-his1 29163  ax-his3 29165  ax-his4 29166

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-sup 9058  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-z 12177  df-uz 12439  df-rp 12587  df-seq 13575  df-exp 13636  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-hnorm 29049  df-nmfn 29926  df-lnfn 29929

This theorem is referenced by:  lnfncnbd  30138