Theorem nmbdfnlb 32084

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 6921	. . . . . 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 6926	. . . . 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 6922	. . . . . 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 7465	. . . . 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 5179	. . . 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 340	. . 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 2832	. . . . . 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 2829	. . . . . 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 631	. . . . 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 2832	. . . . . 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 6922	. . . . . . 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 2829	. . . . . 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 631	. . . . 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 32019	. . . . . 6 ⊢ ( ℋ × {0}) ∈ LinFn
15		nmfn0 32021	. . . . . . 7 ⊢ (normfn‘( ℋ × {0})) = 0
16		0re 11294	. . . . . . 7 ⊢ 0 ∈ ℝ
17	15, 16	eqeltri 2840	. . . . . 6 ⊢ (normfn‘( ℋ × {0})) ∈ ℝ
18	14, 17	pm3.2i 470	. . . . 5 ⊢ (( ℋ × {0}) ∈ LinFn ∧ (normfn‘( ℋ × {0})) ∈ ℝ)
19	9, 13, 18	elimhyp 4613	. . . 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 32083	. . 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 4606	. 2 ⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ) → (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))))
22	21	3impia 1117	1 ⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ifcif 4548  {csn 4648   class class class wbr 5166   × cxp 5698  ‘cfv 6575  (class class class)co 7450  ℝcr 11185  0cc0 11186   · cmul 11191   ≤ cle 11327  abscabs 15285   ℋchba 30953  norm_ℎcno 30957  norm_fncnmf 30985  LinFnclf 30988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772  ax-cnex 11242  ax-resscn 11243  ax-1cn 11244  ax-icn 11245  ax-addcl 11246  ax-addrcl 11247  ax-mulcl 11248  ax-mulrcl 11249  ax-mulcom 11250  ax-addass 11251  ax-mulass 11252  ax-distr 11253  ax-i2m1 11254  ax-1ne0 11255  ax-1rid 11256  ax-rnegex 11257  ax-rrecex 11258  ax-cnre 11259  ax-pre-lttri 11260  ax-pre-lttrn 11261  ax-pre-ltadd 11262  ax-pre-mulgt0 11263  ax-pre-sup 11264  ax-hilex 31033  ax-hfvadd 31034  ax-hv0cl 31037  ax-hvaddid 31038  ax-hfvmul 31039  ax-hvmulid 31040  ax-hvmul0 31044  ax-hfi 31113  ax-his1 31116  ax-his3 31118  ax-his4 31119

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6334  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-riota 7406  df-ov 7453  df-oprab 7454  df-mpo 7455  df-om 7906  df-2nd 8033  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-rdg 8468  df-er 8765  df-map 8888  df-en 9006  df-dom 9007  df-sdom 9008  df-sup 9513  df-pnf 11328  df-mnf 11329  df-xr 11330  df-ltxr 11331  df-le 11332  df-sub 11524  df-neg 11525  df-div 11950  df-nn 12296  df-2 12358  df-3 12359  df-n0 12556  df-z 12642  df-uz 12906  df-rp 13060  df-seq 14055  df-exp 14115  df-cj 15150  df-re 15151  df-im 15152  df-sqrt 15286  df-abs 15287  df-hnorm 31002  df-nmfn 31879  df-lnfn 31882

This theorem is referenced by:  lnfncnbd  32091