Theorem nmbdfnlb 31977

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 6889	. . . . . 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 6894	. . . . 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 6890	. . . . . 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 7428	. . . . 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 5156	. . . 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 339	. . 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 2814	. . . . . 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 2811	. . . . . 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 630	. . . . 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 2814	. . . . . 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 6890	. . . . . . 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 2811	. . . . . 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 630	. . . . 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 31912	. . . . . 6 ⊢ ( ℋ × {0}) ∈ LinFn
15		nmfn0 31914	. . . . . . 7 ⊢ (normfn‘( ℋ × {0})) = 0
16		0re 11254	. . . . . . 7 ⊢ 0 ∈ ℝ
17	15, 16	eqeltri 2822	. . . . . 6 ⊢ (normfn‘( ℋ × {0})) ∈ ℝ
18	14, 17	pm3.2i 469	. . . . 5 ⊢ (( ℋ × {0}) ∈ LinFn ∧ (normfn‘( ℋ × {0})) ∈ ℝ)
19	9, 13, 18	elimhyp 4588	. . . 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 31976	. . 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 4581	. 2 ⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ) → (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))))
22	21	3impia 1114	1 ⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  ifcif 4523  {csn 4623   class class class wbr 5143   × cxp 5670  ‘cfv 6543  (class class class)co 7413  ℝcr 11145  0cc0 11146   · cmul 11151   ≤ cle 11287  abscabs 15231   ℋchba 30846  norm_ℎcno 30850  norm_fncnmf 30878  LinFnclf 30881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7735  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224  ax-hilex 30926  ax-hfvadd 30927  ax-hv0cl 30930  ax-hvaddid 30931  ax-hfvmul 30932  ax-hvmulid 30933  ax-hvmul0 30937  ax-hfi 31006  ax-his1 31009  ax-his3 31011  ax-his4 31012

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6302  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8723  df-map 8846  df-en 8964  df-dom 8965  df-sdom 8966  df-sup 9475  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12256  df-2 12318  df-3 12319  df-n0 12516  df-z 12602  df-uz 12866  df-rp 13020  df-seq 14013  df-exp 14073  df-cj 15096  df-re 15097  df-im 15098  df-sqrt 15232  df-abs 15233  df-hnorm 30895  df-nmfn 31772  df-lnfn 31775

This theorem is referenced by:  lnfncnbd  31984