Theorem nmbdfnlb 29825

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 6662	. . . . . 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 6667	. . . . 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 6663	. . . . . 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 7164	. . . . 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 5072	. . . 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 343	. . 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 2899	. . . . . 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 2896	. . . . . 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 632	. . . . 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 2899	. . . . . 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 6663	. . . . . . 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 2896	. . . . . 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 632	. . . . 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 29760	. . . . . 6 ⊢ ( ℋ × {0}) ∈ LinFn
15		nmfn0 29762	. . . . . . 7 ⊢ (normfn‘( ℋ × {0})) = 0
16		0re 10636	. . . . . . 7 ⊢ 0 ∈ ℝ
17	15, 16	eqeltri 2908	. . . . . 6 ⊢ (normfn‘( ℋ × {0})) ∈ ℝ
18	14, 17	pm3.2i 473	. . . . 5 ⊢ (( ℋ × {0}) ∈ LinFn ∧ (normfn‘( ℋ × {0})) ∈ ℝ)
19	9, 13, 18	elimhyp 4523	. . . 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 29824	. . 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 4516	. 2 ⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ) → (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))))
22	21	3impia 1112	1 ⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1082   = wceq 1536   ∈ wcel 2113  ifcif 4460  {csn 4560   class class class wbr 5059   × cxp 5546  ‘cfv 6348  (class class class)co 7149  ℝcr 10529  0cc0 10530   · cmul 10535   ≤ cle 10669  abscabs 14588   ℋchba 28694  norm_ℎcno 28698  norm_fncnmf 28726  LinFnclf 28729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-hilex 28774  ax-hfvadd 28775  ax-hv0cl 28778  ax-hvaddid 28779  ax-hfvmul 28780  ax-hvmulid 28781  ax-hvmul0 28785  ax-hfi 28854  ax-his1 28857  ax-his3 28859  ax-his4 28860

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-nel 3123  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7574  df-2nd 7683  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-er 8282  df-map 8401  df-en 8503  df-dom 8504  df-sdom 8505  df-sup 8899  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11632  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12384  df-seq 13367  df-exp 13427  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-hnorm 28743  df-nmfn 29620  df-lnfn 29623

This theorem is referenced by:  lnfncnbd  29832