Theorem nmbdfnlb 31280

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 6886	. . . . . 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 6891	. . . . 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 6887	. . . . . 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 7418	. . . . 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 5159	. . . 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 341	. . 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 2822	. . . . . 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 2819	. . . . . 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 2822	. . . . . 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 6887	. . . . . . 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 2819	. . . . . 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 31215	. . . . . 6 ⊢ ( ℋ × {0}) ∈ LinFn
15		nmfn0 31217	. . . . . . 7 ⊢ (normfn‘( ℋ × {0})) = 0
16		0re 11211	. . . . . . 7 ⊢ 0 ∈ ℝ
17	15, 16	eqeltri 2830	. . . . . 6 ⊢ (normfn‘( ℋ × {0})) ∈ ℝ
18	14, 17	pm3.2i 472	. . . . 5 ⊢ (( ℋ × {0}) ∈ LinFn ∧ (normfn‘( ℋ × {0})) ∈ ℝ)
19	9, 13, 18	elimhyp 4591	. . . 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 31279	. . 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 4584	. 2 ⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ) → (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))))
22	21	3impia 1118	1 ⊢ ((𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ifcif 4526  {csn 4626   class class class wbr 5146   × cxp 5672  ‘cfv 6539  (class class class)co 7403  ℝcr 11104  0cc0 11105   · cmul 11110   ≤ cle 11244  abscabs 15176   ℋchba 30149  norm_ℎcno 30153  norm_fncnmf 30181  LinFnclf 30184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-pre-sup 11183  ax-hilex 30229  ax-hfvadd 30230  ax-hv0cl 30233  ax-hvaddid 30234  ax-hfvmul 30235  ax-hvmulid 30236  ax-hvmul0 30240  ax-hfi 30309  ax-his1 30312  ax-his3 30314  ax-his4 30315

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6296  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7359  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7850  df-2nd 7970  df-frecs 8260  df-wrecs 8291  df-recs 8365  df-rdg 8404  df-er 8698  df-map 8817  df-en 8935  df-dom 8936  df-sdom 8937  df-sup 9432  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11441  df-neg 11442  df-div 11867  df-nn 12208  df-2 12270  df-3 12271  df-n0 12468  df-z 12554  df-uz 12818  df-rp 12970  df-seq 13962  df-exp 14023  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-hnorm 30198  df-nmfn 31075  df-lnfn 31078

This theorem is referenced by:  lnfncnbd  31287