Theorem nmfnleub2 29702

Description: An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
nmfnleub2	⊢ ((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥))) → (normfn‘𝑇) ≤ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑇

Proof of Theorem nmfnleub2

Step	Hyp	Ref	Expression
1		normcl 28901	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
2	1	ad2antlr 725	. . . . . . . . . 10 ⊢ ((((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘𝑥) ∈ ℝ)
3		simpllr 774	. . . . . . . . . 10 ⊢ ((((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
4		simpr 487	. . . . . . . . . 10 ⊢ ((((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘𝑥) ≤ 1)
5		1re 10640	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
6		lemul2a 11494	. . . . . . . . . . 11 ⊢ ((((normℎ‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ (normℎ‘𝑥) ≤ 1) → (𝐴 · (normℎ‘𝑥)) ≤ (𝐴 · 1))
7	5, 6	mp3anl2 1452	. . . . . . . . . 10 ⊢ ((((normℎ‘𝑥) ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ (normℎ‘𝑥) ≤ 1) → (𝐴 · (normℎ‘𝑥)) ≤ (𝐴 · 1))
8	2, 3, 4, 7	syl21anc 835	. . . . . . . . 9 ⊢ ((((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (𝐴 · (normℎ‘𝑥)) ≤ (𝐴 · 1))
9		ax-1rid 10606	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
10	9	ad2antrl 726	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → (𝐴 · 1) = 𝐴)
11	10	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (𝐴 · 1) = 𝐴)
12	8, 11	breqtrd 5091	. . . . . . . 8 ⊢ ((((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (𝐴 · (normℎ‘𝑥)) ≤ 𝐴)
13		ffvelrn 6848	. . . . . . . . . . . 12 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℂ)
14	13	abscld 14795	. . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝑥 ∈ ℋ) → (abs‘(𝑇‘𝑥)) ∈ ℝ)
15	14	adantlr 713	. . . . . . . . . 10 ⊢ (((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ 𝑥 ∈ ℋ) → (abs‘(𝑇‘𝑥)) ∈ ℝ)
16		remulcl 10621	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ (normℎ‘𝑥) ∈ ℝ) → (𝐴 · (normℎ‘𝑥)) ∈ ℝ)
17	1, 16	sylan2 594	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℋ) → (𝐴 · (normℎ‘𝑥)) ∈ ℝ)
18	17	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑥 ∈ ℋ) → (𝐴 · (normℎ‘𝑥)) ∈ ℝ)
19	18	adantll 712	. . . . . . . . . 10 ⊢ (((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ 𝑥 ∈ ℋ) → (𝐴 · (normℎ‘𝑥)) ∈ ℝ)
20		simplrl 775	. . . . . . . . . 10 ⊢ (((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ 𝑥 ∈ ℋ) → 𝐴 ∈ ℝ)
21		letr 10733	. . . . . . . . . 10 ⊢ (((abs‘(𝑇‘𝑥)) ∈ ℝ ∧ (𝐴 · (normℎ‘𝑥)) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (((abs‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥)) ∧ (𝐴 · (normℎ‘𝑥)) ≤ 𝐴) → (abs‘(𝑇‘𝑥)) ≤ 𝐴))
22	15, 19, 20, 21	syl3anc 1367	. . . . . . . . 9 ⊢ (((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ 𝑥 ∈ ℋ) → (((abs‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥)) ∧ (𝐴 · (normℎ‘𝑥)) ≤ 𝐴) → (abs‘(𝑇‘𝑥)) ≤ 𝐴))
23	22	adantr 483	. . . . . . . 8 ⊢ ((((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (((abs‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥)) ∧ (𝐴 · (normℎ‘𝑥)) ≤ 𝐴) → (abs‘(𝑇‘𝑥)) ≤ 𝐴))
24	12, 23	mpan2d 692	. . . . . . 7 ⊢ ((((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → ((abs‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥)) → (abs‘(𝑇‘𝑥)) ≤ 𝐴))
25	24	ex 415	. . . . . 6 ⊢ (((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ 𝑥 ∈ ℋ) → ((normℎ‘𝑥) ≤ 1 → ((abs‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥)) → (abs‘(𝑇‘𝑥)) ≤ 𝐴)))
26	25	com23 86	. . . . 5 ⊢ (((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ 𝑥 ∈ ℋ) → ((abs‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴)))
27	26	ralimdva 3177	. . . 4 ⊢ ((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → (∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥)) → ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴)))
28	27	imp 409	. . 3 ⊢ (((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ ∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥))) → ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴))
29		rexr 10686	. . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*)
30	29	adantr 483	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ*)
31		nmfnleub 29701	. . . . 5 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴)))
32	30, 31	sylan2 594	. . . 4 ⊢ ((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → ((normfn‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴)))
33	32	biimpar 480	. . 3 ⊢ (((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴)) → (normfn‘𝑇) ≤ 𝐴)
34	28, 33	syldan 593	. 2 ⊢ (((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ ∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥))) → (normfn‘𝑇) ≤ 𝐴)
35	34	3impa 1106	1 ⊢ ((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ ℋ (abs‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥))) → (normfn‘𝑇) ≤ 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138   class class class wbr 5065  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155  ℂcc 10534  ℝcr 10535  0cc0 10536  1c1 10537   · cmul 10541  ℝ^*cxr 10673   ≤ cle 10675  abscabs 14592   ℋchba 28695  norm_ℎcno 28699  norm_fncnmf 28727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614  ax-hilex 28775  ax-hv0cl 28779  ax-hvmul0 28786  ax-hfi 28855  ax-his1 28858  ax-his3 28860  ax-his4 28861

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-sup 8905  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-rp 12389  df-seq 13369  df-exp 13429  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-hnorm 28744  df-nmfn 29621

This theorem is referenced by: (None)