Theorem nmfnleub 29704

Description: An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (Revised by Mario Carneiro, 7-Sep-2014.) (New usage is discouraged.)

Assertion

Ref	Expression
nmfnleub	⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑇

Proof of Theorem nmfnleub

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmfnval 29655	. . . 4 ⊢ (𝑇: ℋ⟶ℂ → (normfn‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))}, ℝ*, < ))
2	1	adantr 483	. . 3 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → (normfn‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))}, ℝ*, < ))
3	2	breq1d 5078	. 2 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn‘𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴))
4		nmfnsetre 29656	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))} ⊆ ℝ)
5		ressxr 10687	. . . . 5 ⊢ ℝ ⊆ ℝ*
6	4, 5	sstrdi 3981	. . . 4 ⊢ (𝑇: ℋ⟶ℂ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))} ⊆ ℝ*)
7		supxrleub 12722	. . . 4 ⊢ (({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))} ⊆ ℝ* ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴))
8	6, 7	sylan 582	. . 3 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴))
9		ancom 463	. . . . . . 7 ⊢ (((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥))) ↔ (𝑦 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1))
10		eqeq1 2827	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑦 = (abs‘(𝑇‘𝑥)) ↔ 𝑧 = (abs‘(𝑇‘𝑥))))
11	10	anbi1d 631	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑦 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) ↔ (𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
12	9, 11	syl5bb 285	. . . . . 6 ⊢ (𝑦 = 𝑧 → (((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥))) ↔ (𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
13	12	rexbidv 3299	. . . . 5 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
14	13	ralab 3686	. . . 4 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴 ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴))
15		ralcom4 3237	. . . . 5 ⊢ (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ∀𝑧∀𝑥 ∈ ℋ ((𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴))
16		impexp 453	. . . . . . . 8 ⊢ (((𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (abs‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
17	16	albii 1820	. . . . . . 7 ⊢ (∀𝑧((𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ∀𝑧(𝑧 = (abs‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
18		fvex 6685	. . . . . . . 8 ⊢ (abs‘(𝑇‘𝑥)) ∈ V
19		breq1 5071	. . . . . . . . 9 ⊢ (𝑧 = (abs‘(𝑇‘𝑥)) → (𝑧 ≤ 𝐴 ↔ (abs‘(𝑇‘𝑥)) ≤ 𝐴))
20	19	imbi2d 343	. . . . . . . 8 ⊢ (𝑧 = (abs‘(𝑇‘𝑥)) → (((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴) ↔ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴)))
21	18, 20	ceqsalv 3534	. . . . . . 7 ⊢ (∀𝑧(𝑧 = (abs‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)) ↔ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴))
22	17, 21	bitri 277	. . . . . 6 ⊢ (∀𝑧((𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴))
23	22	ralbii 3167	. . . . 5 ⊢ (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴))
24		r19.23v 3281	. . . . . 6 ⊢ (∀𝑥 ∈ ℋ ((𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ (∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴))
25	24	albii 1820	. . . . 5 ⊢ (∀𝑧∀𝑥 ∈ ℋ ((𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴))
26	15, 23, 25	3bitr3i 303	. . . 4 ⊢ (∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴))
27	14, 26	bitr4i 280	. . 3 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))}𝑧 ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴))
28	8, 27	syl6bb 289	. 2 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴)))
29	3, 28	bitrd 281	1 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537   ∈ wcel 2114  {cab 2801  ∀wral 3140  ∃wrex 3141   ⊆ wss 3938   class class class wbr 5068  ⟶wf 6353  ‘cfv 6357  supcsup 8906  ℂcc 10537  ℝcr 10538  1c1 10540  ℝ^*cxr 10676   < clt 10677   ≤ cle 10678  abscabs 14595   ℋchba 28698  norm_ℎcno 28702  norm_fncnmf 28730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617  ax-hilex 28778

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  df-sup 8908  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-rp 12393  df-seq 13373  df-exp 13433  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-nmfn 29624

This theorem is referenced by:  nmfnleub2  29705