Theorem nmfnleub 32004

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 31955	. . . 4 ⊢ (𝑇: ℋ⟶ℂ → (normfn‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))}, ℝ*, < ))
2	1	adantr 480	. . 3 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → (normfn‘𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))}, ℝ*, < ))
3	2	breq1d 5109	. 2 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn‘𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴))
4		nmfnsetre 31956	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))} ⊆ ℝ)
5		ressxr 11180	. . . . 5 ⊢ ℝ ⊆ ℝ*
6	4, 5	sstrdi 3947	. . . 4 ⊢ (𝑇: ℋ⟶ℂ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))} ⊆ ℝ*)
7		supxrleub 13245	. . . 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 581	. . 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 460	. . . . . . 7 ⊢ (((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥))) ↔ (𝑦 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1))
10		eqeq1 2741	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑦 = (abs‘(𝑇‘𝑥)) ↔ 𝑧 = (abs‘(𝑇‘𝑥))))
11	10	anbi1d 632	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑦 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) ↔ (𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
12	9, 11	bitrid 283	. . . . . 6 ⊢ (𝑦 = 𝑧 → (((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥))) ↔ (𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
13	12	rexbidv 3161	. . . . 5 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
14	13	ralab 3652	. . . 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 3263	. . . . 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 450	. . . . . . . 8 ⊢ (((𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (abs‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
17	16	albii 1821	. . . . . . 7 ⊢ (∀𝑧((𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ∀𝑧(𝑧 = (abs‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
18		fvex 6848	. . . . . . . 8 ⊢ (abs‘(𝑇‘𝑥)) ∈ V
19		breq1 5102	. . . . . . . . 9 ⊢ (𝑧 = (abs‘(𝑇‘𝑥)) → (𝑧 ≤ 𝐴 ↔ (abs‘(𝑇‘𝑥)) ≤ 𝐴))
20	19	imbi2d 340	. . . . . . . 8 ⊢ (𝑧 = (abs‘(𝑇‘𝑥)) → (((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴) ↔ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴)))
21	18, 20	ceqsalv 3481	. . . . . . 7 ⊢ (∀𝑧(𝑧 = (abs‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)) ↔ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴))
22	17, 21	bitri 275	. . . . . 6 ⊢ (∀𝑧((𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴))
23	22	ralbii 3083	. . . . 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 3164	. . . . . 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 1821	. . . . 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 301	. . . 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 278	. . 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	bitrdi 287	. 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 279	1 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3061   ⊆ wss 3902   class class class wbr 5099  ⟶wf 6489  ‘cfv 6493  supcsup 9347  ℂcc 11028  ℝcr 11029  1c1 11031  ℝ^*cxr 11169   < clt 11170   ≤ cle 11171  abscabs 15161   ℋchba 30998  norm_ℎcno 31002  norm_fncnmf 31030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108  ax-hilex 31078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-sup 9349  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12150  df-2 12212  df-3 12213  df-n0 12406  df-z 12493  df-uz 12756  df-rp 12910  df-seq 13929  df-exp 13989  df-cj 15026  df-re 15027  df-im 15028  df-sqrt 15162  df-abs 15163  df-nmfn 31924

This theorem is referenced by:  nmfnleub2  32005