Theorem nmfnleub 28168

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 28119	. . . 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 4593	. 2 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn‘𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))}, ℝ*, < ) ≤ 𝐴))
4		nmfnsetre 28120	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))} ⊆ ℝ)
5		ressxr 9962	. . . . 5 ⊢ ℝ ⊆ ℝ*
6	4, 5	syl6ss 3580	. . . 4 ⊢ (𝑇: ℋ⟶ℂ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥)))} ⊆ ℝ*)
7		supxrleub 12028	. . . 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 487	. . 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 465	. . . . . . 7 ⊢ (((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥))) ↔ (𝑦 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1))
10		eqeq1 2614	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑦 = (abs‘(𝑇‘𝑥)) ↔ 𝑧 = (abs‘(𝑇‘𝑥))))
11	10	anbi1d 737	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑦 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) ↔ (𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
12	9, 11	syl5bb 271	. . . . . 6 ⊢ (𝑦 = 𝑧 → (((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥))) ↔ (𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
13	12	rexbidv 3034	. . . . 5 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1)))
14	13	ralab 3334	. . . 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 3197	. . . . 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 461	. . . . . . . 8 ⊢ (((𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ (𝑧 = (abs‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
17	16	albii 1737	. . . . . . 7 ⊢ (∀𝑧((𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ∀𝑧(𝑧 = (abs‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)))
18		fvex 6113	. . . . . . . 8 ⊢ (abs‘(𝑇‘𝑥)) ∈ V
19		breq1 4586	. . . . . . . . 9 ⊢ (𝑧 = (abs‘(𝑇‘𝑥)) → (𝑧 ≤ 𝐴 ↔ (abs‘(𝑇‘𝑥)) ≤ 𝐴))
20	19	imbi2d 329	. . . . . . . 8 ⊢ (𝑧 = (abs‘(𝑇‘𝑥)) → (((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴) ↔ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴)))
21	18, 20	ceqsalv 3206	. . . . . . 7 ⊢ (∀𝑧(𝑧 = (abs‘(𝑇‘𝑥)) → ((normℎ‘𝑥) ≤ 1 → 𝑧 ≤ 𝐴)) ↔ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴))
22	17, 21	bitri 263	. . . . . 6 ⊢ (∀𝑧((𝑧 = (abs‘(𝑇‘𝑥)) ∧ (normℎ‘𝑥) ≤ 1) → 𝑧 ≤ 𝐴) ↔ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴))
23	22	ralbii 2963	. . . . 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 3005	. . . . . 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 1737	. . . . 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 289	. . . 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 266	. . 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 275	. 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 267	1 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   class class class wbr 4583  ⟶wf 5800  ‘cfv 5804  supcsup 8229  ℂcc 9813  ℝcr 9814  1c1 9816  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954  abscabs 13822   ℋchil 27160  norm_ℎcno 27164  norm_fncnmf 27192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-hilex 27240

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-nmfn 28088

This theorem is referenced by:  nmfnleub2  28169