Theorem nmfnlb 31944

Description: A lower bound for a functional norm. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
nmfnlb	⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝑇‘𝐴)) ≤ (normfn‘𝑇))

Proof of Theorem nmfnlb

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

Step	Hyp	Ref	Expression
1		nmfnsetre 31897	. . . . 5 ⊢ (𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ⊆ ℝ)
2		ressxr 11306	. . . . 5 ⊢ ℝ ⊆ ℝ*
3	1, 2	sstrdi 3995	. . . 4 ⊢ (𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ⊆ ℝ*)
4	3	3ad2ant1 1133	. . 3 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ⊆ ℝ*)
5		fveq2 6905	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (normℎ‘𝑦) = (normℎ‘𝐴))
6	5	breq1d 5152	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘𝐴) ≤ 1))
7		2fveq3 6910	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (abs‘(𝑇‘𝑦)) = (abs‘(𝑇‘𝐴)))
8	7	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦)) ↔ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝐴))))
9	6, 8	anbi12d 632	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦))) ↔ ((normℎ‘𝐴) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝐴)))))
10		eqid 2736	. . . . . . . 8 ⊢ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝐴))
11	10	biantru 529	. . . . . . 7 ⊢ ((normℎ‘𝐴) ≤ 1 ↔ ((normℎ‘𝐴) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝐴))))
12	9, 11	bitr4di 289	. . . . . 6 ⊢ (𝑦 = 𝐴 → (((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦))) ↔ (normℎ‘𝐴) ≤ 1))
13	12	rspcev 3621	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦))))
14		fvex 6918	. . . . . 6 ⊢ (abs‘(𝑇‘𝐴)) ∈ V
15		eqeq1 2740	. . . . . . . 8 ⊢ (𝑥 = (abs‘(𝑇‘𝐴)) → (𝑥 = (abs‘(𝑇‘𝑦)) ↔ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦))))
16	15	anbi2d 630	. . . . . . 7 ⊢ (𝑥 = (abs‘(𝑇‘𝐴)) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦)))))
17	16	rexbidv 3178	. . . . . 6 ⊢ (𝑥 = (abs‘(𝑇‘𝐴)) → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦)))))
18	14, 17	elab 3678	. . . . 5 ⊢ ((abs‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (abs‘(𝑇‘𝐴)) = (abs‘(𝑇‘𝑦))))
19	13, 18	sylibr 234	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))})
20	19	3adant1 1130	. . 3 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))})
21		supxrub 13367	. . 3 ⊢ (({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ⊆ ℝ* ∧ (abs‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}) → (abs‘(𝑇‘𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ))
22	4, 20, 21	syl2anc 584	. 2 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝑇‘𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ))
23		nmfnval 31896	. . 3 ⊢ (𝑇: ℋ⟶ℂ → (normfn‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ))
24	23	3ad2ant1 1133	. 2 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normfn‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < ))
25	22, 24	breqtrrd 5170	1 ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝑇‘𝐴)) ≤ (normfn‘𝑇))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  {cab 2713  ∃wrex 3069   ⊆ wss 3950   class class class wbr 5142  ⟶wf 6556  ‘cfv 6560  supcsup 9481  ℂcc 11154  ℝcr 11155  1c1 11157  ℝ^*cxr 11295   < clt 11296   ≤ cle 11297  abscabs 15274   ℋchba 30939  norm_ℎcno 30943  norm_fncnmf 30971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234  ax-hilex 31019

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-sup 9483  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-rp 13036  df-seq 14044  df-exp 14104  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-nmfn 31865

This theorem is referenced by:  nmfnge0  31947  nmbdfnlbi  32069  nmcfnlbi  32072