Theorem nmoplb 28766

Description: A lower bound for an operator norm. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
nmoplb	⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ≤ (normop‘𝑇))

Proof of Theorem nmoplb

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

Step	Hyp	Ref	Expression
1		nmopsetretHIL 28723	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ)
2		ressxr 10083	. . . . 5 ⊢ ℝ ⊆ ℝ*
3	1, 2	syl6ss 3615	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*)
4	3	3ad2ant1 1082	. . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*)
5		fveq2 6191	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (normℎ‘𝑦) = (normℎ‘𝐴))
6	5	breq1d 4663	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘𝐴) ≤ 1))
7		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑇‘𝑦) = (𝑇‘𝐴))
8	7	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (normℎ‘(𝑇‘𝑦)) = (normℎ‘(𝑇‘𝐴)))
9	8	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦)) ↔ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝐴))))
10	6, 9	anbi12d 747	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝐴) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝐴)))))
11		eqid 2622	. . . . . . . 8 ⊢ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝐴))
12	11	biantru 526	. . . . . . 7 ⊢ ((normℎ‘𝐴) ≤ 1 ↔ ((normℎ‘𝐴) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝐴))))
13	10, 12	syl6bbr 278	. . . . . 6 ⊢ (𝑦 = 𝐴 → (((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))) ↔ (normℎ‘𝐴) ≤ 1))
14	13	rspcev 3309	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))))
15		fvex 6201	. . . . . 6 ⊢ (normℎ‘(𝑇‘𝐴)) ∈ V
16		eqeq1 2626	. . . . . . . 8 ⊢ (𝑥 = (normℎ‘(𝑇‘𝐴)) → (𝑥 = (normℎ‘(𝑇‘𝑦)) ↔ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))))
17	16	anbi2d 740	. . . . . . 7 ⊢ (𝑥 = (normℎ‘(𝑇‘𝐴)) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦)))))
18	17	rexbidv 3052	. . . . . 6 ⊢ (𝑥 = (normℎ‘(𝑇‘𝐴)) → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦)))))
19	15, 18	elab 3350	. . . . 5 ⊢ ((normℎ‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))))
20	14, 19	sylibr 224	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))})
21	20	3adant1 1079	. . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))})
22		supxrub 12154	. . 3 ⊢ (({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ* ∧ (normℎ‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}) → (normℎ‘(𝑇‘𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
23	4, 21, 22	syl2anc 693	. 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
24		nmopval 28715	. . 3 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
25	24	3ad2ant1 1082	. 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
26	23, 25	breqtrrd 4681	1 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ≤ (normop‘𝑇))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608  ∃wrex 2913   ⊆ wss 3574   class class class wbr 4653  ⟶wf 5884  ‘cfv 5888  supcsup 8346  ℝcr 9935  1c1 9937  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075   ℋchil 27776  norm_ℎcno 27780  norm_opcnop 27802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-grpo 27347  df-gid 27348  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-hnorm 27825  df-hba 27826  df-hvsub 27828  df-nmop 28698

This theorem is referenced by:  nmopge0  28770  nmbdoplbi  28883  nmcoplbi  28887  nmophmi  28890  nmoptrii  28953  nmopcoi  28954