Theorem nmoplb 31428

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 31385	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ)
2		ressxr 11263	. . . . 5 ⊢ ℝ ⊆ ℝ*
3	1, 2	sstrdi 3994	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*)
4	3	3ad2ant1 1132	. . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*)
5		fveq2 6891	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (normℎ‘𝑦) = (normℎ‘𝐴))
6	5	breq1d 5158	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘𝐴) ≤ 1))
7		2fveq3 6896	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (normℎ‘(𝑇‘𝑦)) = (normℎ‘(𝑇‘𝐴)))
8	7	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦)) ↔ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝐴))))
9	6, 8	anbi12d 630	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝐴) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝐴)))))
10		eqid 2731	. . . . . . . 8 ⊢ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝐴))
11	10	biantru 529	. . . . . . 7 ⊢ ((normℎ‘𝐴) ≤ 1 ↔ ((normℎ‘𝐴) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝐴))))
12	9, 11	bitr4di 289	. . . . . 6 ⊢ (𝑦 = 𝐴 → (((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))) ↔ (normℎ‘𝐴) ≤ 1))
13	12	rspcev 3612	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))))
14		fvex 6904	. . . . . 6 ⊢ (normℎ‘(𝑇‘𝐴)) ∈ V
15		eqeq1 2735	. . . . . . . 8 ⊢ (𝑥 = (normℎ‘(𝑇‘𝐴)) → (𝑥 = (normℎ‘(𝑇‘𝑦)) ↔ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))))
16	15	anbi2d 628	. . . . . . 7 ⊢ (𝑥 = (normℎ‘(𝑇‘𝐴)) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦)))))
17	16	rexbidv 3177	. . . . . 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)))))
18	14, 17	elab 3668	. . . . 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))))
19	13, 18	sylibr 233	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))})
20	19	3adant1 1129	. . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))})
21		supxrub 13308	. . 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*, < ))
22	4, 20, 21	syl2anc 583	. 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
23		nmopval 31377	. . 3 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
24	23	3ad2ant1 1132	. 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
25	22, 24	breqtrrd 5176	1 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ≤ (normop‘𝑇))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  {cab 2708  ∃wrex 3069   ⊆ wss 3948   class class class wbr 5148  ⟶wf 6539  ‘cfv 6543  supcsup 9439  ℝcr 11113  1c1 11115  ℝ^*cxr 11252   < clt 11253   ≤ cle 11254   ℋchba 30440  norm_ℎcno 30444  norm_opcnop 30466

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192  ax-hilex 30520  ax-hfvadd 30521  ax-hvcom 30522  ax-hvass 30523  ax-hv0cl 30524  ax-hvaddid 30525  ax-hfvmul 30526  ax-hvmulid 30527  ax-hvmulass 30528  ax-hvdistr1 30529  ax-hvdistr2 30530  ax-hvmul0 30531  ax-hfi 30600  ax-his1 30603  ax-his2 30604  ax-his3 30605  ax-his4 30606

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-sup 9441  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-n0 12478  df-z 12564  df-uz 12828  df-rp 12980  df-seq 13972  df-exp 14033  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-grpo 30014  df-gid 30015  df-ablo 30066  df-vc 30080  df-nv 30113  df-va 30116  df-ba 30117  df-sm 30118  df-0v 30119  df-nmcv 30121  df-hnorm 30489  df-hba 30490  df-hvsub 30492  df-nmop 31360

This theorem is referenced by:  nmopge0  31432  nmbdoplbi  31545  nmcoplbi  31549  nmophmi  31552  nmoptrii  31615  nmopcoi  31616