Theorem nmoplb 29375

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 29332	. . . . 5 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ)
2		ressxr 10531	. . . . 5 ⊢ ℝ ⊆ ℝ*
3	1, 2	syl6ss 3901	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*)
4	3	3ad2ant1 1126	. . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ*)
5		fveq2 6538	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (normℎ‘𝑦) = (normℎ‘𝐴))
6	5	breq1d 4972	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘𝐴) ≤ 1))
7		2fveq3 6543	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (normℎ‘(𝑇‘𝑦)) = (normℎ‘(𝑇‘𝐴)))
8	7	eqeq2d 2805	. . . . . . . 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 2795	. . . . . . . 8 ⊢ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝐴))
11	10	biantru 530	. . . . . . 7 ⊢ ((normℎ‘𝐴) ≤ 1 ↔ ((normℎ‘𝐴) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝐴))))
12	9, 11	syl6bbr 290	. . . . . 6 ⊢ (𝑦 = 𝐴 → (((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))) ↔ (normℎ‘𝐴) ≤ 1))
13	12	rspcev 3559	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘𝑦))))
14		fvex 6551	. . . . . 6 ⊢ (normℎ‘(𝑇‘𝐴)) ∈ V
15		eqeq1 2799	. . . . . . . 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 3260	. . . . . 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 3605	. . . . 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 235	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))})
20	19	3adant1 1123	. . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))})
21		supxrub 12567	. . 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 584	. 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 29324	. . 3 ⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < ))
24	23	3ad2ant1 1126	. 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 4990	1 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ≤ (normop‘𝑇))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  {cab 2775  ∃wrex 3106   ⊆ wss 3859   class class class wbr 4962  ⟶wf 6221  ‘cfv 6225  supcsup 8750  ℝcr 10382  1c1 10384  ℝ^*cxr 10520   < clt 10521   ≤ cle 10522   ℋchba 28387  norm_ℎcno 28391  norm_opcnop 28413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461  ax-hilex 28467  ax-hfvadd 28468  ax-hvcom 28469  ax-hvass 28470  ax-hv0cl 28471  ax-hvaddid 28472  ax-hfvmul 28473  ax-hvmulid 28474  ax-hvmulass 28475  ax-hvdistr1 28476  ax-hvdistr2 28477  ax-hvmul0 28478  ax-hfi 28547  ax-his1 28550  ax-his2 28551  ax-his3 28552  ax-his4 28553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-sdom 8360  df-sup 8752  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-n0 11746  df-z 11830  df-uz 12094  df-rp 12240  df-seq 13220  df-exp 13280  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-grpo 27961  df-gid 27962  df-ablo 28013  df-vc 28027  df-nv 28060  df-va 28063  df-ba 28064  df-sm 28065  df-0v 28066  df-nmcv 28068  df-hnorm 28436  df-hba 28437  df-hvsub 28439  df-nmop 29307

This theorem is referenced by:  nmopge0  29379  nmbdoplbi  29492  nmcoplbi  29496  nmophmi  29499  nmoptrii  29562  nmopcoi  29563