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Theorem nmopub 29223
Description: An upper bound for an operator norm. (Contributed by NM, 7-Mar-2006.) (New usage is discouraged.)
Assertion
Ref Expression
nmopub ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑇

Proof of Theorem nmopub
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmopval 29171 . . . 4 (𝑇: ℋ⟶ ℋ → (normop𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ))
21adantr 472 . . 3 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (normop𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ))
32breq1d 4819 . 2 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴))
4 nmopsetretALT 29178 . . . . 5 (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))} ⊆ ℝ)
5 ressxr 10337 . . . . 5 ℝ ⊆ ℝ*
64, 5syl6ss 3773 . . . 4 (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))} ⊆ ℝ*)
7 supxrleub 12358 . . . 4 (({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))} ⊆ ℝ*𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}𝑧𝐴))
86, 7sylan 575 . . 3 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}𝑧𝐴))
9 ancom 452 . . . . . . 7 (((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥))) ↔ (𝑦 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1))
10 eqeq1 2769 . . . . . . . 8 (𝑦 = 𝑧 → (𝑦 = (norm‘(𝑇𝑥)) ↔ 𝑧 = (norm‘(𝑇𝑥))))
1110anbi1d 623 . . . . . . 7 (𝑦 = 𝑧 → ((𝑦 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) ↔ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
129, 11syl5bb 274 . . . . . 6 (𝑦 = 𝑧 → (((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥))) ↔ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
1312rexbidv 3199 . . . . 5 (𝑦 = 𝑧 → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥))) ↔ ∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
1413ralab 3524 . . . 4 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
15 ralcom4 3377 . . . . 5 (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧𝑥 ∈ ℋ ((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
16 impexp 441 . . . . . . . 8 (((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ (𝑧 = (norm‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)))
1716albii 1914 . . . . . . 7 (∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧(𝑧 = (norm‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)))
18 fvex 6388 . . . . . . . 8 (norm‘(𝑇𝑥)) ∈ V
19 breq1 4812 . . . . . . . . 9 (𝑧 = (norm‘(𝑇𝑥)) → (𝑧𝐴 ↔ (norm‘(𝑇𝑥)) ≤ 𝐴))
2019imbi2d 331 . . . . . . . 8 (𝑧 = (norm‘(𝑇𝑥)) → (((norm𝑥) ≤ 1 → 𝑧𝐴) ↔ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))
2118, 20ceqsalv 3386 . . . . . . 7 (∀𝑧(𝑧 = (norm‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)) ↔ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
2217, 21bitri 266 . . . . . 6 (∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
2322ralbii 3127 . . . . 5 (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
24 r19.23v 3170 . . . . . 6 (∀𝑥 ∈ ℋ ((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ (∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2524albii 1914 . . . . 5 (∀𝑧𝑥 ∈ ℋ ((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2615, 23, 253bitr3i 292 . . . 4 (∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2714, 26bitr4i 269 . . 3 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
288, 27syl6bb 278 . 2 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))
293, 28bitrd 270 1 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wcel 2155  {cab 2751  wral 3055  wrex 3056  wss 3732   class class class wbr 4809  wf 6064  cfv 6068  supcsup 8553  cr 10188  1c1 10190  *cxr 10327   < clt 10328  cle 10329  chba 28232  normcno 28236  normopcnop 28258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-hilex 28312  ax-hv0cl 28316  ax-hvmul0 28323  ax-hfi 28392  ax-his1 28395  ax-his3 28397  ax-his4 28398
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-sup 8555  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-seq 13009  df-exp 13068  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-hnorm 28281  df-nmop 29154
This theorem is referenced by:  nmopub2tALT  29224  nmophmi  29346  nmopadjlem  29404  nmoptrii  29409  nmopcoi  29410  nmopcoadji  29416
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