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Theorem nmopub 31711
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 31659 . . . 4 (𝑇: ℋ⟶ ℋ → (normop𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ))
21adantr 480 . . 3 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (normop𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ))
32breq1d 5152 . 2 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴))
4 nmopsetretALT 31666 . . . . 5 (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))} ⊆ ℝ)
5 ressxr 11282 . . . . 5 ℝ ⊆ ℝ*
64, 5sstrdi 3990 . . . 4 (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))} ⊆ ℝ*)
7 supxrleub 13331 . . . 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 579 . . 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 460 . . . . . . 7 (((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥))) ↔ (𝑦 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1))
10 eqeq1 2731 . . . . . . . 8 (𝑦 = 𝑧 → (𝑦 = (norm‘(𝑇𝑥)) ↔ 𝑧 = (norm‘(𝑇𝑥))))
1110anbi1d 629 . . . . . . 7 (𝑦 = 𝑧 → ((𝑦 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) ↔ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
129, 11bitrid 283 . . . . . 6 (𝑦 = 𝑧 → (((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥))) ↔ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
1312rexbidv 3173 . . . . 5 (𝑦 = 𝑧 → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥))) ↔ ∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
1413ralab 3684 . . . 4 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
15 ralcom4 3278 . . . . 5 (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧𝑥 ∈ ℋ ((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
16 impexp 450 . . . . . . . 8 (((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ (𝑧 = (norm‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)))
1716albii 1814 . . . . . . 7 (∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧(𝑧 = (norm‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)))
18 fvex 6904 . . . . . . . 8 (norm‘(𝑇𝑥)) ∈ V
19 breq1 5145 . . . . . . . . 9 (𝑧 = (norm‘(𝑇𝑥)) → (𝑧𝐴 ↔ (norm‘(𝑇𝑥)) ≤ 𝐴))
2019imbi2d 340 . . . . . . . 8 (𝑧 = (norm‘(𝑇𝑥)) → (((norm𝑥) ≤ 1 → 𝑧𝐴) ↔ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))
2118, 20ceqsalv 3507 . . . . . . 7 (∀𝑧(𝑧 = (norm‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)) ↔ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
2217, 21bitri 275 . . . . . 6 (∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
2322ralbii 3088 . . . . 5 (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
24 r19.23v 3177 . . . . . 6 (∀𝑥 ∈ ℋ ((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ (∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2524albii 1814 . . . . 5 (∀𝑧𝑥 ∈ ℋ ((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2615, 23, 253bitr3i 301 . . . 4 (∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2714, 26bitr4i 278 . . 3 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
288, 27bitrdi 287 . 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 279 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 205  wa 395  wal 1532   = wceq 1534  wcel 2099  {cab 2704  wral 3056  wrex 3065  wss 3944   class class class wbr 5142  wf 6538  cfv 6542  supcsup 9457  cr 11131  1c1 11133  *cxr 11271   < clt 11272  cle 11273  chba 30722  normcno 30726  normopcnop 30748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209  ax-pre-sup 11210  ax-hilex 30802  ax-hv0cl 30806  ax-hvmul0 30813  ax-hfi 30882  ax-his1 30885  ax-his3 30887  ax-his4 30888
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-sup 9459  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-div 11896  df-nn 12237  df-2 12299  df-3 12300  df-n0 12497  df-z 12583  df-uz 12847  df-rp 13001  df-seq 13993  df-exp 14053  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-hnorm 30771  df-nmop 31642
This theorem is referenced by:  nmopub2tALT  31712  nmophmi  31834  nmopadjlem  31892  nmoptrii  31897  nmopcoi  31898  nmopcoadji  31904
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