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Theorem nmoubi 28553
Description: An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoubi.1 𝑋 = (BaseSet‘𝑈)
nmoubi.y 𝑌 = (BaseSet‘𝑊)
nmoubi.l 𝐿 = (normCV𝑈)
nmoubi.m 𝑀 = (normCV𝑊)
nmoubi.3 𝑁 = (𝑈 normOpOLD 𝑊)
nmoubi.u 𝑈 ∈ NrmCVec
nmoubi.w 𝑊 ∈ NrmCVec
Assertion
Ref Expression
nmoubi ((𝑇:𝑋𝑌𝐴 ∈ ℝ*) → ((𝑁𝑇) ≤ 𝐴 ↔ ∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐿   𝑥,𝑈   𝑥,𝑊   𝑥,𝑌   𝑥,𝑀   𝑥,𝑇   𝑥,𝑋
Allowed substitution hint:   𝑁(𝑥)

Proof of Theorem nmoubi
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmoubi.u . . . . . 6 𝑈 ∈ NrmCVec
2 nmoubi.w . . . . . 6 𝑊 ∈ NrmCVec
3 nmoubi.1 . . . . . . 7 𝑋 = (BaseSet‘𝑈)
4 nmoubi.y . . . . . . 7 𝑌 = (BaseSet‘𝑊)
5 nmoubi.l . . . . . . 7 𝐿 = (normCV𝑈)
6 nmoubi.m . . . . . . 7 𝑀 = (normCV𝑊)
7 nmoubi.3 . . . . . . 7 𝑁 = (𝑈 normOpOLD 𝑊)
83, 4, 5, 6, 7nmooval 28544 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝑁𝑇) = sup({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}, ℝ*, < ))
91, 2, 8mp3an12 1448 . . . . 5 (𝑇:𝑋𝑌 → (𝑁𝑇) = sup({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}, ℝ*, < ))
109breq1d 5052 . . . 4 (𝑇:𝑋𝑌 → ((𝑁𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴))
1110adantr 484 . . 3 ((𝑇:𝑋𝑌𝐴 ∈ ℝ*) → ((𝑁𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴))
124, 6nmosetre 28545 . . . . . 6 ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))} ⊆ ℝ)
132, 12mpan 689 . . . . 5 (𝑇:𝑋𝑌 → {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))} ⊆ ℝ)
14 ressxr 10674 . . . . 5 ℝ ⊆ ℝ*
1513, 14sstrdi 3954 . . . 4 (𝑇:𝑋𝑌 → {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))} ⊆ ℝ*)
16 supxrleub 12707 . . . 4 (({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))} ⊆ ℝ*𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}𝑧𝐴))
1715, 16sylan 583 . . 3 ((𝑇:𝑋𝑌𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}𝑧𝐴))
1811, 17bitrd 282 . 2 ((𝑇:𝑋𝑌𝐴 ∈ ℝ*) → ((𝑁𝑇) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}𝑧𝐴))
19 eqeq1 2826 . . . . . 6 (𝑦 = 𝑧 → (𝑦 = (𝑀‘(𝑇𝑥)) ↔ 𝑧 = (𝑀‘(𝑇𝑥))))
2019anbi2d 631 . . . . 5 (𝑦 = 𝑧 → (((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥))) ↔ ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥)))))
2120rexbidv 3283 . . . 4 (𝑦 = 𝑧 → (∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥))) ↔ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥)))))
2221ralab 3659 . . 3 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑧(∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
23 ralcom4 3223 . . . 4 (∀𝑥𝑋𝑧(((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ∀𝑧𝑥𝑋 (((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
24 ancomst 468 . . . . . . . 8 ((((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ((𝑧 = (𝑀‘(𝑇𝑥)) ∧ (𝐿𝑥) ≤ 1) → 𝑧𝐴))
25 impexp 454 . . . . . . . 8 (((𝑧 = (𝑀‘(𝑇𝑥)) ∧ (𝐿𝑥) ≤ 1) → 𝑧𝐴) ↔ (𝑧 = (𝑀‘(𝑇𝑥)) → ((𝐿𝑥) ≤ 1 → 𝑧𝐴)))
2624, 25bitri 278 . . . . . . 7 ((((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ (𝑧 = (𝑀‘(𝑇𝑥)) → ((𝐿𝑥) ≤ 1 → 𝑧𝐴)))
2726albii 1821 . . . . . 6 (∀𝑧(((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ∀𝑧(𝑧 = (𝑀‘(𝑇𝑥)) → ((𝐿𝑥) ≤ 1 → 𝑧𝐴)))
28 fvex 6665 . . . . . . 7 (𝑀‘(𝑇𝑥)) ∈ V
29 breq1 5045 . . . . . . . 8 (𝑧 = (𝑀‘(𝑇𝑥)) → (𝑧𝐴 ↔ (𝑀‘(𝑇𝑥)) ≤ 𝐴))
3029imbi2d 344 . . . . . . 7 (𝑧 = (𝑀‘(𝑇𝑥)) → (((𝐿𝑥) ≤ 1 → 𝑧𝐴) ↔ ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴)))
3128, 30ceqsalv 3507 . . . . . 6 (∀𝑧(𝑧 = (𝑀‘(𝑇𝑥)) → ((𝐿𝑥) ≤ 1 → 𝑧𝐴)) ↔ ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴))
3227, 31bitri 278 . . . . 5 (∀𝑧(((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴))
3332ralbii 3157 . . . 4 (∀𝑥𝑋𝑧(((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴))
34 r19.23v 3265 . . . . 5 (∀𝑥𝑋 (((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ (∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
3534albii 1821 . . . 4 (∀𝑧𝑥𝑋 (((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ∀𝑧(∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
3623, 33, 353bitr3i 304 . . 3 (∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴) ↔ ∀𝑧(∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
3722, 36bitr4i 281 . 2 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴))
3818, 37syl6bb 290 1 ((𝑇:𝑋𝑌𝐴 ∈ ℝ*) → ((𝑁𝑇) ≤ 𝐴 ↔ ∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wcel 2114  {cab 2800  wral 3130  wrex 3131  wss 3908   class class class wbr 5042  wf 6330  cfv 6334  (class class class)co 7140  supcsup 8892  cr 10525  1c1 10527  *cxr 10663   < clt 10664  cle 10665  NrmCVeccnv 28365  BaseSetcba 28367  normCVcnmcv 28371   normOpOLD cnmoo 28522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-po 5451  df-so 5452  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-sup 8894  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-vc 28340  df-nv 28373  df-va 28376  df-ba 28377  df-sm 28378  df-0v 28379  df-nmcv 28381  df-nmoo 28526
This theorem is referenced by:  nmoub3i  28554  nmobndi  28556  ubthlem2  28652
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