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Theorem nmoubi 28476
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 28467 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝑁𝑇) = sup({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}, ℝ*, < ))
91, 2, 8mp3an12 1442 . . . . 5 (𝑇:𝑋𝑌 → (𝑁𝑇) = sup({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}, ℝ*, < ))
109breq1d 5067 . . . 4 (𝑇:𝑋𝑌 → ((𝑁𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴))
1110adantr 481 . . 3 ((𝑇:𝑋𝑌𝐴 ∈ ℝ*) → ((𝑁𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴))
124, 6nmosetre 28468 . . . . . 6 ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))} ⊆ ℝ)
132, 12mpan 686 . . . . 5 (𝑇:𝑋𝑌 → {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))} ⊆ ℝ)
14 ressxr 10673 . . . . 5 ℝ ⊆ ℝ*
1513, 14sstrdi 3976 . . . 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 580 . . 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 280 . 2 ((𝑇:𝑋𝑌𝐴 ∈ ℝ*) → ((𝑁𝑇) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}𝑧𝐴))
19 eqeq1 2822 . . . . . 6 (𝑦 = 𝑧 → (𝑦 = (𝑀‘(𝑇𝑥)) ↔ 𝑧 = (𝑀‘(𝑇𝑥))))
2019anbi2d 628 . . . . 5 (𝑦 = 𝑧 → (((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥))) ↔ ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥)))))
2120rexbidv 3294 . . . 4 (𝑦 = 𝑧 → (∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥))) ↔ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥)))))
2221ralab 3681 . . 3 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑧(∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
23 ralcom4 3232 . . . 4 (∀𝑥𝑋𝑧(((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ∀𝑧𝑥𝑋 (((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
24 ancomst 465 . . . . . . . 8 ((((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ((𝑧 = (𝑀‘(𝑇𝑥)) ∧ (𝐿𝑥) ≤ 1) → 𝑧𝐴))
25 impexp 451 . . . . . . . 8 (((𝑧 = (𝑀‘(𝑇𝑥)) ∧ (𝐿𝑥) ≤ 1) → 𝑧𝐴) ↔ (𝑧 = (𝑀‘(𝑇𝑥)) → ((𝐿𝑥) ≤ 1 → 𝑧𝐴)))
2624, 25bitri 276 . . . . . . 7 ((((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ (𝑧 = (𝑀‘(𝑇𝑥)) → ((𝐿𝑥) ≤ 1 → 𝑧𝐴)))
2726albii 1811 . . . . . 6 (∀𝑧(((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ∀𝑧(𝑧 = (𝑀‘(𝑇𝑥)) → ((𝐿𝑥) ≤ 1 → 𝑧𝐴)))
28 fvex 6676 . . . . . . 7 (𝑀‘(𝑇𝑥)) ∈ V
29 breq1 5060 . . . . . . . 8 (𝑧 = (𝑀‘(𝑇𝑥)) → (𝑧𝐴 ↔ (𝑀‘(𝑇𝑥)) ≤ 𝐴))
3029imbi2d 342 . . . . . . 7 (𝑧 = (𝑀‘(𝑇𝑥)) → (((𝐿𝑥) ≤ 1 → 𝑧𝐴) ↔ ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴)))
3128, 30ceqsalv 3530 . . . . . 6 (∀𝑧(𝑧 = (𝑀‘(𝑇𝑥)) → ((𝐿𝑥) ≤ 1 → 𝑧𝐴)) ↔ ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴))
3227, 31bitri 276 . . . . 5 (∀𝑧(((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴))
3332ralbii 3162 . . . 4 (∀𝑥𝑋𝑧(((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴))
34 r19.23v 3276 . . . . 5 (∀𝑥𝑋 (((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ (∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
3534albii 1811 . . . 4 (∀𝑧𝑥𝑋 (((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ∀𝑧(∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
3623, 33, 353bitr3i 302 . . 3 (∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴) ↔ ∀𝑧(∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
3722, 36bitr4i 279 . 2 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴))
3818, 37syl6bb 288 1 ((𝑇:𝑋𝑌𝐴 ∈ ℝ*) → ((𝑁𝑇) ≤ 𝐴 ↔ ∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wcel 2105  {cab 2796  wral 3135  wrex 3136  wss 3933   class class class wbr 5057  wf 6344  cfv 6348  (class class class)co 7145  supcsup 8892  cr 10524  1c1 10526  *cxr 10662   < clt 10663  cle 10664  NrmCVeccnv 28288  BaseSetcba 28290  normCVcnmcv 28294   normOpOLD cnmoo 28445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-vc 28263  df-nv 28296  df-va 28299  df-ba 28300  df-sm 28301  df-0v 28302  df-nmcv 28304  df-nmoo 28449
This theorem is referenced by:  nmoub3i  28477  nmobndi  28479  ubthlem2  28575
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