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Theorem nmoubi 27011
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 27002 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝑁𝑇) = sup({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}, ℝ*, < ))
91, 2, 8mp3an12 1406 . . . . 5 (𝑇:𝑋𝑌 → (𝑁𝑇) = sup({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}, ℝ*, < ))
109breq1d 4593 . . . 4 (𝑇:𝑋𝑌 → ((𝑁𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴))
1110adantr 480 . . 3 ((𝑇:𝑋𝑌𝐴 ∈ ℝ*) → ((𝑁𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴))
124, 6nmosetre 27003 . . . . . 6 ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))} ⊆ ℝ)
132, 12mpan 702 . . . . 5 (𝑇:𝑋𝑌 → {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))} ⊆ ℝ)
14 ressxr 9962 . . . . 5 ℝ ⊆ ℝ*
1513, 14syl6ss 3580 . . . 4 (𝑇:𝑋𝑌 → {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))} ⊆ ℝ*)
16 supxrleub 12028 . . . 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 487 . . 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 267 . 2 ((𝑇:𝑋𝑌𝐴 ∈ ℝ*) → ((𝑁𝑇) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}𝑧𝐴))
19 eqeq1 2614 . . . . . 6 (𝑦 = 𝑧 → (𝑦 = (𝑀‘(𝑇𝑥)) ↔ 𝑧 = (𝑀‘(𝑇𝑥))))
2019anbi2d 736 . . . . 5 (𝑦 = 𝑧 → (((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥))) ↔ ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥)))))
2120rexbidv 3034 . . . 4 (𝑦 = 𝑧 → (∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥))) ↔ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥)))))
2221ralab 3334 . . 3 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑧(∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
23 ralcom4 3197 . . . 4 (∀𝑥𝑋𝑧(((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ∀𝑧𝑥𝑋 (((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
24 ancomst 467 . . . . . . . 8 ((((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ((𝑧 = (𝑀‘(𝑇𝑥)) ∧ (𝐿𝑥) ≤ 1) → 𝑧𝐴))
25 impexp 461 . . . . . . . 8 (((𝑧 = (𝑀‘(𝑇𝑥)) ∧ (𝐿𝑥) ≤ 1) → 𝑧𝐴) ↔ (𝑧 = (𝑀‘(𝑇𝑥)) → ((𝐿𝑥) ≤ 1 → 𝑧𝐴)))
2624, 25bitri 263 . . . . . . 7 ((((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ (𝑧 = (𝑀‘(𝑇𝑥)) → ((𝐿𝑥) ≤ 1 → 𝑧𝐴)))
2726albii 1737 . . . . . 6 (∀𝑧(((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ∀𝑧(𝑧 = (𝑀‘(𝑇𝑥)) → ((𝐿𝑥) ≤ 1 → 𝑧𝐴)))
28 fvex 6113 . . . . . . 7 (𝑀‘(𝑇𝑥)) ∈ V
29 breq1 4586 . . . . . . . 8 (𝑧 = (𝑀‘(𝑇𝑥)) → (𝑧𝐴 ↔ (𝑀‘(𝑇𝑥)) ≤ 𝐴))
3029imbi2d 329 . . . . . . 7 (𝑧 = (𝑀‘(𝑇𝑥)) → (((𝐿𝑥) ≤ 1 → 𝑧𝐴) ↔ ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴)))
3128, 30ceqsalv 3206 . . . . . 6 (∀𝑧(𝑧 = (𝑀‘(𝑇𝑥)) → ((𝐿𝑥) ≤ 1 → 𝑧𝐴)) ↔ ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴))
3227, 31bitri 263 . . . . 5 (∀𝑧(((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴))
3332ralbii 2963 . . . 4 (∀𝑥𝑋𝑧(((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴))
34 r19.23v 3005 . . . . 5 (∀𝑥𝑋 (((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ (∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
3534albii 1737 . . . 4 (∀𝑧𝑥𝑋 (((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴) ↔ ∀𝑧(∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
3623, 33, 353bitr3i 289 . . 3 (∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴) ↔ ∀𝑧(∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑧 = (𝑀‘(𝑇𝑥))) → 𝑧𝐴))
3722, 36bitr4i 266 . 2 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥𝑋 ((𝐿𝑥) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴))
3818, 37syl6bb 275 1 ((𝑇:𝑋𝑌𝐴 ∈ ℝ*) → ((𝑁𝑇) ≤ 𝐴 ↔ ∀𝑥𝑋 ((𝐿𝑥) ≤ 1 → (𝑀‘(𝑇𝑥)) ≤ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wal 1473   = wceq 1475  wcel 1977  {cab 2596  wral 2896  wrex 2897  wss 3540   class class class wbr 4583  wf 5800  cfv 5804  (class class class)co 6549  supcsup 8229  cr 9814  1c1 9816  *cxr 9952   < clt 9953  cle 9954  NrmCVeccnv 26823  BaseSetcba 26825  normCVcnmcv 26829   normOpOLD cnmoo 26980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-nmcv 26839  df-nmoo 26984
This theorem is referenced by:  nmoub3i  27012  nmobndi  27014  ubthlem2  27111
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