MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmosetre Structured version   Visualization version   GIF version

Theorem nmosetre 30783
Description: The set in the supremum of the operator norm definition df-nmoo 30764 is a set of reals. (Contributed by NM, 13-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmosetre.2 𝑌 = (BaseSet‘𝑊)
nmosetre.4 𝑁 = (normCV𝑊)
Assertion
Ref Expression
nmosetre ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → {𝑥 ∣ ∃𝑧𝑋 ((𝑀𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇𝑧)))} ⊆ ℝ)
Distinct variable groups:   𝑥,𝑧,𝑇   𝑥,𝑊,𝑧   𝑥,𝑋,𝑧   𝑥,𝑌,𝑧
Allowed substitution hints:   𝑀(𝑥,𝑧)   𝑁(𝑥,𝑧)

Proof of Theorem nmosetre
StepHypRef Expression
1 ffvelcdm 7101 . . . . . . . 8 ((𝑇:𝑋𝑌𝑧𝑋) → (𝑇𝑧) ∈ 𝑌)
2 nmosetre.2 . . . . . . . . 9 𝑌 = (BaseSet‘𝑊)
3 nmosetre.4 . . . . . . . . 9 𝑁 = (normCV𝑊)
42, 3nvcl 30680 . . . . . . . 8 ((𝑊 ∈ NrmCVec ∧ (𝑇𝑧) ∈ 𝑌) → (𝑁‘(𝑇𝑧)) ∈ ℝ)
51, 4sylan2 593 . . . . . . 7 ((𝑊 ∈ NrmCVec ∧ (𝑇:𝑋𝑌𝑧𝑋)) → (𝑁‘(𝑇𝑧)) ∈ ℝ)
65anassrs 467 . . . . . 6 (((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) ∧ 𝑧𝑋) → (𝑁‘(𝑇𝑧)) ∈ ℝ)
7 eleq1 2829 . . . . . 6 (𝑥 = (𝑁‘(𝑇𝑧)) → (𝑥 ∈ ℝ ↔ (𝑁‘(𝑇𝑧)) ∈ ℝ))
86, 7imbitrrid 246 . . . . 5 (𝑥 = (𝑁‘(𝑇𝑧)) → (((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) ∧ 𝑧𝑋) → 𝑥 ∈ ℝ))
98impcom 407 . . . 4 ((((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) ∧ 𝑧𝑋) ∧ 𝑥 = (𝑁‘(𝑇𝑧))) → 𝑥 ∈ ℝ)
109adantrl 716 . . 3 ((((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) ∧ 𝑧𝑋) ∧ ((𝑀𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇𝑧)))) → 𝑥 ∈ ℝ)
1110rexlimdva2 3157 . 2 ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (∃𝑧𝑋 ((𝑀𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇𝑧))) → 𝑥 ∈ ℝ))
1211abssdv 4068 1 ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → {𝑥 ∣ ∃𝑧𝑋 ((𝑀𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇𝑧)))} ⊆ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  wss 3951   class class class wbr 5143  wf 6557  cfv 6561  cr 11154  1c1 11156  cle 11296  NrmCVeccnv 30603  BaseSetcba 30605  normCVcnmcv 30609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-1st 8014  df-2nd 8015  df-vc 30578  df-nv 30611  df-va 30614  df-ba 30615  df-sm 30616  df-0v 30617  df-nmcv 30619
This theorem is referenced by:  nmoxr  30785  nmooge0  30786  nmorepnf  30787  nmoolb  30790  nmoubi  30791  nmlno0lem  30812  nmopsetretHIL  31883
  Copyright terms: Public domain W3C validator