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Theorem nmosetre 28533
Description: The set in the supremum of the operator norm definition df-nmoo 28514 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 ffvelrn 6842 . . . . . . . 8 ((𝑇:𝑋𝑌𝑧𝑋) → (𝑇𝑧) ∈ 𝑌)
2 nmosetre.2 . . . . . . . . 9 𝑌 = (BaseSet‘𝑊)
3 nmosetre.4 . . . . . . . . 9 𝑁 = (normCV𝑊)
42, 3nvcl 28430 . . . . . . . 8 ((𝑊 ∈ NrmCVec ∧ (𝑇𝑧) ∈ 𝑌) → (𝑁‘(𝑇𝑧)) ∈ ℝ)
51, 4sylan2 594 . . . . . . 7 ((𝑊 ∈ NrmCVec ∧ (𝑇:𝑋𝑌𝑧𝑋)) → (𝑁‘(𝑇𝑧)) ∈ ℝ)
65anassrs 470 . . . . . 6 (((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) ∧ 𝑧𝑋) → (𝑁‘(𝑇𝑧)) ∈ ℝ)
7 eleq1 2898 . . . . . 6 (𝑥 = (𝑁‘(𝑇𝑧)) → (𝑥 ∈ ℝ ↔ (𝑁‘(𝑇𝑧)) ∈ ℝ))
86, 7syl5ibr 248 . . . . 5 (𝑥 = (𝑁‘(𝑇𝑧)) → (((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) ∧ 𝑧𝑋) → 𝑥 ∈ ℝ))
98impcom 410 . . . 4 ((((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) ∧ 𝑧𝑋) ∧ 𝑥 = (𝑁‘(𝑇𝑧))) → 𝑥 ∈ ℝ)
109adantrl 714 . . 3 ((((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) ∧ 𝑧𝑋) ∧ ((𝑀𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇𝑧)))) → 𝑥 ∈ ℝ)
1110rexlimdva2 3285 . 2 ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (∃𝑧𝑋 ((𝑀𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇𝑧))) → 𝑥 ∈ ℝ))
1211abssdv 4043 1 ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → {𝑥 ∣ ∃𝑧𝑋 ((𝑀𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇𝑧)))} ⊆ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wcel 2107  {cab 2797  wrex 3137  wss 3934   class class class wbr 5057  wf 6344  cfv 6348  cr 10528  1c1 10530  cle 10668  NrmCVeccnv 28353  BaseSetcba 28355  normCVcnmcv 28359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  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-ov 7151  df-oprab 7152  df-1st 7681  df-2nd 7682  df-vc 28328  df-nv 28361  df-va 28364  df-ba 28365  df-sm 28366  df-0v 28367  df-nmcv 28369
This theorem is referenced by:  nmoxr  28535  nmooge0  28536  nmorepnf  28537  nmoolb  28540  nmoubi  28541  nmlno0lem  28562  nmopsetretHIL  29633
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