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Theorem nmosetre 30602
Description: The set in the supremum of the operator norm definition df-nmoo 30583 is a set of reals. (Contributed by NM, 13-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmosetre.2 π‘Œ = (BaseSetβ€˜π‘Š)
nmosetre.4 𝑁 = (normCVβ€˜π‘Š)
Assertion
Ref Expression
nmosetre ((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ {π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((π‘€β€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘β€˜(π‘‡β€˜π‘§)))} βŠ† ℝ)
Distinct variable groups:   π‘₯,𝑧,𝑇   π‘₯,π‘Š,𝑧   π‘₯,𝑋,𝑧   π‘₯,π‘Œ,𝑧
Allowed substitution hints:   𝑀(π‘₯,𝑧)   𝑁(π‘₯,𝑧)
Proof of Theorem nmosetre
StepHypRef Expression
1 ffvelcdm 7096 . . . . . . . 8 ((𝑇:π‘‹βŸΆπ‘Œ ∧ 𝑧 ∈ 𝑋) β†’ (π‘‡β€˜π‘§) ∈ π‘Œ)
2 nmosetre.2 . . . . . . . . 9 π‘Œ = (BaseSetβ€˜π‘Š)
3 nmosetre.4 . . . . . . . . 9 𝑁 = (normCVβ€˜π‘Š)
42, 3nvcl 30499 . . . . . . . 8 ((π‘Š ∈ NrmCVec ∧ (π‘‡β€˜π‘§) ∈ π‘Œ) β†’ (π‘β€˜(π‘‡β€˜π‘§)) ∈ ℝ)
51, 4sylan2 591 . . . . . . 7 ((π‘Š ∈ NrmCVec ∧ (𝑇:π‘‹βŸΆπ‘Œ ∧ 𝑧 ∈ 𝑋)) β†’ (π‘β€˜(π‘‡β€˜π‘§)) ∈ ℝ)
65anassrs 466 . . . . . 6 (((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) ∧ 𝑧 ∈ 𝑋) β†’ (π‘β€˜(π‘‡β€˜π‘§)) ∈ ℝ)
7 eleq1 2817 . . . . . 6 (π‘₯ = (π‘β€˜(π‘‡β€˜π‘§)) β†’ (π‘₯ ∈ ℝ ↔ (π‘β€˜(π‘‡β€˜π‘§)) ∈ ℝ))
86, 7imbitrrid 245 . . . . 5 (π‘₯ = (π‘β€˜(π‘‡β€˜π‘§)) β†’ (((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) ∧ 𝑧 ∈ 𝑋) β†’ π‘₯ ∈ ℝ))
98impcom 406 . . . 4 ((((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) ∧ 𝑧 ∈ 𝑋) ∧ π‘₯ = (π‘β€˜(π‘‡β€˜π‘§))) β†’ π‘₯ ∈ ℝ)
109adantrl 714 . . 3 ((((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) ∧ 𝑧 ∈ 𝑋) ∧ ((π‘€β€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘β€˜(π‘‡β€˜π‘§)))) β†’ π‘₯ ∈ ℝ)
1110rexlimdva2 3154 . 2 ((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ (βˆƒπ‘§ ∈ 𝑋 ((π‘€β€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘β€˜(π‘‡β€˜π‘§))) β†’ π‘₯ ∈ ℝ))
1211abssdv 4065 1 ((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ {π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((π‘€β€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘β€˜(π‘‡β€˜π‘§)))} βŠ† ℝ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2705  βˆƒwrex 3067   βŠ† wss 3949   class class class wbr 5152  βŸΆwf 6549  β€˜cfv 6553  β„cr 11147  1c1 11149   ≀ cle 11289  NrmCVeccnv 30422  BaseSetcba 30424  normCVcnmcv 30428
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-1st 8001  df-2nd 8002  df-vc 30397  df-nv 30430  df-va 30433  df-ba 30434  df-sm 30435  df-0v 30436  df-nmcv 30438
This theorem is referenced by:  nmoxr  30604  nmooge0  30605  nmorepnf  30606  nmoolb  30609  nmoubi  30610  nmlno0lem  30631  nmopsetretHIL  31702
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