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Theorem nmosetre 29748
Description: The set in the supremum of the operator norm definition df-nmoo 29729 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 7033 . . . . . . . 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 29645 . . . . . . . 8 ((π‘Š ∈ NrmCVec ∧ (π‘‡β€˜π‘§) ∈ π‘Œ) β†’ (π‘β€˜(π‘‡β€˜π‘§)) ∈ ℝ)
51, 4sylan2 594 . . . . . . 7 ((π‘Š ∈ NrmCVec ∧ (𝑇:π‘‹βŸΆπ‘Œ ∧ 𝑧 ∈ 𝑋)) β†’ (π‘β€˜(π‘‡β€˜π‘§)) ∈ ℝ)
65anassrs 469 . . . . . 6 (((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) ∧ 𝑧 ∈ 𝑋) β†’ (π‘β€˜(π‘‡β€˜π‘§)) ∈ ℝ)
7 eleq1 2822 . . . . . 6 (π‘₯ = (π‘β€˜(π‘‡β€˜π‘§)) β†’ (π‘₯ ∈ ℝ ↔ (π‘β€˜(π‘‡β€˜π‘§)) ∈ ℝ))
86, 7syl5ibr 246 . . . . 5 (π‘₯ = (π‘β€˜(π‘‡β€˜π‘§)) β†’ (((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) ∧ 𝑧 ∈ 𝑋) β†’ π‘₯ ∈ ℝ))
98impcom 409 . . . 4 ((((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) ∧ 𝑧 ∈ 𝑋) ∧ π‘₯ = (π‘β€˜(π‘‡β€˜π‘§))) β†’ π‘₯ ∈ ℝ)
109adantrl 715 . . 3 ((((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) ∧ 𝑧 ∈ 𝑋) ∧ ((π‘€β€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘β€˜(π‘‡β€˜π‘§)))) β†’ π‘₯ ∈ ℝ)
1110rexlimdva2 3151 . 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 4026 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 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3070   βŠ† wss 3911   class class class wbr 5106  βŸΆwf 6493  β€˜cfv 6497  β„cr 11055  1c1 11057   ≀ cle 11195  NrmCVeccnv 29568  BaseSetcba 29570  normCVcnmcv 29574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-1st 7922  df-2nd 7923  df-vc 29543  df-nv 29576  df-va 29579  df-ba 29580  df-sm 29581  df-0v 29582  df-nmcv 29584
This theorem is referenced by:  nmoxr  29750  nmooge0  29751  nmorepnf  29752  nmoolb  29755  nmoubi  29756  nmlno0lem  29777  nmopsetretHIL  30848
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