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Theorem nmosetre 30526
Description: The set in the supremum of the operator norm definition df-nmoo 30507 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 7077 . . . . . . . 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 30423 . . . . . . . 8 ((π‘Š ∈ NrmCVec ∧ (π‘‡β€˜π‘§) ∈ π‘Œ) β†’ (π‘β€˜(π‘‡β€˜π‘§)) ∈ ℝ)
51, 4sylan2 592 . . . . . . 7 ((π‘Š ∈ NrmCVec ∧ (𝑇:π‘‹βŸΆπ‘Œ ∧ 𝑧 ∈ 𝑋)) β†’ (π‘β€˜(π‘‡β€˜π‘§)) ∈ ℝ)
65anassrs 467 . . . . . 6 (((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) ∧ 𝑧 ∈ 𝑋) β†’ (π‘β€˜(π‘‡β€˜π‘§)) ∈ ℝ)
7 eleq1 2815 . . . . . 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 407 . . . 4 ((((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) ∧ 𝑧 ∈ 𝑋) ∧ π‘₯ = (π‘β€˜(π‘‡β€˜π‘§))) β†’ π‘₯ ∈ ℝ)
109adantrl 713 . . 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 4060 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 395   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆƒwrex 3064   βŠ† wss 3943   class class class wbr 5141  βŸΆwf 6533  β€˜cfv 6537  β„cr 11111  1c1 11113   ≀ cle 11253  NrmCVeccnv 30346  BaseSetcba 30348  normCVcnmcv 30352
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-1st 7974  df-2nd 7975  df-vc 30321  df-nv 30354  df-va 30357  df-ba 30358  df-sm 30359  df-0v 30360  df-nmcv 30362
This theorem is referenced by:  nmoxr  30528  nmooge0  30529  nmorepnf  30530  nmoolb  30533  nmoubi  30534  nmlno0lem  30555  nmopsetretHIL  31626
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