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Theorem nmosetre 30017
Description: The set in the supremum of the operator norm definition df-nmoo 29998 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 7084 . . . . . . . 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 29914 . . . . . . . 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, 7imbitrrid 245 . . . . 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 3158 . 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 4066 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 3071   βŠ† wss 3949   class class class wbr 5149  βŸΆwf 6540  β€˜cfv 6544  β„cr 11109  1c1 11111   ≀ cle 11249  NrmCVeccnv 29837  BaseSetcba 29839  normCVcnmcv 29843
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 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-1st 7975  df-2nd 7976  df-vc 29812  df-nv 29845  df-va 29848  df-ba 29849  df-sm 29850  df-0v 29851  df-nmcv 29853
This theorem is referenced by:  nmoxr  30019  nmooge0  30020  nmorepnf  30021  nmoolb  30024  nmoubi  30025  nmlno0lem  30046  nmopsetretHIL  31117
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