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Theorem nmooval 29747
Description: The operator norm function. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoofval.1 𝑋 = (BaseSetβ€˜π‘ˆ)
nmoofval.2 π‘Œ = (BaseSetβ€˜π‘Š)
nmoofval.3 𝐿 = (normCVβ€˜π‘ˆ)
nmoofval.4 𝑀 = (normCVβ€˜π‘Š)
nmoofval.6 𝑁 = (π‘ˆ normOpOLD π‘Š)
Assertion
Ref Expression
nmooval ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ (π‘β€˜π‘‡) = sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘§)))}, ℝ*, < ))
Distinct variable groups:   π‘₯,𝑧,π‘ˆ   π‘₯,π‘Š,𝑧   𝑧,𝑋   π‘₯,π‘Œ   π‘₯,𝑇,𝑧
Allowed substitution hints:   𝐿(π‘₯,𝑧)   𝑀(π‘₯,𝑧)   𝑁(π‘₯,𝑧)   𝑋(π‘₯)   π‘Œ(𝑧)

Proof of Theorem nmooval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 nmoofval.2 . . . . 5 π‘Œ = (BaseSetβ€˜π‘Š)
21fvexi 6857 . . . 4 π‘Œ ∈ V
3 nmoofval.1 . . . . 5 𝑋 = (BaseSetβ€˜π‘ˆ)
43fvexi 6857 . . . 4 𝑋 ∈ V
52, 4elmap 8812 . . 3 (𝑇 ∈ (π‘Œ ↑m 𝑋) ↔ 𝑇:π‘‹βŸΆπ‘Œ)
6 nmoofval.3 . . . . . 6 𝐿 = (normCVβ€˜π‘ˆ)
7 nmoofval.4 . . . . . 6 𝑀 = (normCVβ€˜π‘Š)
8 nmoofval.6 . . . . . 6 𝑁 = (π‘ˆ normOpOLD π‘Š)
93, 1, 6, 7, 8nmoofval 29746 . . . . 5 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ 𝑁 = (𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )))
109fveq1d 6845 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ (π‘β€˜π‘‡) = ((𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < ))β€˜π‘‡))
11 fveq1 6842 . . . . . . . . . . 11 (𝑑 = 𝑇 β†’ (π‘‘β€˜π‘§) = (π‘‡β€˜π‘§))
1211fveq2d 6847 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ (π‘€β€˜(π‘‘β€˜π‘§)) = (π‘€β€˜(π‘‡β€˜π‘§)))
1312eqeq2d 2744 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)) ↔ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘§))))
1413anbi2d 630 . . . . . . . 8 (𝑑 = 𝑇 β†’ (((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§))) ↔ ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘§)))))
1514rexbidv 3172 . . . . . . 7 (𝑑 = 𝑇 β†’ (βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§))) ↔ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘§)))))
1615abbidv 2802 . . . . . 6 (𝑑 = 𝑇 β†’ {π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))} = {π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘§)))})
1716supeq1d 9387 . . . . 5 (𝑑 = 𝑇 β†’ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < ) = sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘§)))}, ℝ*, < ))
18 eqid 2733 . . . . 5 (𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )) = (𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < ))
19 xrltso 13066 . . . . . 6 < Or ℝ*
2019supex 9404 . . . . 5 sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘§)))}, ℝ*, < ) ∈ V
2117, 18, 20fvmpt 6949 . . . 4 (𝑇 ∈ (π‘Œ ↑m 𝑋) β†’ ((𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < ))β€˜π‘‡) = sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘§)))}, ℝ*, < ))
2210, 21sylan9eq 2793 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) ∧ 𝑇 ∈ (π‘Œ ↑m 𝑋)) β†’ (π‘β€˜π‘‡) = sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘§)))}, ℝ*, < ))
235, 22sylan2br 596 . 2 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ (π‘β€˜π‘‡) = sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘§)))}, ℝ*, < ))
24233impa 1111 1 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ (π‘β€˜π‘‡) = sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘§)))}, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3070   class class class wbr 5106   ↦ cmpt 5189  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358   ↑m cmap 8768  supcsup 9381  1c1 11057  β„*cxr 11193   < clt 11194   ≀ cle 11195  NrmCVeccnv 29568  BaseSetcba 29570  normCVcnmcv 29574   normOpOLD cnmoo 29725
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-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-pre-lttri 11130  ax-pre-lttrn 11131
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  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-pw 4563  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-po 5546  df-so 5547  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-mpo 7363  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9383  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-nmoo 29729
This theorem is referenced by:  nmoxr  29750  nmooge0  29751  nmorepnf  29752  nmoolb  29755  nmoubi  29756  nmoo0  29775  nmlno0lem  29777
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