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Theorem nmfnleub 31672
Description: An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (Revised by Mario Carneiro, 7-Sep-2014.) (New usage is discouraged.)
Assertion
Ref Expression
nmfnleub ((𝑇: β„‹βŸΆβ„‚ ∧ 𝐴 ∈ ℝ*) β†’ ((normfnβ€˜π‘‡) ≀ 𝐴 ↔ βˆ€π‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 β†’ (absβ€˜(π‘‡β€˜π‘₯)) ≀ 𝐴)))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝑇

Proof of Theorem nmfnleub
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmfnval 31623 . . . 4 (𝑇: β„‹βŸΆβ„‚ β†’ (normfnβ€˜π‘‡) = sup({𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))}, ℝ*, < ))
21adantr 480 . . 3 ((𝑇: β„‹βŸΆβ„‚ ∧ 𝐴 ∈ ℝ*) β†’ (normfnβ€˜π‘‡) = sup({𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))}, ℝ*, < ))
32breq1d 5149 . 2 ((𝑇: β„‹βŸΆβ„‚ ∧ 𝐴 ∈ ℝ*) β†’ ((normfnβ€˜π‘‡) ≀ 𝐴 ↔ sup({𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))}, ℝ*, < ) ≀ 𝐴))
4 nmfnsetre 31624 . . . . 5 (𝑇: β„‹βŸΆβ„‚ β†’ {𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))} βŠ† ℝ)
5 ressxr 11257 . . . . 5 ℝ βŠ† ℝ*
64, 5sstrdi 3987 . . . 4 (𝑇: β„‹βŸΆβ„‚ β†’ {𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))} βŠ† ℝ*)
7 supxrleub 13306 . . . 4 (({𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))} βŠ† ℝ* ∧ 𝐴 ∈ ℝ*) β†’ (sup({𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))}, ℝ*, < ) ≀ 𝐴 ↔ βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))}𝑧 ≀ 𝐴))
86, 7sylan 579 . . 3 ((𝑇: β„‹βŸΆβ„‚ ∧ 𝐴 ∈ ℝ*) β†’ (sup({𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))}, ℝ*, < ) ≀ 𝐴 ↔ βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))}𝑧 ≀ 𝐴))
9 ancom 460 . . . . . . 7 (((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯))) ↔ (𝑦 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1))
10 eqeq1 2728 . . . . . . . 8 (𝑦 = 𝑧 β†’ (𝑦 = (absβ€˜(π‘‡β€˜π‘₯)) ↔ 𝑧 = (absβ€˜(π‘‡β€˜π‘₯))))
1110anbi1d 629 . . . . . . 7 (𝑦 = 𝑧 β†’ ((𝑦 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1) ↔ (𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1)))
129, 11bitrid 283 . . . . . 6 (𝑦 = 𝑧 β†’ (((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯))) ↔ (𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1)))
1312rexbidv 3170 . . . . 5 (𝑦 = 𝑧 β†’ (βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯))) ↔ βˆƒπ‘₯ ∈ β„‹ (𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1)))
1413ralab 3680 . . . 4 (βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))}𝑧 ≀ 𝐴 ↔ βˆ€π‘§(βˆƒπ‘₯ ∈ β„‹ (𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1) β†’ 𝑧 ≀ 𝐴))
15 ralcom4 3275 . . . . 5 (βˆ€π‘₯ ∈ β„‹ βˆ€π‘§((𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1) β†’ 𝑧 ≀ 𝐴) ↔ βˆ€π‘§βˆ€π‘₯ ∈ β„‹ ((𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1) β†’ 𝑧 ≀ 𝐴))
16 impexp 450 . . . . . . . 8 (((𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1) β†’ 𝑧 ≀ 𝐴) ↔ (𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) β†’ ((normβ„Žβ€˜π‘₯) ≀ 1 β†’ 𝑧 ≀ 𝐴)))
1716albii 1813 . . . . . . 7 (βˆ€π‘§((𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1) β†’ 𝑧 ≀ 𝐴) ↔ βˆ€π‘§(𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) β†’ ((normβ„Žβ€˜π‘₯) ≀ 1 β†’ 𝑧 ≀ 𝐴)))
18 fvex 6895 . . . . . . . 8 (absβ€˜(π‘‡β€˜π‘₯)) ∈ V
19 breq1 5142 . . . . . . . . 9 (𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) β†’ (𝑧 ≀ 𝐴 ↔ (absβ€˜(π‘‡β€˜π‘₯)) ≀ 𝐴))
2019imbi2d 340 . . . . . . . 8 (𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) β†’ (((normβ„Žβ€˜π‘₯) ≀ 1 β†’ 𝑧 ≀ 𝐴) ↔ ((normβ„Žβ€˜π‘₯) ≀ 1 β†’ (absβ€˜(π‘‡β€˜π‘₯)) ≀ 𝐴)))
2118, 20ceqsalv 3504 . . . . . . 7 (βˆ€π‘§(𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) β†’ ((normβ„Žβ€˜π‘₯) ≀ 1 β†’ 𝑧 ≀ 𝐴)) ↔ ((normβ„Žβ€˜π‘₯) ≀ 1 β†’ (absβ€˜(π‘‡β€˜π‘₯)) ≀ 𝐴))
2217, 21bitri 275 . . . . . 6 (βˆ€π‘§((𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1) β†’ 𝑧 ≀ 𝐴) ↔ ((normβ„Žβ€˜π‘₯) ≀ 1 β†’ (absβ€˜(π‘‡β€˜π‘₯)) ≀ 𝐴))
2322ralbii 3085 . . . . 5 (βˆ€π‘₯ ∈ β„‹ βˆ€π‘§((𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1) β†’ 𝑧 ≀ 𝐴) ↔ βˆ€π‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 β†’ (absβ€˜(π‘‡β€˜π‘₯)) ≀ 𝐴))
24 r19.23v 3174 . . . . . 6 (βˆ€π‘₯ ∈ β„‹ ((𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1) β†’ 𝑧 ≀ 𝐴) ↔ (βˆƒπ‘₯ ∈ β„‹ (𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1) β†’ 𝑧 ≀ 𝐴))
2524albii 1813 . . . . 5 (βˆ€π‘§βˆ€π‘₯ ∈ β„‹ ((𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1) β†’ 𝑧 ≀ 𝐴) ↔ βˆ€π‘§(βˆƒπ‘₯ ∈ β„‹ (𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1) β†’ 𝑧 ≀ 𝐴))
2615, 23, 253bitr3i 301 . . . 4 (βˆ€π‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 β†’ (absβ€˜(π‘‡β€˜π‘₯)) ≀ 𝐴) ↔ βˆ€π‘§(βˆƒπ‘₯ ∈ β„‹ (𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1) β†’ 𝑧 ≀ 𝐴))
2714, 26bitr4i 278 . . 3 (βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))}𝑧 ≀ 𝐴 ↔ βˆ€π‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 β†’ (absβ€˜(π‘‡β€˜π‘₯)) ≀ 𝐴))
288, 27bitrdi 287 . 2 ((𝑇: β„‹βŸΆβ„‚ ∧ 𝐴 ∈ ℝ*) β†’ (sup({𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))}, ℝ*, < ) ≀ 𝐴 ↔ βˆ€π‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 β†’ (absβ€˜(π‘‡β€˜π‘₯)) ≀ 𝐴)))
293, 28bitrd 279 1 ((𝑇: β„‹βŸΆβ„‚ ∧ 𝐴 ∈ ℝ*) β†’ ((normfnβ€˜π‘‡) ≀ 𝐴 ↔ βˆ€π‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 β†’ (absβ€˜(π‘‡β€˜π‘₯)) ≀ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395  βˆ€wal 1531   = wceq 1533   ∈ wcel 2098  {cab 2701  βˆ€wral 3053  βˆƒwrex 3062   βŠ† wss 3941   class class class wbr 5139  βŸΆwf 6530  β€˜cfv 6534  supcsup 9432  β„‚cc 11105  β„cr 11106  1c1 11108  β„*cxr 11246   < clt 11247   ≀ cle 11248  abscabs 15183   β„‹chba 30666  normβ„Žcno 30670  normfncnmf 30698
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 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185  ax-hilex 30746
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-sup 9434  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-n0 12472  df-z 12558  df-uz 12822  df-rp 12976  df-seq 13968  df-exp 14029  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-nmfn 31592
This theorem is referenced by:  nmfnleub2  31673
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