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Theorem nmfnleub 30916
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 30867 . . . 4 (𝑇: β„‹βŸΆβ„‚ β†’ (normfnβ€˜π‘‡) = sup({𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))}, ℝ*, < ))
21adantr 482 . . 3 ((𝑇: β„‹βŸΆβ„‚ ∧ 𝐴 ∈ ℝ*) β†’ (normfnβ€˜π‘‡) = sup({𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))}, ℝ*, < ))
32breq1d 5119 . 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 30868 . . . . 5 (𝑇: β„‹βŸΆβ„‚ β†’ {𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))} βŠ† ℝ)
5 ressxr 11207 . . . . 5 ℝ βŠ† ℝ*
64, 5sstrdi 3960 . . . 4 (𝑇: β„‹βŸΆβ„‚ β†’ {𝑦 ∣ βˆƒπ‘₯ ∈ β„‹ ((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯)))} βŠ† ℝ*)
7 supxrleub 13254 . . . 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 581 . . 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 462 . . . . . . 7 (((normβ„Žβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (absβ€˜(π‘‡β€˜π‘₯))) ↔ (𝑦 = (absβ€˜(π‘‡β€˜π‘₯)) ∧ (normβ„Žβ€˜π‘₯) ≀ 1))
10 eqeq1 2737 . . . . . . . 8 (𝑦 = 𝑧 β†’ (𝑦 = (absβ€˜(π‘‡β€˜π‘₯)) ↔ 𝑧 = (absβ€˜(π‘‡β€˜π‘₯))))
1110anbi1d 631 . . . . . . 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 3172 . . . . 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 3653 . . . 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 3268 . . . . 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 452 . . . . . . . 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 1822 . . . . . . 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 6859 . . . . . . . 8 (absβ€˜(π‘‡β€˜π‘₯)) ∈ V
19 breq1 5112 . . . . . . . . 9 (𝑧 = (absβ€˜(π‘‡β€˜π‘₯)) β†’ (𝑧 ≀ 𝐴 ↔ (absβ€˜(π‘‡β€˜π‘₯)) ≀ 𝐴))
2019imbi2d 341 . . . . . . . 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 3483 . . . . . . 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 3093 . . . . 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 3176 . . . . . 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 1822 . . . . 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 397  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3914   class class class wbr 5109  βŸΆwf 6496  β€˜cfv 6500  supcsup 9384  β„‚cc 11057  β„cr 11058  1c1 11060  β„*cxr 11196   < clt 11197   ≀ cle 11198  abscabs 15128   β„‹chba 29910  normβ„Žcno 29914  normfncnmf 29942
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-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137  ax-hilex 29990
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 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-sup 9386  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-z 12508  df-uz 12772  df-rp 12924  df-seq 13916  df-exp 13977  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-nmfn 30836
This theorem is referenced by:  nmfnleub2  30917
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