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Theorem nmfnsetn0 23230
Description: The set in the supremum of the functional norm definition df-nmfn 23197 is nonempty. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
nmfnsetn0  |-  ( abs `  ( T `  0h ) )  e.  {
x  |  E. y  e.  ~H  ( ( normh `  y )  <_  1  /\  x  =  ( abs `  ( T `  y ) ) ) }
Distinct variable group:    x, y, T

Proof of Theorem nmfnsetn0
StepHypRef Expression
1 ax-hv0cl 22355 . . 3  |-  0h  e.  ~H
2 norm0 22479 . . . . 5  |-  ( normh `  0h )  =  0
3 0le1 9484 . . . . 5  |-  0  <_  1
42, 3eqbrtri 4173 . . . 4  |-  ( normh `  0h )  <_  1
5 eqid 2388 . . . 4  |-  ( abs `  ( T `  0h ) )  =  ( abs `  ( T `
 0h ) )
64, 5pm3.2i 442 . . 3  |-  ( (
normh `  0h )  <_ 
1  /\  ( abs `  ( T `  0h ) )  =  ( abs `  ( T `
 0h ) ) )
7 fveq2 5669 . . . . . 6  |-  ( y  =  0h  ->  ( normh `  y )  =  ( normh `  0h )
)
87breq1d 4164 . . . . 5  |-  ( y  =  0h  ->  (
( normh `  y )  <_  1  <->  ( normh `  0h )  <_  1 ) )
9 fveq2 5669 . . . . . . 7  |-  ( y  =  0h  ->  ( T `  y )  =  ( T `  0h ) )
109fveq2d 5673 . . . . . 6  |-  ( y  =  0h  ->  ( abs `  ( T `  y ) )  =  ( abs `  ( T `  0h )
) )
1110eqeq2d 2399 . . . . 5  |-  ( y  =  0h  ->  (
( abs `  ( T `  0h )
)  =  ( abs `  ( T `  y
) )  <->  ( abs `  ( T `  0h ) )  =  ( abs `  ( T `
 0h ) ) ) )
128, 11anbi12d 692 . . . 4  |-  ( y  =  0h  ->  (
( ( normh `  y
)  <_  1  /\  ( abs `  ( T `
 0h ) )  =  ( abs `  ( T `  y )
) )  <->  ( ( normh `  0h )  <_ 
1  /\  ( abs `  ( T `  0h ) )  =  ( abs `  ( T `
 0h ) ) ) ) )
1312rspcev 2996 . . 3  |-  ( ( 0h  e.  ~H  /\  ( ( normh `  0h )  <_  1  /\  ( abs `  ( T `  0h ) )  =  ( abs `  ( T `
 0h ) ) ) )  ->  E. y  e.  ~H  ( ( normh `  y )  <_  1  /\  ( abs `  ( T `  0h )
)  =  ( abs `  ( T `  y
) ) ) )
141, 6, 13mp2an 654 . 2  |-  E. y  e.  ~H  ( ( normh `  y )  <_  1  /\  ( abs `  ( T `  0h )
)  =  ( abs `  ( T `  y
) ) )
15 fvex 5683 . . 3  |-  ( abs `  ( T `  0h ) )  e.  _V
16 eqeq1 2394 . . . . 5  |-  ( x  =  ( abs `  ( T `  0h )
)  ->  ( x  =  ( abs `  ( T `  y )
)  <->  ( abs `  ( T `  0h )
)  =  ( abs `  ( T `  y
) ) ) )
1716anbi2d 685 . . . 4  |-  ( x  =  ( abs `  ( T `  0h )
)  ->  ( (
( normh `  y )  <_  1  /\  x  =  ( abs `  ( T `  y )
) )  <->  ( ( normh `  y )  <_ 
1  /\  ( abs `  ( T `  0h ) )  =  ( abs `  ( T `
 y ) ) ) ) )
1817rexbidv 2671 . . 3  |-  ( x  =  ( abs `  ( T `  0h )
)  ->  ( E. y  e.  ~H  (
( normh `  y )  <_  1  /\  x  =  ( abs `  ( T `  y )
) )  <->  E. y  e.  ~H  ( ( normh `  y )  <_  1  /\  ( abs `  ( T `  0h )
)  =  ( abs `  ( T `  y
) ) ) ) )
1915, 18elab 3026 . 2  |-  ( ( abs `  ( T `
 0h ) )  e.  { x  |  E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  x  =  ( abs `  ( T `  y
) ) ) }  <->  E. y  e.  ~H  ( ( normh `  y
)  <_  1  /\  ( abs `  ( T `
 0h ) )  =  ( abs `  ( T `  y )
) ) )
2014, 19mpbir 201 1  |-  ( abs `  ( T `  0h ) )  e.  {
x  |  E. y  e.  ~H  ( ( normh `  y )  <_  1  /\  x  =  ( abs `  ( T `  y ) ) ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2374   E.wrex 2651   class class class wbr 4154   ` cfv 5395   0cc0 8924   1c1 8925    <_ cle 9055   abscabs 11967   ~Hchil 22271   normhcno 22275   0hc0v 22276
This theorem is referenced by:  nmfnrepnf  23232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-hv0cl 22355  ax-hvmul0 22362  ax-hfi 22430  ax-his3 22435
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-hnorm 22320
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