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Theorem lnfn0i 23393
Description: The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
lnfnl.1  |-  T  e. 
LinFn
Assertion
Ref Expression
lnfn0i  |-  ( T `
 0h )  =  0

Proof of Theorem lnfn0i
StepHypRef Expression
1 ax-hv0cl 22354 . . . 4  |-  0h  e.  ~H
2 lnfnl.1 . . . . . 6  |-  T  e. 
LinFn
32lnfnfi 23392 . . . . 5  |-  T : ~H
--> CC
43ffvelrni 5808 . . . 4  |-  ( 0h  e.  ~H  ->  ( T `  0h )  e.  CC )
51, 4ax-mp 8 . . 3  |-  ( T `
 0h )  e.  CC
6 pncan 9243 . . 3  |-  ( ( ( T `  0h )  e.  CC  /\  ( T `  0h )  e.  CC )  ->  (
( ( T `  0h )  +  ( T `  0h )
)  -  ( T `
 0h ) )  =  ( T `  0h ) )
75, 5, 6mp2an 654 . 2  |-  ( ( ( T `  0h )  +  ( T `  0h ) )  -  ( T `  0h )
)  =  ( T `
 0h )
8 ax-1cn 8981 . . . . . . 7  |-  1  e.  CC
92lnfnli 23391 . . . . . . 7  |-  ( ( 1  e.  CC  /\  0h  e.  ~H  /\  0h  e.  ~H )  ->  ( T `  ( (
1  .h  0h )  +h  0h ) )  =  ( ( 1  x.  ( T `  0h ) )  +  ( T `  0h )
) )
108, 1, 1, 9mp3an 1279 . . . . . 6  |-  ( T `
 ( ( 1  .h  0h )  +h 
0h ) )  =  ( ( 1  x.  ( T `  0h ) )  +  ( T `  0h )
)
118, 1hvmulcli 22365 . . . . . . . . 9  |-  ( 1  .h  0h )  e. 
~H
12 ax-hvaddid 22355 . . . . . . . . 9  |-  ( ( 1  .h  0h )  e.  ~H  ->  ( (
1  .h  0h )  +h  0h )  =  ( 1  .h  0h )
)
1311, 12ax-mp 8 . . . . . . . 8  |-  ( ( 1  .h  0h )  +h  0h )  =  ( 1  .h  0h )
14 ax-hvmulid 22357 . . . . . . . . 9  |-  ( 0h  e.  ~H  ->  (
1  .h  0h )  =  0h )
151, 14ax-mp 8 . . . . . . . 8  |-  ( 1  .h  0h )  =  0h
1613, 15eqtri 2407 . . . . . . 7  |-  ( ( 1  .h  0h )  +h  0h )  =  0h
1716fveq2i 5671 . . . . . 6  |-  ( T `
 ( ( 1  .h  0h )  +h 
0h ) )  =  ( T `  0h )
1810, 17eqtr3i 2409 . . . . 5  |-  ( ( 1  x.  ( T `
 0h ) )  +  ( T `  0h ) )  =  ( T `  0h )
195mulid2i 9026 . . . . . 6  |-  ( 1  x.  ( T `  0h ) )  =  ( T `  0h )
2019oveq1i 6030 . . . . 5  |-  ( ( 1  x.  ( T `
 0h ) )  +  ( T `  0h ) )  =  ( ( T `  0h )  +  ( T `  0h ) )
2118, 20eqtr3i 2409 . . . 4  |-  ( T `
 0h )  =  ( ( T `  0h )  +  ( T `  0h )
)
2221oveq1i 6030 . . 3  |-  ( ( T `  0h )  -  ( T `  0h ) )  =  ( ( ( T `  0h )  +  ( T `  0h )
)  -  ( T `
 0h ) )
235subidi 9303 . . 3  |-  ( ( T `  0h )  -  ( T `  0h ) )  =  0
2422, 23eqtr3i 2409 . 2  |-  ( ( ( T `  0h )  +  ( T `  0h ) )  -  ( T `  0h )
)  =  0
257, 24eqtr3i 2409 1  |-  ( T `
 0h )  =  0
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   CCcc 8921   0cc0 8923   1c1 8924    + caddc 8926    x. cmul 8928    - cmin 9223   ~Hchil 22270    +h cva 22271    .h csm 22272   0hc0v 22275   LinFnclf 22305
This theorem is referenced by:  lnfnmuli  23395  lnfn0  23398  nmbdfnlbi  23400  nmcfnexi  23402  nmcfnlbi  23403  nlelshi  23411
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-hilex 22350  ax-hv0cl 22354  ax-hvaddid 22355  ax-hfvmul 22356  ax-hvmulid 22357
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-ltxr 9058  df-sub 9225  df-lnfn 23199
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