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Theorem lnfn0i 23498
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 22459 . . . 4  |-  0h  e.  ~H
2 lnfnl.1 . . . . . 6  |-  T  e. 
LinFn
32lnfnfi 23497 . . . . 5  |-  T : ~H
--> CC
43ffvelrni 5828 . . . 4  |-  ( 0h  e.  ~H  ->  ( T `  0h )  e.  CC )
51, 4ax-mp 8 . . 3  |-  ( T `
 0h )  e.  CC
6 pncan 9267 . . 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 9004 . . . . . . 7  |-  1  e.  CC
92lnfnli 23496 . . . . . . 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 22470 . . . . . . . . 9  |-  ( 1  .h  0h )  e. 
~H
12 ax-hvaddid 22460 . . . . . . . . 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 22462 . . . . . . . . 9  |-  ( 0h  e.  ~H  ->  (
1  .h  0h )  =  0h )
151, 14ax-mp 8 . . . . . . . 8  |-  ( 1  .h  0h )  =  0h
1613, 15eqtri 2424 . . . . . . 7  |-  ( ( 1  .h  0h )  +h  0h )  =  0h
1716fveq2i 5690 . . . . . 6  |-  ( T `
 ( ( 1  .h  0h )  +h 
0h ) )  =  ( T `  0h )
1810, 17eqtr3i 2426 . . . . 5  |-  ( ( 1  x.  ( T `
 0h ) )  +  ( T `  0h ) )  =  ( T `  0h )
195mulid2i 9049 . . . . . 6  |-  ( 1  x.  ( T `  0h ) )  =  ( T `  0h )
2019oveq1i 6050 . . . . 5  |-  ( ( 1  x.  ( T `
 0h ) )  +  ( T `  0h ) )  =  ( ( T `  0h )  +  ( T `  0h ) )
2118, 20eqtr3i 2426 . . . 4  |-  ( T `
 0h )  =  ( ( T `  0h )  +  ( T `  0h )
)
2221oveq1i 6050 . . 3  |-  ( ( T `  0h )  -  ( T `  0h ) )  =  ( ( ( T `  0h )  +  ( T `  0h )
)  -  ( T `
 0h ) )
235subidi 9327 . . 3  |-  ( ( T `  0h )  -  ( T `  0h ) )  =  0
2422, 23eqtr3i 2426 . 2  |-  ( ( ( T `  0h )  +  ( T `  0h ) )  -  ( T `  0h )
)  =  0
257, 24eqtr3i 2426 1  |-  ( T `
 0h )  =  0
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247   ~Hchil 22375    +h cva 22376    .h csm 22377   0hc0v 22380   LinFnclf 22410
This theorem is referenced by:  lnfnmuli  23500  lnfn0  23503  nmbdfnlbi  23505  nmcfnexi  23507  nmcfnlbi  23508  nlelshi  23516
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-hilex 22455  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-lnfn 23304
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