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Theorem lnfn0i 26833
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 25792 . . . 4  |-  0h  e.  ~H
2 lnfnl.1 . . . . . 6  |-  T  e. 
LinFn
32lnfnfi 26832 . . . . 5  |-  T : ~H
--> CC
43ffvelrni 6015 . . . 4  |-  ( 0h  e.  ~H  ->  ( T `  0h )  e.  CC )
51, 4ax-mp 5 . . 3  |-  ( T `
 0h )  e.  CC
65, 5pncan3oi 9841 . 2  |-  ( ( ( T `  0h )  +  ( T `  0h ) )  -  ( T `  0h )
)  =  ( T `
 0h )
7 ax-1cn 9553 . . . . . . 7  |-  1  e.  CC
82lnfnli 26831 . . . . . . 7  |-  ( ( 1  e.  CC  /\  0h  e.  ~H  /\  0h  e.  ~H )  ->  ( T `  ( (
1  .h  0h )  +h  0h ) )  =  ( ( 1  x.  ( T `  0h ) )  +  ( T `  0h )
) )
97, 1, 1, 8mp3an 1325 . . . . . 6  |-  ( T `
 ( ( 1  .h  0h )  +h 
0h ) )  =  ( ( 1  x.  ( T `  0h ) )  +  ( T `  0h )
)
107, 1hvmulcli 25803 . . . . . . . . 9  |-  ( 1  .h  0h )  e. 
~H
11 ax-hvaddid 25793 . . . . . . . . 9  |-  ( ( 1  .h  0h )  e.  ~H  ->  ( (
1  .h  0h )  +h  0h )  =  ( 1  .h  0h )
)
1210, 11ax-mp 5 . . . . . . . 8  |-  ( ( 1  .h  0h )  +h  0h )  =  ( 1  .h  0h )
13 ax-hvmulid 25795 . . . . . . . . 9  |-  ( 0h  e.  ~H  ->  (
1  .h  0h )  =  0h )
141, 13ax-mp 5 . . . . . . . 8  |-  ( 1  .h  0h )  =  0h
1512, 14eqtri 2472 . . . . . . 7  |-  ( ( 1  .h  0h )  +h  0h )  =  0h
1615fveq2i 5859 . . . . . 6  |-  ( T `
 ( ( 1  .h  0h )  +h 
0h ) )  =  ( T `  0h )
179, 16eqtr3i 2474 . . . . 5  |-  ( ( 1  x.  ( T `
 0h ) )  +  ( T `  0h ) )  =  ( T `  0h )
185mulid2i 9602 . . . . . 6  |-  ( 1  x.  ( T `  0h ) )  =  ( T `  0h )
1918oveq1i 6291 . . . . 5  |-  ( ( 1  x.  ( T `
 0h ) )  +  ( T `  0h ) )  =  ( ( T `  0h )  +  ( T `  0h ) )
2017, 19eqtr3i 2474 . . . 4  |-  ( T `
 0h )  =  ( ( T `  0h )  +  ( T `  0h )
)
2120oveq1i 6291 . . 3  |-  ( ( T `  0h )  -  ( T `  0h ) )  =  ( ( ( T `  0h )  +  ( T `  0h )
)  -  ( T `
 0h ) )
225subidi 9895 . . 3  |-  ( ( T `  0h )  -  ( T `  0h ) )  =  0
2321, 22eqtr3i 2474 . 2  |-  ( ( ( T `  0h )  +  ( T `  0h ) )  -  ( T `  0h )
)  =  0
246, 23eqtr3i 2474 1  |-  ( T `
 0h )  =  0
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500    - cmin 9810   ~Hchil 25708    +h cva 25709    .h csm 25710   0hc0v 25713   LinFnclf 25743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-hilex 25788  ax-hv0cl 25792  ax-hvaddid 25793  ax-hfvmul 25794  ax-hvmulid 25795
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-ltxr 9636  df-sub 9812  df-lnfn 26639
This theorem is referenced by:  lnfnmuli  26835  lnfn0  26838  nmbdfnlbi  26840  nmcfnexi  26842  nmcfnlbi  26843  nlelshi  26851
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