HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  lnfn0i Structured version   Unicode version

Theorem lnfn0i 27159
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 26118 . . . 4  |-  0h  e.  ~H
2 lnfnl.1 . . . . . 6  |-  T  e. 
LinFn
32lnfnfi 27158 . . . . 5  |-  T : ~H
--> CC
43ffvelrni 6006 . . . 4  |-  ( 0h  e.  ~H  ->  ( T `  0h )  e.  CC )
51, 4ax-mp 5 . . 3  |-  ( T `
 0h )  e.  CC
65, 5pncan3oi 9827 . 2  |-  ( ( ( T `  0h )  +  ( T `  0h ) )  -  ( T `  0h )
)  =  ( T `
 0h )
7 ax-1cn 9539 . . . . . . 7  |-  1  e.  CC
82lnfnli 27157 . . . . . . 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 1322 . . . . . 6  |-  ( T `
 ( ( 1  .h  0h )  +h 
0h ) )  =  ( ( 1  x.  ( T `  0h ) )  +  ( T `  0h )
)
107, 1hvmulcli 26129 . . . . . . . . 9  |-  ( 1  .h  0h )  e. 
~H
11 ax-hvaddid 26119 . . . . . . . . 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 26121 . . . . . . . . 9  |-  ( 0h  e.  ~H  ->  (
1  .h  0h )  =  0h )
141, 13ax-mp 5 . . . . . . . 8  |-  ( 1  .h  0h )  =  0h
1512, 14eqtri 2483 . . . . . . 7  |-  ( ( 1  .h  0h )  +h  0h )  =  0h
1615fveq2i 5851 . . . . . 6  |-  ( T `
 ( ( 1  .h  0h )  +h 
0h ) )  =  ( T `  0h )
179, 16eqtr3i 2485 . . . . 5  |-  ( ( 1  x.  ( T `
 0h ) )  +  ( T `  0h ) )  =  ( T `  0h )
185mulid2i 9588 . . . . . 6  |-  ( 1  x.  ( T `  0h ) )  =  ( T `  0h )
1918oveq1i 6280 . . . . 5  |-  ( ( 1  x.  ( T `
 0h ) )  +  ( T `  0h ) )  =  ( ( T `  0h )  +  ( T `  0h ) )
2017, 19eqtr3i 2485 . . . 4  |-  ( T `
 0h )  =  ( ( T `  0h )  +  ( T `  0h )
)
2120oveq1i 6280 . . 3  |-  ( ( T `  0h )  -  ( T `  0h ) )  =  ( ( ( T `  0h )  +  ( T `  0h )
)  -  ( T `
 0h ) )
225subidi 9881 . . 3  |-  ( ( T `  0h )  -  ( T `  0h ) )  =  0
2321, 22eqtr3i 2485 . 2  |-  ( ( ( T `  0h )  +  ( T `  0h ) )  -  ( T `  0h )
)  =  0
246, 23eqtr3i 2485 1  |-  ( T `
 0h )  =  0
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9796   ~Hchil 26034    +h cva 26035    .h csm 26036   0hc0v 26039   LinFnclf 26069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-hilex 26114  ax-hv0cl 26118  ax-hvaddid 26119  ax-hfvmul 26120  ax-hvmulid 26121
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9798  df-lnfn 26965
This theorem is referenced by:  lnfnmuli  27161  lnfn0  27164  nmbdfnlbi  27166  nmcfnexi  27168  nmcfnlbi  27169  nlelshi  27177
  Copyright terms: Public domain W3C validator