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Theorem lnfn0i 22583
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 21544 . . . 4  |-  0h  e.  ~H
2 lnfnl.1 . . . . . 6  |-  T  e. 
LinFn
32lnfnfi 22582 . . . . 5  |-  T : ~H
--> CC
43ffvelrni 5598 . . . 4  |-  ( 0h  e.  ~H  ->  ( T `  0h )  e.  CC )
51, 4ax-mp 10 . . 3  |-  ( T `
 0h )  e.  CC
6 pncan 9025 . . 3  |-  ( ( ( T `  0h )  e.  CC  /\  ( T `  0h )  e.  CC )  ->  (
( ( T `  0h )  +  ( T `  0h )
)  -  ( T `
 0h ) )  =  ( T `  0h ) )
75, 5, 6mp2an 656 . 2  |-  ( ( ( T `  0h )  +  ( T `  0h ) )  -  ( T `  0h )
)  =  ( T `
 0h )
8 ax-1cn 8763 . . . . . . 7  |-  1  e.  CC
92lnfnli 22581 . . . . . . 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 1282 . . . . . 6  |-  ( T `
 ( ( 1  .h  0h )  +h 
0h ) )  =  ( ( 1  x.  ( T `  0h ) )  +  ( T `  0h )
)
118, 1hvmulcli 21555 . . . . . . . . 9  |-  ( 1  .h  0h )  e. 
~H
12 ax-hvaddid 21545 . . . . . . . . 9  |-  ( ( 1  .h  0h )  e.  ~H  ->  ( (
1  .h  0h )  +h  0h )  =  ( 1  .h  0h )
)
1311, 12ax-mp 10 . . . . . . . 8  |-  ( ( 1  .h  0h )  +h  0h )  =  ( 1  .h  0h )
14 ax-hvmulid 21547 . . . . . . . . 9  |-  ( 0h  e.  ~H  ->  (
1  .h  0h )  =  0h )
151, 14ax-mp 10 . . . . . . . 8  |-  ( 1  .h  0h )  =  0h
1613, 15eqtri 2278 . . . . . . 7  |-  ( ( 1  .h  0h )  +h  0h )  =  0h
1716fveq2i 5461 . . . . . 6  |-  ( T `
 ( ( 1  .h  0h )  +h 
0h ) )  =  ( T `  0h )
1810, 17eqtr3i 2280 . . . . 5  |-  ( ( 1  x.  ( T `
 0h ) )  +  ( T `  0h ) )  =  ( T `  0h )
195mulid2i 8808 . . . . . 6  |-  ( 1  x.  ( T `  0h ) )  =  ( T `  0h )
2019oveq1i 5802 . . . . 5  |-  ( ( 1  x.  ( T `
 0h ) )  +  ( T `  0h ) )  =  ( ( T `  0h )  +  ( T `  0h ) )
2118, 20eqtr3i 2280 . . . 4  |-  ( T `
 0h )  =  ( ( T `  0h )  +  ( T `  0h )
)
2221oveq1i 5802 . . 3  |-  ( ( T `  0h )  -  ( T `  0h ) )  =  ( ( ( T `  0h )  +  ( T `  0h )
)  -  ( T `
 0h ) )
235subidi 9085 . . 3  |-  ( ( T `  0h )  -  ( T `  0h ) )  =  0
2422, 23eqtr3i 2280 . 2  |-  ( ( ( T `  0h )  +  ( T `  0h ) )  -  ( T `  0h )
)  =  0
257, 24eqtr3i 2280 1  |-  ( T `
 0h )  =  0
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621   ` cfv 4673  (class class class)co 5792   CCcc 8703   0cc0 8705   1c1 8706    + caddc 8708    x. cmul 8710    - cmin 9005   ~Hchil 21460    +h cva 21461    .h csm 21462   0hc0v 21465   LinFnclf 21495
This theorem is referenced by:  lnfnmuli  22585  lnfn0  22588  nmbdfnlbi  22590  nmcfnexi  22592  nmcfnlbi  22593  nlelshi  22601
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-hilex 21540  ax-hv0cl 21544  ax-hvaddid 21545  ax-hfvmul 21546  ax-hvmulid 21547
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-ltxr 8840  df-sub 9007  df-lnfn 22389
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