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Theorem lnfnmuli 26625
Description: Multiplicative property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
lnfnl.1  |-  T  e. 
LinFn
Assertion
Ref Expression
lnfnmuli  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( T `  ( A  .h  B )
)  =  ( A  x.  ( T `  B ) ) )

Proof of Theorem lnfnmuli
StepHypRef Expression
1 ax-hv0cl 25582 . . 3  |-  0h  e.  ~H
2 lnfnl.1 . . . 4  |-  T  e. 
LinFn
32lnfnli 26621 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  0h  e.  ~H )  ->  ( T `  ( ( A  .h  B )  +h  0h ) )  =  ( ( A  x.  ( T `  B ) )  +  ( T `
 0h ) ) )
41, 3mp3an3 1308 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( T `  (
( A  .h  B
)  +h  0h )
)  =  ( ( A  x.  ( T `
 B ) )  +  ( T `  0h ) ) )
5 hvmulcl 25592 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
6 ax-hvaddid 25583 . . . 4  |-  ( ( A  .h  B )  e.  ~H  ->  (
( A  .h  B
)  +h  0h )  =  ( A  .h  B ) )
75, 6syl 16 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  .h  B )  +h  0h )  =  ( A  .h  B ) )
87fveq2d 5861 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( T `  (
( A  .h  B
)  +h  0h )
)  =  ( T `
 ( A  .h  B ) ) )
92lnfn0i 26623 . . . 4  |-  ( T `
 0h )  =  0
109oveq2i 6286 . . 3  |-  ( ( A  x.  ( T `
 B ) )  +  ( T `  0h ) )  =  ( ( A  x.  ( T `  B )
)  +  0 )
112lnfnfi 26622 . . . . . 6  |-  T : ~H
--> CC
1211ffvelrni 6011 . . . . 5  |-  ( B  e.  ~H  ->  ( T `  B )  e.  CC )
13 mulcl 9565 . . . . 5  |-  ( ( A  e.  CC  /\  ( T `  B )  e.  CC )  -> 
( A  x.  ( T `  B )
)  e.  CC )
1412, 13sylan2 474 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  x.  ( T `  B )
)  e.  CC )
1514addid1d 9768 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  x.  ( T `  B ) )  +  0 )  =  ( A  x.  ( T `  B ) ) )
1610, 15syl5eq 2513 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  x.  ( T `  B ) )  +  ( T `
 0h ) )  =  ( A  x.  ( T `  B ) ) )
174, 8, 163eqtr3d 2509 1  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( T `  ( A  .h  B )
)  =  ( A  x.  ( T `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   CCcc 9479   0cc0 9481    + caddc 9484    x. cmul 9486   ~Hchil 25498    +h cva 25499    .h csm 25500   0hc0v 25503   LinFnclf 25533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 25578  ax-hv0cl 25582  ax-hvaddid 25583  ax-hfvmul 25584  ax-hvmulid 25585
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9796  df-lnfn 26429
This theorem is referenced by:  lnfnaddmuli  26626  lnfnmul  26629  nmbdfnlbi  26630  nmcfnexi  26632  nmcfnlbi  26633  nlelshi  26641  riesz3i  26643
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