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Theorem lnfnsubi 26641
Description: Subtraction property for a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
lnfnl.1  |-  T  e. 
LinFn
Assertion
Ref Expression
lnfnsubi  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( ( T `
 A )  -  ( T `  B ) ) )

Proof of Theorem lnfnsubi
StepHypRef Expression
1 neg1cn 10635 . . 3  |-  -u 1  e.  CC
2 lnfnl.1 . . . 4  |-  T  e. 
LinFn
32lnfnaddmuli 26640 . . 3  |-  ( (
-u 1  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  ( -u 1  .h  B ) ) )  =  ( ( T `
 A )  +  ( -u 1  x.  ( T `  B
) ) ) )
41, 3mp3an1 1311 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  ( -u 1  .h  B ) ) )  =  ( ( T `
 A )  +  ( -u 1  x.  ( T `  B
) ) ) )
5 hvsubval 25609 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
65fveq2d 5868 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( T `  ( A  +h  ( -u 1  .h  B ) ) ) )
72lnfnfi 26636 . . . 4  |-  T : ~H
--> CC
87ffvelrni 6018 . . 3  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
97ffvelrni 6018 . . 3  |-  ( B  e.  ~H  ->  ( T `  B )  e.  CC )
10 mulm1 9994 . . . . . 6  |-  ( ( T `  B )  e.  CC  ->  ( -u 1  x.  ( T `
 B ) )  =  -u ( T `  B ) )
1110oveq2d 6298 . . . . 5  |-  ( ( T `  B )  e.  CC  ->  (
( T `  A
)  +  ( -u
1  x.  ( T `
 B ) ) )  =  ( ( T `  A )  +  -u ( T `  B ) ) )
1211adantl 466 . . . 4  |-  ( ( ( T `  A
)  e.  CC  /\  ( T `  B )  e.  CC )  -> 
( ( T `  A )  +  (
-u 1  x.  ( T `  B )
) )  =  ( ( T `  A
)  +  -u ( T `  B )
) )
13 negsub 9863 . . . 4  |-  ( ( ( T `  A
)  e.  CC  /\  ( T `  B )  e.  CC )  -> 
( ( T `  A )  +  -u ( T `  B ) )  =  ( ( T `  A )  -  ( T `  B ) ) )
1412, 13eqtr2d 2509 . . 3  |-  ( ( ( T `  A
)  e.  CC  /\  ( T `  B )  e.  CC )  -> 
( ( T `  A )  -  ( T `  B )
)  =  ( ( T `  A )  +  ( -u 1  x.  ( T `  B
) ) ) )
158, 9, 14syl2an 477 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( T `  A )  -  ( T `  B )
)  =  ( ( T `  A )  +  ( -u 1  x.  ( T `  B
) ) ) )
164, 6, 153eqtr4d 2518 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( ( T `
 A )  -  ( T `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   CCcc 9486   1c1 9489    + caddc 9491    x. cmul 9493    - cmin 9801   -ucneg 9802   ~Hchil 25512    +h cva 25513    .h csm 25514    -h cmv 25518   LinFnclf 25547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-hilex 25592  ax-hv0cl 25596  ax-hvaddid 25597  ax-hfvmul 25598  ax-hvmulid 25599
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-ltxr 9629  df-sub 9803  df-neg 9804  df-hvsub 25564  df-lnfn 26443
This theorem is referenced by:  lnfnconi  26650  riesz3i  26657
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