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Theorem lnfnsubi 23549
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 10067 . . 3  |-  -u 1  e.  CC
2 lnfnl.1 . . . 4  |-  T  e. 
LinFn
32lnfnaddmuli 23548 . . 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 1266 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  ( -u 1  .h  B ) ) )  =  ( ( T `
 A )  +  ( -u 1  x.  ( T `  B
) ) ) )
5 hvsubval 22519 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
65fveq2d 5732 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( T `  ( A  +h  ( -u 1  .h  B ) ) ) )
72lnfnfi 23544 . . . 4  |-  T : ~H
--> CC
87ffvelrni 5869 . . 3  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
97ffvelrni 5869 . . 3  |-  ( B  e.  ~H  ->  ( T `  B )  e.  CC )
10 mulm1 9475 . . . . . 6  |-  ( ( T `  B )  e.  CC  ->  ( -u 1  x.  ( T `
 B ) )  =  -u ( T `  B ) )
1110oveq2d 6097 . . . . 5  |-  ( ( T `  B )  e.  CC  ->  (
( T `  A
)  +  ( -u
1  x.  ( T `
 B ) ) )  =  ( ( T `  A )  +  -u ( T `  B ) ) )
1211adantl 453 . . . 4  |-  ( ( ( T `  A
)  e.  CC  /\  ( T `  B )  e.  CC )  -> 
( ( T `  A )  +  (
-u 1  x.  ( T `  B )
) )  =  ( ( T `  A
)  +  -u ( T `  B )
) )
13 negsub 9349 . . . 4  |-  ( ( ( T `  A
)  e.  CC  /\  ( T `  B )  e.  CC )  -> 
( ( T `  A )  +  -u ( T `  B ) )  =  ( ( T `  A )  -  ( T `  B ) ) )
1412, 13eqtr2d 2469 . . 3  |-  ( ( ( T `  A
)  e.  CC  /\  ( T `  B )  e.  CC )  -> 
( ( T `  A )  -  ( T `  B )
)  =  ( ( T `  A )  +  ( -u 1  x.  ( T `  B
) ) ) )
158, 9, 14syl2an 464 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( T `  A )  -  ( T `  B )
)  =  ( ( T `  A )  +  ( -u 1  x.  ( T `  B
) ) ) )
164, 6, 153eqtr4d 2478 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( ( T `
 A )  -  ( T `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   -ucneg 9292   ~Hchil 22422    +h cva 22423    .h csm 22424    -h cmv 22428   LinFnclf 22457
This theorem is referenced by:  lnfnconi  23558  riesz3i  23565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-hilex 22502  ax-hv0cl 22506  ax-hvaddid 22507  ax-hfvmul 22508  ax-hvmulid 22509
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-neg 9294  df-hvsub 22474  df-lnfn 23351
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