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Theorem lnfnsubi 25401
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 10417 . . 3  |-  -u 1  e.  CC
2 lnfnl.1 . . . 4  |-  T  e. 
LinFn
32lnfnaddmuli 25400 . . 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 1301 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  ( -u 1  .h  B ) ) )  =  ( ( T `
 A )  +  ( -u 1  x.  ( T `  B
) ) ) )
5 hvsubval 24369 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
65fveq2d 5690 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( T `  ( A  +h  ( -u 1  .h  B ) ) ) )
72lnfnfi 25396 . . . 4  |-  T : ~H
--> CC
87ffvelrni 5837 . . 3  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
97ffvelrni 5837 . . 3  |-  ( B  e.  ~H  ->  ( T `  B )  e.  CC )
10 mulm1 9778 . . . . . 6  |-  ( ( T `  B )  e.  CC  ->  ( -u 1  x.  ( T `
 B ) )  =  -u ( T `  B ) )
1110oveq2d 6102 . . . . 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 9649 . . . 4  |-  ( ( ( T `  A
)  e.  CC  /\  ( T `  B )  e.  CC )  -> 
( ( T `  A )  +  -u ( T `  B ) )  =  ( ( T `  A )  -  ( T `  B ) ) )
1412, 13eqtr2d 2471 . . 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 2480 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 1369    e. wcel 1756   ` cfv 5413  (class class class)co 6086   CCcc 9272   1c1 9275    + caddc 9277    x. cmul 9279    - cmin 9587   -ucneg 9588   ~Hchil 24272    +h cva 24273    .h csm 24274    -h cmv 24278   LinFnclf 24307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-hilex 24352  ax-hv0cl 24356  ax-hvaddid 24357  ax-hfvmul 24358  ax-hvmulid 24359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-ltxr 9415  df-sub 9589  df-neg 9590  df-hvsub 24324  df-lnfn 25203
This theorem is referenced by:  lnfnconi  25410  riesz3i  25417
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