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Theorem lnfnsubi 27378
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 10680 . . 3  |-  -u 1  e.  CC
2 lnfnl.1 . . . 4  |-  T  e. 
LinFn
32lnfnaddmuli 27377 . . 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 1313 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  ( -u 1  .h  B ) ) )  =  ( ( T `
 A )  +  ( -u 1  x.  ( T `  B
) ) ) )
5 hvsubval 26347 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
65fveq2d 5853 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( T `  ( A  +h  ( -u 1  .h  B ) ) ) )
72lnfnfi 27373 . . . 4  |-  T : ~H
--> CC
87ffvelrni 6008 . . 3  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
97ffvelrni 6008 . . 3  |-  ( B  e.  ~H  ->  ( T `  B )  e.  CC )
10 mulm1 10039 . . . . . 6  |-  ( ( T `  B )  e.  CC  ->  ( -u 1  x.  ( T `
 B ) )  =  -u ( T `  B ) )
1110oveq2d 6294 . . . . 5  |-  ( ( T `  B )  e.  CC  ->  (
( T `  A
)  +  ( -u
1  x.  ( T `
 B ) ) )  =  ( ( T `  A )  +  -u ( T `  B ) ) )
1211adantl 464 . . . 4  |-  ( ( ( T `  A
)  e.  CC  /\  ( T `  B )  e.  CC )  -> 
( ( T `  A )  +  (
-u 1  x.  ( T `  B )
) )  =  ( ( T `  A
)  +  -u ( T `  B )
) )
13 negsub 9903 . . . 4  |-  ( ( ( T `  A
)  e.  CC  /\  ( T `  B )  e.  CC )  -> 
( ( T `  A )  +  -u ( T `  B ) )  =  ( ( T `  A )  -  ( T `  B ) ) )
1412, 13eqtr2d 2444 . . 3  |-  ( ( ( T `  A
)  e.  CC  /\  ( T `  B )  e.  CC )  -> 
( ( T `  A )  -  ( T `  B )
)  =  ( ( T `  A )  +  ( -u 1  x.  ( T `  B
) ) ) )
158, 9, 14syl2an 475 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( T `  A )  -  ( T `  B )
)  =  ( ( T `  A )  +  ( -u 1  x.  ( T `  B
) ) ) )
164, 6, 153eqtr4d 2453 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 367    = wceq 1405    e. wcel 1842   ` cfv 5569  (class class class)co 6278   CCcc 9520   1c1 9523    + caddc 9525    x. cmul 9527    - cmin 9841   -ucneg 9842   ~Hchil 26250    +h cva 26251    .h csm 26252    -h cmv 26256   LinFnclf 26285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-hilex 26330  ax-hv0cl 26334  ax-hvaddid 26335  ax-hfvmul 26336  ax-hvmulid 26337
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-ltxr 9663  df-sub 9843  df-neg 9844  df-hvsub 26302  df-lnfn 27180
This theorem is referenced by:  lnfnconi  27387  riesz3i  27394
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