HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  lnfnsubi Structured version   Unicode version

Theorem lnfnsubi 25595
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 10529 . . 3  |-  -u 1  e.  CC
2 lnfnl.1 . . . 4  |-  T  e. 
LinFn
32lnfnaddmuli 25594 . . 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 1302 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  ( -u 1  .h  B ) ) )  =  ( ( T `
 A )  +  ( -u 1  x.  ( T `  B
) ) ) )
5 hvsubval 24563 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
65fveq2d 5796 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  -h  B ) )  =  ( T `  ( A  +h  ( -u 1  .h  B ) ) ) )
72lnfnfi 25590 . . . 4  |-  T : ~H
--> CC
87ffvelrni 5944 . . 3  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
97ffvelrni 5944 . . 3  |-  ( B  e.  ~H  ->  ( T `  B )  e.  CC )
10 mulm1 9890 . . . . . 6  |-  ( ( T `  B )  e.  CC  ->  ( -u 1  x.  ( T `
 B ) )  =  -u ( T `  B ) )
1110oveq2d 6209 . . . . 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 9761 . . . 4  |-  ( ( ( T `  A
)  e.  CC  /\  ( T `  B )  e.  CC )  -> 
( ( T `  A )  +  -u ( T `  B ) )  =  ( ( T `  A )  -  ( T `  B ) ) )
1412, 13eqtr2d 2493 . . 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 2502 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 1370    e. wcel 1758   ` cfv 5519  (class class class)co 6193   CCcc 9384   1c1 9387    + caddc 9389    x. cmul 9391    - cmin 9699   -ucneg 9700   ~Hchil 24466    +h cva 24467    .h csm 24468    -h cmv 24472   LinFnclf 24501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-hilex 24546  ax-hv0cl 24550  ax-hvaddid 24551  ax-hfvmul 24552  ax-hvmulid 24553
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-ltxr 9527  df-sub 9701  df-neg 9702  df-hvsub 24518  df-lnfn 25397
This theorem is referenced by:  lnfnconi  25604  riesz3i  25611
  Copyright terms: Public domain W3C validator