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Theorem lnfnl 23426
Description: Basic linearity property of a linear functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
lnfnl  |-  ( ( ( T  e.  LinFn  /\  A  e.  CC )  /\  ( B  e. 
~H  /\  C  e.  ~H ) )  ->  ( T `  ( ( A  .h  B )  +h  C ) )  =  ( ( A  x.  ( T `  B ) )  +  ( T `
 C ) ) )

Proof of Theorem lnfnl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ellnfn 23378 . . . . . 6  |-  ( T  e.  LinFn 
<->  ( T : ~H --> CC  /\  A. x  e.  CC  A. y  e. 
~H  A. z  e.  ~H  ( T `  ( ( x  .h  y )  +h  z ) )  =  ( ( x  x.  ( T `  y ) )  +  ( T `  z
) ) ) )
21simprbi 451 . . . . 5  |-  ( T  e.  LinFn  ->  A. x  e.  CC  A. y  e. 
~H  A. z  e.  ~H  ( T `  ( ( x  .h  y )  +h  z ) )  =  ( ( x  x.  ( T `  y ) )  +  ( T `  z
) ) )
3 oveq1 6080 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  .h  y )  =  ( A  .h  y ) )
43oveq1d 6088 . . . . . . . 8  |-  ( x  =  A  ->  (
( x  .h  y
)  +h  z )  =  ( ( A  .h  y )  +h  z ) )
54fveq2d 5724 . . . . . . 7  |-  ( x  =  A  ->  ( T `  ( (
x  .h  y )  +h  z ) )  =  ( T `  ( ( A  .h  y )  +h  z
) ) )
6 oveq1 6080 . . . . . . . 8  |-  ( x  =  A  ->  (
x  x.  ( T `
 y ) )  =  ( A  x.  ( T `  y ) ) )
76oveq1d 6088 . . . . . . 7  |-  ( x  =  A  ->  (
( x  x.  ( T `  y )
)  +  ( T `
 z ) )  =  ( ( A  x.  ( T `  y ) )  +  ( T `  z
) ) )
85, 7eqeq12d 2449 . . . . . 6  |-  ( x  =  A  ->  (
( T `  (
( x  .h  y
)  +h  z ) )  =  ( ( x  x.  ( T `
 y ) )  +  ( T `  z ) )  <->  ( T `  ( ( A  .h  y )  +h  z
) )  =  ( ( A  x.  ( T `  y )
)  +  ( T `
 z ) ) ) )
9 oveq2 6081 . . . . . . . . 9  |-  ( y  =  B  ->  ( A  .h  y )  =  ( A  .h  B ) )
109oveq1d 6088 . . . . . . . 8  |-  ( y  =  B  ->  (
( A  .h  y
)  +h  z )  =  ( ( A  .h  B )  +h  z ) )
1110fveq2d 5724 . . . . . . 7  |-  ( y  =  B  ->  ( T `  ( ( A  .h  y )  +h  z ) )  =  ( T `  (
( A  .h  B
)  +h  z ) ) )
12 fveq2 5720 . . . . . . . . 9  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
1312oveq2d 6089 . . . . . . . 8  |-  ( y  =  B  ->  ( A  x.  ( T `  y ) )  =  ( A  x.  ( T `  B )
) )
1413oveq1d 6088 . . . . . . 7  |-  ( y  =  B  ->  (
( A  x.  ( T `  y )
)  +  ( T `
 z ) )  =  ( ( A  x.  ( T `  B ) )  +  ( T `  z
) ) )
1511, 14eqeq12d 2449 . . . . . 6  |-  ( y  =  B  ->  (
( T `  (
( A  .h  y
)  +h  z ) )  =  ( ( A  x.  ( T `
 y ) )  +  ( T `  z ) )  <->  ( T `  ( ( A  .h  B )  +h  z
) )  =  ( ( A  x.  ( T `  B )
)  +  ( T `
 z ) ) ) )
16 oveq2 6081 . . . . . . . 8  |-  ( z  =  C  ->  (
( A  .h  B
)  +h  z )  =  ( ( A  .h  B )  +h  C ) )
1716fveq2d 5724 . . . . . . 7  |-  ( z  =  C  ->  ( T `  ( ( A  .h  B )  +h  z ) )  =  ( T `  (
( A  .h  B
)  +h  C ) ) )
18 fveq2 5720 . . . . . . . 8  |-  ( z  =  C  ->  ( T `  z )  =  ( T `  C ) )
1918oveq2d 6089 . . . . . . 7  |-  ( z  =  C  ->  (
( A  x.  ( T `  B )
)  +  ( T `
 z ) )  =  ( ( A  x.  ( T `  B ) )  +  ( T `  C
) ) )
2017, 19eqeq12d 2449 . . . . . 6  |-  ( z  =  C  ->  (
( T `  (
( A  .h  B
)  +h  z ) )  =  ( ( A  x.  ( T `
 B ) )  +  ( T `  z ) )  <->  ( T `  ( ( A  .h  B )  +h  C
) )  =  ( ( A  x.  ( T `  B )
)  +  ( T `
 C ) ) ) )
218, 15, 20rspc3v 3053 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A. x  e.  CC  A. y  e.  ~H  A. z  e.  ~H  ( T `  ( (
x  .h  y )  +h  z ) )  =  ( ( x  x.  ( T `  y ) )  +  ( T `  z
) )  ->  ( T `  ( ( A  .h  B )  +h  C ) )  =  ( ( A  x.  ( T `  B ) )  +  ( T `
 C ) ) ) )
222, 21syl5 30 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( T  e.  LinFn  ->  ( T `  ( ( A  .h  B )  +h  C ) )  =  ( ( A  x.  ( T `  B ) )  +  ( T `
 C ) ) ) )
23223expb 1154 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  ~H  /\  C  e.  ~H )
)  ->  ( T  e.  LinFn  ->  ( T `  ( ( A  .h  B )  +h  C
) )  =  ( ( A  x.  ( T `  B )
)  +  ( T `
 C ) ) ) )
2423impcom 420 . 2  |-  ( ( T  e.  LinFn  /\  ( A  e.  CC  /\  ( B  e.  ~H  /\  C  e.  ~H ) ) )  ->  ( T `  ( ( A  .h  B )  +h  C
) )  =  ( ( A  x.  ( T `  B )
)  +  ( T `
 C ) ) )
2524anassrs 630 1  |-  ( ( ( T  e.  LinFn  /\  A  e.  CC )  /\  ( B  e. 
~H  /\  C  e.  ~H ) )  ->  ( T `  ( ( A  .h  B )  +h  C ) )  =  ( ( A  x.  ( T `  B ) )  +  ( T `
 C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980    + caddc 8985    x. cmul 8987   ~Hchil 22414    +h cva 22415    .h csm 22416   LinFnclf 22449
This theorem is referenced by:  lnfnli  23535
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-hilex 22494
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-lnfn 23343
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