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Theorem lnoadd 24156
Description: Addition property of a linear operator. (Contributed by NM, 7-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnoadd.1  |-  X  =  ( BaseSet `  U )
lnoadd.5  |-  G  =  ( +v `  U
)
lnoadd.6  |-  H  =  ( +v `  W
)
lnoadd.7  |-  L  =  ( U  LnOp  W
)
Assertion
Ref Expression
lnoadd  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  ( A G B ) )  =  ( ( T `  A
) H ( T `
 B ) ) )

Proof of Theorem lnoadd
StepHypRef Expression
1 ax-1cn 9338 . . 3  |-  1  e.  CC
2 lnoadd.1 . . . 4  |-  X  =  ( BaseSet `  U )
3 eqid 2441 . . . 4  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
4 lnoadd.5 . . . 4  |-  G  =  ( +v `  U
)
5 lnoadd.6 . . . 4  |-  H  =  ( +v `  W
)
6 eqid 2441 . . . 4  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
7 eqid 2441 . . . 4  |-  ( .sOLD `  W )  =  ( .sOLD `  W )
8 lnoadd.7 . . . 4  |-  L  =  ( U  LnOp  W
)
92, 3, 4, 5, 6, 7, 8lnolin 24152 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  (
1  e.  CC  /\  A  e.  X  /\  B  e.  X )
)  ->  ( T `  ( ( 1 ( .sOLD `  U
) A ) G B ) )  =  ( ( 1 ( .sOLD `  W
) ( T `  A ) ) H ( T `  B
) ) )
101, 9mp3anr1 1311 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  ( ( 1 ( .sOLD `  U
) A ) G B ) )  =  ( ( 1 ( .sOLD `  W
) ( T `  A ) ) H ( T `  B
) ) )
11 simp1 988 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  U  e.  NrmCVec )
12 simpl 457 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  A  e.  X )
132, 6nvsid 24005 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 ( .sOLD `  U ) A )  =  A )
1411, 12, 13syl2an 477 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( 1 ( .sOLD `  U ) A )  =  A )
1514oveq1d 6104 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( (
1 ( .sOLD `  U ) A ) G B )  =  ( A G B ) )
1615fveq2d 5693 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  ( ( 1 ( .sOLD `  U
) A ) G B ) )  =  ( T `  ( A G B ) ) )
17 simpl2 992 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  W  e.  NrmCVec )
182, 3, 8lnof 24153 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> ( BaseSet `  W
) )
19 ffvelrn 5839 . . . . 5  |-  ( ( T : X --> ( BaseSet `  W )  /\  A  e.  X )  ->  ( T `  A )  e.  ( BaseSet `  W )
)
2018, 12, 19syl2an 477 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  A )  e.  (
BaseSet `  W ) )
213, 7nvsid 24005 . . . 4  |-  ( ( W  e.  NrmCVec  /\  ( T `  A )  e.  ( BaseSet `  W )
)  ->  ( 1 ( .sOLD `  W ) ( T `
 A ) )  =  ( T `  A ) )
2217, 20, 21syl2anc 661 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( 1 ( .sOLD `  W ) ( T `
 A ) )  =  ( T `  A ) )
2322oveq1d 6104 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( (
1 ( .sOLD `  W ) ( T `
 A ) ) H ( T `  B ) )  =  ( ( T `  A ) H ( T `  B ) ) )
2410, 16, 233eqtr3d 2481 1  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  ( A G B ) )  =  ( ( T `  A
) H ( T `
 B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   -->wf 5412   ` cfv 5416  (class class class)co 6089   CCcc 9278   1c1 9281   NrmCVeccnv 23960   +vcpv 23961   BaseSetcba 23962   .sOLDcns 23963    LnOp clno 24138
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-1cn 9338
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-map 7214  df-vc 23922  df-nv 23968  df-va 23971  df-ba 23972  df-sm 23973  df-0v 23974  df-nmcv 23976  df-lno 24142
This theorem is referenced by: (None)
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