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Theorem lnoadd 22108
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 8982 . . 3  |-  1  e.  CC
2 lnoadd.1 . . . 4  |-  X  =  ( BaseSet `  U )
3 eqid 2388 . . . 4  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
4 lnoadd.5 . . . 4  |-  G  =  ( +v `  U
)
5 lnoadd.6 . . . 4  |-  H  =  ( +v `  W
)
6 eqid 2388 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
7 eqid 2388 . . . 4  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
8 lnoadd.7 . . . 4  |-  L  =  ( U  LnOp  W
)
92, 3, 4, 5, 6, 7, 8lnolin 22104 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  (
1  e.  CC  /\  A  e.  X  /\  B  e.  X )
)  ->  ( T `  ( ( 1 ( .s OLD `  U
) A ) G B ) )  =  ( ( 1 ( .s OLD `  W
) ( T `  A ) ) H ( T `  B
) ) )
101, 9mp3anr1 1276 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  ( ( 1 ( .s OLD `  U
) A ) G B ) )  =  ( ( 1 ( .s OLD `  W
) ( T `  A ) ) H ( T `  B
) ) )
11 simp1 957 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  U  e.  NrmCVec )
12 simpl 444 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  A  e.  X )
132, 6nvsid 21957 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 ( .s OLD `  U ) A )  =  A )
1411, 12, 13syl2an 464 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( 1 ( .s OLD `  U
) A )  =  A )
1514oveq1d 6036 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( (
1 ( .s OLD `  U ) A ) G B )  =  ( A G B ) )
1615fveq2d 5673 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  ( ( 1 ( .s OLD `  U
) A ) G B ) )  =  ( T `  ( A G B ) ) )
17 simpl2 961 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  W  e.  NrmCVec )
182, 3, 8lnof 22105 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> ( BaseSet `  W
) )
19 ffvelrn 5808 . . . . 5  |-  ( ( T : X --> ( BaseSet `  W )  /\  A  e.  X )  ->  ( T `  A )  e.  ( BaseSet `  W )
)
2018, 12, 19syl2an 464 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  A )  e.  (
BaseSet `  W ) )
213, 7nvsid 21957 . . . 4  |-  ( ( W  e.  NrmCVec  /\  ( T `  A )  e.  ( BaseSet `  W )
)  ->  ( 1 ( .s OLD `  W
) ( T `  A ) )  =  ( T `  A
) )
2217, 20, 21syl2anc 643 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( 1 ( .s OLD `  W
) ( T `  A ) )  =  ( T `  A
) )
2322oveq1d 6036 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( (
1 ( .s OLD `  W ) ( T `
 A ) ) H ( T `  B ) )  =  ( ( T `  A ) H ( T `  B ) ) )
2410, 16, 233eqtr3d 2428 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   -->wf 5391   ` cfv 5395  (class class class)co 6021   CCcc 8922   1c1 8925   NrmCVeccnv 21912   +vcpv 21913   BaseSetcba 21914   .s
OLDcns 21915    LnOp clno 22090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-1cn 8982
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-map 6957  df-vc 21874  df-nv 21920  df-va 21923  df-ba 21924  df-sm 21925  df-0v 21926  df-nmcv 21928  df-lno 22094
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