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Theorem lnoadd 25799
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 9567 . . 3  |-  1  e.  CC
2 lnoadd.1 . . . 4  |-  X  =  ( BaseSet `  U )
3 eqid 2457 . . . 4  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
4 lnoadd.5 . . . 4  |-  G  =  ( +v `  U
)
5 lnoadd.6 . . . 4  |-  H  =  ( +v `  W
)
6 eqid 2457 . . . 4  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
7 eqid 2457 . . . 4  |-  ( .sOLD `  W )  =  ( .sOLD `  W )
8 lnoadd.7 . . . 4  |-  L  =  ( U  LnOp  W
)
92, 3, 4, 5, 6, 7, 8lnolin 25795 . . 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 1321 . 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 996 . . . . 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 25648 . . . . 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 6311 . . 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 5876 . 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 1000 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  W  e.  NrmCVec )
182, 3, 8lnof 25796 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> ( BaseSet `  W
) )
19 ffvelrn 6030 . . . . 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 25648 . . . 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 6311 . 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 2506 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 973    = wceq 1395    e. wcel 1819   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   1c1 9510   NrmCVeccnv 25603   +vcpv 25604   BaseSetcba 25605   .sOLDcns 25606    LnOp clno 25781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-1cn 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-map 7440  df-vc 25565  df-nv 25611  df-va 25614  df-ba 25615  df-sm 25616  df-0v 25617  df-nmcv 25619  df-lno 25785
This theorem is referenced by: (None)
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