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Theorem lnosub 22108
Description: Subtraction 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
lnosub.1  |-  X  =  ( BaseSet `  U )
lnosub.5  |-  M  =  ( -v `  U
)
lnosub.6  |-  N  =  ( -v `  W
)
lnosub.7  |-  L  =  ( U  LnOp  W
)
Assertion
Ref Expression
lnosub  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  ( A M B ) )  =  ( ( T `  A
) N ( T `
 B ) ) )

Proof of Theorem lnosub
StepHypRef Expression
1 neg1cn 9999 . . . 4  |-  -u 1  e.  CC
2 lnosub.1 . . . . 5  |-  X  =  ( BaseSet `  U )
3 eqid 2387 . . . . 5  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
4 eqid 2387 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
5 eqid 2387 . . . . 5  |-  ( +v
`  W )  =  ( +v `  W
)
6 eqid 2387 . . . . 5  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
7 eqid 2387 . . . . 5  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
8 lnosub.7 . . . . 5  |-  L  =  ( U  LnOp  W
)
92, 3, 4, 5, 6, 7, 8lnolin 22103 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( -u 1  e.  CC  /\  B  e.  X  /\  A  e.  X )
)  ->  ( T `  ( ( -u 1
( .s OLD `  U
) B ) ( +v `  U ) A ) )  =  ( ( -u 1
( .s OLD `  W
) ( T `  B ) ) ( +v `  W ) ( T `  A
) ) )
101, 9mp3anr1 1276 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( B  e.  X  /\  A  e.  X )
)  ->  ( T `  ( ( -u 1
( .s OLD `  U
) B ) ( +v `  U ) A ) )  =  ( ( -u 1
( .s OLD `  W
) ( T `  B ) ) ( +v `  W ) ( T `  A
) ) )
1110ancom2s 778 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  ( ( -u 1
( .s OLD `  U
) B ) ( +v `  U ) A ) )  =  ( ( -u 1
( .s OLD `  W
) ( T `  B ) ) ( +v `  W ) ( T `  A
) ) )
12 lnosub.5 . . . . . 6  |-  M  =  ( -v `  U
)
132, 4, 6, 12nvmval2 21972 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( ( -u 1
( .s OLD `  U
) B ) ( +v `  U ) A ) )
14133expb 1154 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A M B )  =  ( ( -u 1 ( .s OLD `  U
) B ) ( +v `  U ) A ) )
15143ad2antl1 1119 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A M B )  =  ( ( -u 1 ( .s OLD `  U
) B ) ( +v `  U ) A ) )
1615fveq2d 5672 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  ( A M B ) )  =  ( T `  ( (
-u 1 ( .s
OLD `  U ) B ) ( +v
`  U ) A ) ) )
17 simpl2 961 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  W  e.  NrmCVec )
182, 3, 8lnof 22104 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> ( BaseSet `  W
) )
19 simpl 444 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  ->  A  e.  X )
20 ffvelrn 5807 . . . 4  |-  ( ( T : X --> ( BaseSet `  W )  /\  A  e.  X )  ->  ( T `  A )  e.  ( BaseSet `  W )
)
2118, 19, 20syl2an 464 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  A )  e.  (
BaseSet `  W ) )
22 simpr 448 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  ->  B  e.  X )
23 ffvelrn 5807 . . . 4  |-  ( ( T : X --> ( BaseSet `  W )  /\  B  e.  X )  ->  ( T `  B )  e.  ( BaseSet `  W )
)
2418, 22, 23syl2an 464 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  B )  e.  (
BaseSet `  W ) )
25 lnosub.6 . . . 4  |-  N  =  ( -v `  W
)
263, 5, 7, 25nvmval2 21972 . . 3  |-  ( ( W  e.  NrmCVec  /\  ( T `  A )  e.  ( BaseSet `  W )  /\  ( T `  B
)  e.  ( BaseSet `  W ) )  -> 
( ( T `  A ) N ( T `  B ) )  =  ( (
-u 1 ( .s
OLD `  W )
( T `  B
) ) ( +v
`  W ) ( T `  A ) ) )
2717, 21, 24, 26syl3anc 1184 . 2  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( T `  A ) N ( T `  B ) )  =  ( ( -u 1
( .s OLD `  W
) ( T `  B ) ) ( +v `  W ) ( T `  A
) ) )
2811, 16, 273eqtr4d 2429 1  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  ( A M B ) )  =  ( ( T `  A
) N ( T `
 B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   -->wf 5390   ` cfv 5394  (class class class)co 6020   CCcc 8921   1c1 8924   -ucneg 9224   NrmCVeccnv 21911   +vcpv 21912   BaseSetcba 21913   .s
OLDcns 21914   -vcnsb 21916    LnOp clno 22089
This theorem is referenced by:  blometi  22152  blocnilem  22153  ubthlem2  22221
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-ltxr 9058  df-sub 9225  df-neg 9226  df-grpo 21627  df-gid 21628  df-ginv 21629  df-gdiv 21630  df-ablo 21718  df-vc 21873  df-nv 21919  df-va 21922  df-ba 21923  df-sm 21924  df-0v 21925  df-vs 21926  df-nmcv 21927  df-lno 22093
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