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Theorem lnosub 22252
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 10059 . . . 4  |-  -u 1  e.  CC
2 lnosub.1 . . . . 5  |-  X  =  ( BaseSet `  U )
3 eqid 2435 . . . . 5  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
4 eqid 2435 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
5 eqid 2435 . . . . 5  |-  ( +v
`  W )  =  ( +v `  W
)
6 eqid 2435 . . . . 5  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
7 eqid 2435 . . . . 5  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
8 lnosub.7 . . . . 5  |-  L  =  ( U  LnOp  W
)
92, 3, 4, 5, 6, 7, 8lnolin 22247 . . . 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 22116 . . . . 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 5724 . 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 22248 . . . 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 5860 . . . 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 5860 . . . 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 22116 . . 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 2477 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 1652    e. wcel 1725   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   1c1 8983   -ucneg 9284   NrmCVeccnv 22055   +vcpv 22056   BaseSetcba 22057   .s
OLDcns 22058   -vcnsb 22060    LnOp clno 22233
This theorem is referenced by:  blometi  22296  blocnilem  22297  ubthlem2  22365
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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  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-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-sub 9285  df-neg 9286  df-grpo 21771  df-gid 21772  df-ginv 21773  df-gdiv 21774  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-vs 22070  df-nmcv 22071  df-lno 22237
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