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Theorem lnosub 21353
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 9829 . . . 4  |-  -u 1  e.  CC
2 lnosub.1 . . . . 5  |-  X  =  ( BaseSet `  U )
3 eqid 2296 . . . . 5  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
4 eqid 2296 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
5 eqid 2296 . . . . 5  |-  ( +v
`  W )  =  ( +v `  W
)
6 eqid 2296 . . . . 5  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
7 eqid 2296 . . . . 5  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
8 lnosub.7 . . . . 5  |-  L  =  ( U  LnOp  W
)
92, 3, 4, 5, 6, 7, 8lnolin 21348 . . . 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 1274 . . 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 777 . 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 21217 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( ( -u 1
( .s OLD `  U
) B ) ( +v `  U ) A ) )
14133expb 1152 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A M B )  =  ( ( -u 1 ( .s OLD `  U
) B ) ( +v `  U ) A ) )
15143ad2antl1 1117 . . 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 5545 . 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 959 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  W  e.  NrmCVec )
182, 3, 8lnof 21349 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> ( BaseSet `  W
) )
19 simpl 443 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  ->  A  e.  X )
20 ffvelrn 5679 . . . 4  |-  ( ( T : X --> ( BaseSet `  W )  /\  A  e.  X )  ->  ( T `  A )  e.  ( BaseSet `  W )
)
2118, 19, 20syl2an 463 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( T `  A )  e.  (
BaseSet `  W ) )
22 simpr 447 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  ->  B  e.  X )
23 ffvelrn 5679 . . . 4  |-  ( ( T : X --> ( BaseSet `  W )  /\  B  e.  X )  ->  ( T `  B )  e.  ( BaseSet `  W )
)
2418, 22, 23syl2an 463 . . 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 21217 . . 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 1182 . 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 2338 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754   -ucneg 9054   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159   -vcnsb 21161    LnOp clno 21334
This theorem is referenced by:  blometi  21397  blocnilem  21398  ubthlem2  21466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-vs 21171  df-nmcv 21172  df-lno 21338
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