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Theorem nvnncan 22175
Description: Cancellation law for a normed complex vector space. (Contributed by NM, 17-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvsubsub23.1  |-  X  =  ( BaseSet `  U )
nvsubsub23.3  |-  M  =  ( -v `  U
)
Assertion
Ref Expression
nvnncan  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( A M B ) )  =  B )

Proof of Theorem nvnncan
StepHypRef Expression
1 nvsubsub23.1 . . . 4  |-  X  =  ( BaseSet `  U )
2 nvsubsub23.3 . . . 4  |-  M  =  ( -v `  U
)
31, 2nvmcl 22159 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  e.  X )
4 eqid 2442 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
5 eqid 2442 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
61, 4, 5, 2nvmval 22154 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  ( A M B )  e.  X )  ->  ( A M ( A M B ) )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U )
( A M B ) ) ) )
73, 6syld3an3 1230 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( A M B ) )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U )
( A M B ) ) ) )
8 simp1 958 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  U  e.  NrmCVec )
9 neg1cn 10098 . . . . . . 7  |-  -u 1  e.  CC
109a1i 11 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  -u 1  e.  CC )
11 simp2 959 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
121, 5nvscl 22138 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
139, 12mp3an2 1268 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
14133adant2 977 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
151, 4, 5nvdi 22142 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  A  e.  X  /\  ( -u 1 ( .s
OLD `  U ) B )  e.  X
) )  ->  ( -u 1 ( .s OLD `  U ) ( A ( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) )  =  ( (
-u 1 ( .s
OLD `  U ) A ) ( +v
`  U ) (
-u 1 ( .s
OLD `  U )
( -u 1 ( .s
OLD `  U ) B ) ) ) )
168, 10, 11, 14, 15syl13anc 1187 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) ( A ( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) )  =  ( (
-u 1 ( .s
OLD `  U ) A ) ( +v
`  U ) (
-u 1 ( .s
OLD `  U )
( -u 1 ( .s
OLD `  U ) B ) ) ) )
171, 4, 5, 2nvmval 22154 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) B ) ) )
1817oveq2d 6126 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) ( A M B ) )  =  ( -u 1
( .s OLD `  U
) ( A ( +v `  U ) ( -u 1 ( .s OLD `  U
) B ) ) ) )
19 ax-1cn 9079 . . . . . . . . . . . 12  |-  1  e.  CC
2019, 19mul2negi 9512 . . . . . . . . . . 11  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
21 1t1e1 10157 . . . . . . . . . . 11  |-  ( 1  x.  1 )  =  1
2220, 21eqtri 2462 . . . . . . . . . 10  |-  ( -u
1  x.  -u 1
)  =  1
2322oveq1i 6120 . . . . . . . . 9  |-  ( (
-u 1  x.  -u 1
) ( .s OLD `  U ) B )  =  ( 1 ( .s OLD `  U
) B )
241, 5nvsid 22139 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1 ( .s OLD `  U ) B )  =  B )
2523, 24syl5eq 2486 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( -u 1  x.  -u 1
) ( .s OLD `  U ) B )  =  B )
261, 5nvsass 22140 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  -u 1  e.  CC  /\  B  e.  X )
)  ->  ( ( -u 1  x.  -u 1
) ( .s OLD `  U ) B )  =  ( -u 1
( .s OLD `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
279, 26mp3anr1 1277 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( -u 1  e.  CC  /\  B  e.  X )
)  ->  ( ( -u 1  x.  -u 1
) ( .s OLD `  U ) B )  =  ( -u 1
( .s OLD `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
289, 27mpanr1 666 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( -u 1  x.  -u 1
) ( .s OLD `  U ) B )  =  ( -u 1
( .s OLD `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
2925, 28eqtr3d 2476 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  B  =  ( -u 1
( .s OLD `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
30293adant2 977 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  B  =  ( -u 1
( .s OLD `  U
) ( -u 1
( .s OLD `  U
) B ) ) )
3130oveq2d 6126 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( -u 1 ( .s
OLD `  U ) A ) ( +v
`  U ) B )  =  ( (
-u 1 ( .s
OLD `  U ) A ) ( +v
`  U ) (
-u 1 ( .s
OLD `  U )
( -u 1 ( .s
OLD `  U ) B ) ) ) )
3216, 18, 313eqtr4d 2484 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) ( A M B ) )  =  ( ( -u
1 ( .s OLD `  U ) A ) ( +v `  U
) B ) )
3332oveq2d 6126 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) ( A M B ) ) )  =  ( A ( +v `  U
) ( ( -u
1 ( .s OLD `  U ) A ) ( +v `  U
) B ) ) )
341, 5nvscl 22138 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  A  e.  X )  ->  ( -u 1 ( .s OLD `  U ) A )  e.  X )
359, 34mp3an2 1268 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( -u 1 ( .s OLD `  U ) A )  e.  X )
36353adant3 978 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) A )  e.  X )
37 simp3 960 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
381, 4nvass 22132 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  ( -u 1 ( .s
OLD `  U ) A )  e.  X  /\  B  e.  X
) )  ->  (
( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) A ) ) ( +v `  U ) B )  =  ( A ( +v `  U ) ( (
-u 1 ( .s
OLD `  U ) A ) ( +v
`  U ) B ) ) )
398, 11, 36, 37, 38syl13anc 1187 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) A ) ) ( +v `  U ) B )  =  ( A ( +v `  U ) ( (
-u 1 ( .s
OLD `  U ) A ) ( +v
`  U ) B ) ) )
40 eqid 2442 . . . . . 6  |-  ( 0vec `  U )  =  (
0vec `  U )
411, 4, 5, 40nvrinv 22165 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) )  =  ( 0vec `  U ) )
42413adant3 978 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) )  =  ( 0vec `  U ) )
4342oveq1d 6125 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( A ( +v
`  U ) (
-u 1 ( .s
OLD `  U ) A ) ) ( +v `  U ) B )  =  ( ( 0vec `  U
) ( +v `  U ) B ) )
4433, 39, 433eqtr2d 2480 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) ( A M B ) ) )  =  ( (
0vec `  U )
( +v `  U
) B ) )
451, 4, 40nv0lid 22148 . . 3  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U )
( +v `  U
) B )  =  B )
46453adant2 977 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( 0vec `  U )
( +v `  U
) B )  =  B )
477, 44, 463eqtrd 2478 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M ( A M B ) )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   ` cfv 5483  (class class class)co 6110   CCcc 9019   1c1 9022    x. cmul 9026   -ucneg 9323   NrmCVeccnv 22094   +vcpv 22095   BaseSetcba 22096   .s
OLDcns 22097   0veccn0v 22098   -vcnsb 22099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-ltxr 9156  df-sub 9324  df-neg 9325  df-grpo 21810  df-gid 21811  df-ginv 21812  df-gdiv 21813  df-ablo 21901  df-vc 22056  df-nv 22102  df-va 22105  df-ba 22106  df-sm 22107  df-0v 22108  df-vs 22109  df-nmcv 22110
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