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Theorem nvsubsub23 22174
Description: Swap subtrahend and result of vector subtraction. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvsubsub23.1  |-  X  =  ( BaseSet `  U )
nvsubsub23.3  |-  M  =  ( -v `  U
)
Assertion
Ref Expression
nvsubsub23  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M B )  =  C  <->  ( A M C )  =  B ) )

Proof of Theorem nvsubsub23
StepHypRef Expression
1 nvsubsub23.1 . . . . 5  |-  X  =  ( BaseSet `  U )
2 eqid 2442 . . . . 5  |-  ( +v
`  U )  =  ( +v `  U
)
31, 2nvcom 22131 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  C  e.  X )  ->  ( B ( +v `  U ) C )  =  ( C ( +v `  U ) B ) )
433adant3r1 1163 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B
( +v `  U
) C )  =  ( C ( +v
`  U ) B ) )
54eqeq1d 2450 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( B ( +v `  U ) C )  =  A  <->  ( C
( +v `  U
) B )  =  A ) )
6 nvsubsub23.3 . . 3  |-  M  =  ( -v `  U
)
71, 2, 6nvsubadd 22167 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M B )  =  C  <->  ( B ( +v `  U ) C )  =  A ) )
8 3ancomb 946 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  <->  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)
91, 2, 6nvsubadd 22167 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)  ->  ( ( A M C )  =  B  <->  ( C ( +v `  U ) B )  =  A ) )
108, 9sylan2b 463 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M C )  =  B  <->  ( C ( +v `  U ) B )  =  A ) )
115, 7, 103bitr4d 278 1  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M B )  =  C  <->  ( A M C )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   ` cfv 5483  (class class class)co 6110   NrmCVeccnv 22094   +vcpv 22095   BaseSetcba 22096   -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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