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Theorem nvm 21254
Description: Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvm.1  |-  X  =  ( BaseSet `  U )
nvm.2  |-  G  =  ( +v `  U
)
nvm.3  |-  M  =  ( -v `  U
)
nvm.6  |-  N  =  (  /g  `  G
)
Assertion
Ref Expression
nvm  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A N B ) )

Proof of Theorem nvm
StepHypRef Expression
1 nvm.2 . . . . 5  |-  G  =  ( +v `  U
)
2 nvm.3 . . . . 5  |-  M  =  ( -v `  U
)
31, 2vsfval 21246 . . . 4  |-  M  =  (  /g  `  G
)
4 nvm.6 . . . 4  |-  N  =  (  /g  `  G
)
53, 4eqtr4i 2339 . . 3  |-  M  =  N
65oveqi 5913 . 2  |-  ( A M B )  =  ( A N B )
76a1i 10 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A N B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1633    e. wcel 1701   ` cfv 5292  (class class class)co 5900    /g cgs 20909   NrmCVeccnv 21195   +vcpv 21196   BaseSetcba 21197   -vcnsb 21200
This theorem is referenced by:  nvmval  21255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-grpo 20911  df-gdiv 20914  df-va 21206  df-vs 21210
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