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Theorem nvsubadd 21213
Description: Relationship between vector subtraction and addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvsubadd.1  |-  X  =  ( BaseSet `  U )
nvsubadd.2  |-  G  =  ( +v `  U
)
nvsubadd.3  |-  M  =  ( -v `  U
)
Assertion
Ref Expression
nvsubadd  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M B )  =  C  <->  ( B G C )  =  A ) )

Proof of Theorem nvsubadd
StepHypRef Expression
1 nvsubadd.1 . . . . 5  |-  X  =  ( BaseSet `  U )
2 nvsubadd.2 . . . . 5  |-  G  =  ( +v `  U
)
3 eqid 2283 . . . . 5  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 nvsubadd.3 . . . . 5  |-  M  =  ( -v `  U
)
51, 2, 3, 4nvmval 21200 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A G (
-u 1 ( .s
OLD `  U ) B ) ) )
653adant3r3 1162 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A M B )  =  ( A G ( -u
1 ( .s OLD `  U ) B ) ) )
76eqeq1d 2291 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M B )  =  C  <->  ( A G ( -u 1 ( .s OLD `  U
) B ) )  =  C ) )
8 neg1cn 9813 . . . . . . . . . 10  |-  -u 1  e.  CC
91, 3nvscl 21184 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
108, 9mp3an2 1265 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
11103adant2 974 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
121, 2nvgcl 21176 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  ( -u 1 ( .s OLD `  U ) B )  e.  X )  -> 
( A G (
-u 1 ( .s
OLD `  U ) B ) )  e.  X )
1311, 12syld3an3 1227 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( -u 1
( .s OLD `  U
) B ) )  e.  X )
14133adant3r3 1162 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G ( -u 1
( .s OLD `  U
) B ) )  e.  X )
15 simpr3 963 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
16 simpr2 962 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
1714, 15, 163jca 1132 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G ( -u 1
( .s OLD `  U
) B ) )  e.  X  /\  C  e.  X  /\  B  e.  X ) )
181, 2nvlcan 21182 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  (
( A G (
-u 1 ( .s
OLD `  U ) B ) )  e.  X  /\  C  e.  X  /\  B  e.  X ) )  -> 
( ( B G ( A G (
-u 1 ( .s
OLD `  U ) B ) ) )  =  ( B G C )  <->  ( A G ( -u 1
( .s OLD `  U
) B ) )  =  C ) )
1917, 18syldan 456 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( B G ( A G ( -u 1 ( .s OLD `  U
) B ) ) )  =  ( B G C )  <->  ( A G ( -u 1
( .s OLD `  U
) B ) )  =  C ) )
20 simprr 733 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
21 simprl 732 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  A  e.  X )
2210adantrl 696 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( -u 1
( .s OLD `  U
) B )  e.  X )
2320, 21, 223jca 1132 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( B  e.  X  /\  A  e.  X  /\  ( -u
1 ( .s OLD `  U ) B )  e.  X ) )
241, 2nvadd12 21179 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( B  e.  X  /\  A  e.  X  /\  ( -u 1 ( .s
OLD `  U ) B )  e.  X
) )  ->  ( B G ( A G ( -u 1 ( .s OLD `  U
) B ) ) )  =  ( A G ( B G ( -u 1 ( .s OLD `  U
) B ) ) ) )
2523, 24syldan 456 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( B G ( A G ( -u 1 ( .s OLD `  U
) B ) ) )  =  ( A G ( B G ( -u 1 ( .s OLD `  U
) B ) ) ) )
26 eqid 2283 . . . . . . . . . 10  |-  ( 0vec `  U )  =  (
0vec `  U )
271, 2, 3, 26nvrinv 21211 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( B G ( -u 1
( .s OLD `  U
) B ) )  =  ( 0vec `  U
) )
2827adantrl 696 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( B G ( -u 1
( .s OLD `  U
) B ) )  =  ( 0vec `  U
) )
2928oveq2d 5874 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A G ( B G ( -u 1 ( .s OLD `  U
) B ) ) )  =  ( A G ( 0vec `  U
) ) )
301, 2, 26nv0rid 21193 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( 0vec `  U
) )  =  A )
3130adantrr 697 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A G ( 0vec `  U
) )  =  A )
3225, 29, 313eqtrd 2319 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( B G ( A G ( -u 1 ( .s OLD `  U
) B ) ) )  =  A )
33323adantr3 1116 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B G ( A G ( -u 1 ( .s OLD `  U
) B ) ) )  =  A )
3433eqeq1d 2291 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( B G ( A G ( -u 1 ( .s OLD `  U
) B ) ) )  =  ( B G C )  <->  A  =  ( B G C ) ) )
3519, 34bitr3d 246 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G ( -u 1
( .s OLD `  U
) B ) )  =  C  <->  A  =  ( B G C ) ) )
36 eqcom 2285 . . 3  |-  ( A  =  ( B G C )  <->  ( B G C )  =  A )
3735, 36syl6bb 252 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G ( -u 1
( .s OLD `  U
) B ) )  =  C  <->  ( B G C )  =  A ) )
387, 37bitrd 244 1  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M B )  =  C  <->  ( B G C )  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738   -ucneg 9038   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   0veccn0v 21144   -vcnsb 21145
This theorem is referenced by:  nvsubsub23  21220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156
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