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Theorem nvsubadd 21229
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 2296 . . . . 5  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 nvsubadd.3 . . . . 5  |-  M  =  ( -v `  U
)
51, 2, 3, 4nvmval 21216 . . . 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 2304 . 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 9829 . . . . . . . . . 10  |-  -u 1  e.  CC
91, 3nvscl 21200 . . . . . . . . . 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 21192 . . . . . . . 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 21198 . . . . 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 21195 . . . . . . . 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 2296 . . . . . . . . . 10  |-  ( 0vec `  U )  =  (
0vec `  U )
271, 2, 3, 26nvrinv 21227 . . . . . . . . 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 5890 . . . . . . 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 21209 . . . . . . . 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 2332 . . . . . 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 2304 . . . 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 2298 . . 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 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754   -ucneg 9054   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159   0veccn0v 21160   -vcnsb 21161
This theorem is referenced by:  nvsubsub23  21236
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-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
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