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Theorem nvsubadd 22128
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 2435 . . . . 5  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 nvsubadd.3 . . . . 5  |-  M  =  ( -v `  U
)
51, 2, 3, 4nvmval 22115 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A G (
-u 1 ( .s
OLD `  U ) B ) ) )
653adant3r3 1164 . . 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 2443 . 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 10059 . . . . . . . . . 10  |-  -u 1  e.  CC
91, 3nvscl 22099 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
108, 9mp3an2 1267 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
11103adant2 976 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .s OLD `  U ) B )  e.  X )
121, 2nvgcl 22091 . . . . . . . 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 1229 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( -u 1
( .s OLD `  U
) B ) )  e.  X )
14133adant3r3 1164 . . . . . 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 965 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
16 simpr2 964 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
1714, 15, 163jca 1134 . . . . 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 22097 . . . . 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 457 . . . 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 734 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
21 simprl 733 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  A  e.  X )
2210adantrl 697 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( -u 1
( .s OLD `  U
) B )  e.  X )
2320, 21, 223jca 1134 . . . . . . . 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 22094 . . . . . . . 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 457 . . . . . . 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 2435 . . . . . . . . . 10  |-  ( 0vec `  U )  =  (
0vec `  U )
271, 2, 3, 26nvrinv 22126 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( B G ( -u 1
( .s OLD `  U
) B ) )  =  ( 0vec `  U
) )
2827adantrl 697 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( B G ( -u 1
( .s OLD `  U
) B ) )  =  ( 0vec `  U
) )
2928oveq2d 6089 . . . . . . 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 22108 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( 0vec `  U
) )  =  A )
3130adantrr 698 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A G ( 0vec `  U
) )  =  A )
3225, 29, 313eqtrd 2471 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( B G ( A G ( -u 1 ( .s OLD `  U
) B ) ) )  =  A )
33323adantr3 1118 . . . . 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 2443 . . . 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 247 . . 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 2437 . . 3  |-  ( A  =  ( B G C )  <->  ( B G C )  =  A )
3735, 36syl6bb 253 . 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 245 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980   1c1 8983   -ucneg 9284   NrmCVeccnv 22055   +vcpv 22056   BaseSetcba 22057   .s
OLDcns 22058   0veccn0v 22059   -vcnsb 22060
This theorem is referenced by:  nvsubsub23  22135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-sub 9285  df-neg 9286  df-grpo 21771  df-gid 21772  df-ginv 21773  df-gdiv 21774  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-vs 22070  df-nmcv 22071
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