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Theorem nvsubadd 25748
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 2454 . . . . 5  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
4 nvsubadd.3 . . . . 5  |-  M  =  ( -v `  U
)
51, 2, 3, 4nvmval 25735 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A G (
-u 1 ( .sOLD `  U ) B ) ) )
653adant3r3 1205 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A M B )  =  ( A G ( -u
1 ( .sOLD `  U ) B ) ) )
76eqeq1d 2456 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M B )  =  C  <->  ( A G ( -u 1 ( .sOLD `  U
) B ) )  =  C ) )
8 neg1cn 10635 . . . . . . . . . 10  |-  -u 1  e.  CC
91, 3nvscl 25719 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) B )  e.  X )
108, 9mp3an2 1310 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) B )  e.  X )
11103adant2 1013 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) B )  e.  X )
121, 2nvgcl 25711 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  ( -u 1 ( .sOLD `  U ) B )  e.  X )  -> 
( A G (
-u 1 ( .sOLD `  U ) B ) )  e.  X )
1311, 12syld3an3 1271 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( -u 1
( .sOLD `  U ) B ) )  e.  X )
14133adant3r3 1205 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G ( -u 1
( .sOLD `  U ) B ) )  e.  X )
15 simpr3 1002 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
16 simpr2 1001 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
1714, 15, 163jca 1174 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G ( -u 1
( .sOLD `  U ) B ) )  e.  X  /\  C  e.  X  /\  B  e.  X )
)
181, 2nvlcan 25717 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  (
( A G (
-u 1 ( .sOLD `  U ) B ) )  e.  X  /\  C  e.  X  /\  B  e.  X ) )  -> 
( ( B G ( A G (
-u 1 ( .sOLD `  U ) B ) ) )  =  ( B G C )  <->  ( A G ( -u 1
( .sOLD `  U ) B ) )  =  C ) )
1917, 18syldan 468 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( B G ( A G ( -u 1 ( .sOLD `  U
) B ) ) )  =  ( B G C )  <->  ( A G ( -u 1
( .sOLD `  U ) B ) )  =  C ) )
20 simprr 755 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
21 simprl 754 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  A  e.  X )
2210adantrl 713 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( -u 1
( .sOLD `  U ) B )  e.  X )
2320, 21, 223jca 1174 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( B  e.  X  /\  A  e.  X  /\  ( -u
1 ( .sOLD `  U ) B )  e.  X ) )
241, 2nvadd12 25714 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( B  e.  X  /\  A  e.  X  /\  ( -u 1 ( .sOLD `  U ) B )  e.  X
) )  ->  ( B G ( A G ( -u 1 ( .sOLD `  U
) B ) ) )  =  ( A G ( B G ( -u 1 ( .sOLD `  U
) B ) ) ) )
2523, 24syldan 468 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( B G ( A G ( -u 1 ( .sOLD `  U
) B ) ) )  =  ( A G ( B G ( -u 1 ( .sOLD `  U
) B ) ) ) )
26 eqid 2454 . . . . . . . . . 10  |-  ( 0vec `  U )  =  (
0vec `  U )
271, 2, 3, 26nvrinv 25746 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( B G ( -u 1
( .sOLD `  U ) B ) )  =  ( 0vec `  U ) )
2827adantrl 713 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( B G ( -u 1
( .sOLD `  U ) B ) )  =  ( 0vec `  U ) )
2928oveq2d 6286 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A G ( B G ( -u 1 ( .sOLD `  U
) B ) ) )  =  ( A G ( 0vec `  U
) ) )
301, 2, 26nv0rid 25728 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( 0vec `  U
) )  =  A )
3130adantrr 714 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A G ( 0vec `  U
) )  =  A )
3225, 29, 313eqtrd 2499 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( B G ( A G ( -u 1 ( .sOLD `  U
) B ) ) )  =  A )
33323adantr3 1155 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B G ( A G ( -u 1 ( .sOLD `  U
) B ) ) )  =  A )
3433eqeq1d 2456 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( B G ( A G ( -u 1 ( .sOLD `  U
) B ) ) )  =  ( B G C )  <->  A  =  ( B G C ) ) )
3519, 34bitr3d 255 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G ( -u 1
( .sOLD `  U ) B ) )  =  C  <->  A  =  ( B G C ) ) )
36 eqcom 2463 . . 3  |-  ( A  =  ( B G C )  <->  ( B G C )  =  A )
3735, 36syl6bb 261 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G ( -u 1
( .sOLD `  U ) B ) )  =  C  <->  ( B G C )  =  A ) )
387, 37bitrd 253 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   CCcc 9479   1c1 9482   -ucneg 9797   NrmCVeccnv 25675   +vcpv 25676   BaseSetcba 25677   .sOLDcns 25678   0veccn0v 25679   -vcnsb 25680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9798  df-neg 9799  df-grpo 25391  df-gid 25392  df-ginv 25393  df-gdiv 25394  df-ablo 25482  df-vc 25637  df-nv 25683  df-va 25686  df-ba 25687  df-sm 25688  df-0v 25689  df-vs 25690  df-nmcv 25691
This theorem is referenced by:  nvsubsub23  25755
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