MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvsubadd Structured version   Unicode version

Theorem nvsubadd 24057
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 2443 . . . . 5  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
4 nvsubadd.3 . . . . 5  |-  M  =  ( -v `  U
)
51, 2, 3, 4nvmval 24044 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A M B )  =  ( A G (
-u 1 ( .sOLD `  U ) B ) ) )
653adant3r3 1198 . . 3  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A M B )  =  ( A G ( -u
1 ( .sOLD `  U ) B ) ) )
76eqeq1d 2451 . 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 10446 . . . . . . . . . 10  |-  -u 1  e.  CC
91, 3nvscl 24028 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  -u 1  e.  CC  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) B )  e.  X )
108, 9mp3an2 1302 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) B )  e.  X )
11103adant2 1007 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( -u 1 ( .sOLD `  U ) B )  e.  X )
121, 2nvgcl 24020 . . . . . . . 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 1263 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( -u 1
( .sOLD `  U ) B ) )  e.  X )
14133adant3r3 1198 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G ( -u 1
( .sOLD `  U ) B ) )  e.  X )
15 simpr3 996 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
16 simpr2 995 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
1714, 15, 163jca 1168 . . . . 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 24026 . . . . 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 470 . . . 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 756 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
21 simprl 755 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  A  e.  X )
2210adantrl 715 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( -u 1
( .sOLD `  U ) B )  e.  X )
2320, 21, 223jca 1168 . . . . . . . 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 24023 . . . . . . . 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 470 . . . . . . 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 2443 . . . . . . . . . 10  |-  ( 0vec `  U )  =  (
0vec `  U )
271, 2, 3, 26nvrinv 24055 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( B G ( -u 1
( .sOLD `  U ) B ) )  =  ( 0vec `  U ) )
2827adantrl 715 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( B G ( -u 1
( .sOLD `  U ) B ) )  =  ( 0vec `  U ) )
2928oveq2d 6128 . . . . . . 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 24037 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( 0vec `  U
) )  =  A )
3130adantrr 716 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A G ( 0vec `  U
) )  =  A )
3225, 29, 313eqtrd 2479 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( B G ( A G ( -u 1 ( .sOLD `  U
) B ) ) )  =  A )
33323adantr3 1149 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B G ( A G ( -u 1 ( .sOLD `  U
) B ) ) )  =  A )
3433eqeq1d 2451 . . . 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 2445 . . 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5439  (class class class)co 6112   CCcc 9301   1c1 9304   -ucneg 9617   NrmCVeccnv 23984   +vcpv 23985   BaseSetcba 23986   .sOLDcns 23987   0veccn0v 23988   -vcnsb 23989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-ltxr 9444  df-sub 9618  df-neg 9619  df-grpo 23700  df-gid 23701  df-ginv 23702  df-gdiv 23703  df-ablo 23791  df-vc 23946  df-nv 23992  df-va 23995  df-ba 23996  df-sm 23997  df-0v 23998  df-vs 23999  df-nmcv 24000
This theorem is referenced by:  nvsubsub23  24064
  Copyright terms: Public domain W3C validator