HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hvsubcan Unicode version

Theorem hvsubcan 21649
Description: Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hvsubcan  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  =  ( A  -h  C )  <->  B  =  C ) )

Proof of Theorem hvsubcan
StepHypRef Expression
1 hvsubval 21592 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
213adant3 975 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) ) )
3 hvsubval 21592 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  C
)  =  ( A  +h  ( -u 1  .h  C ) ) )
433adant2 974 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  C )  =  ( A  +h  ( -u 1  .h  C ) ) )
52, 4eqeq12d 2298 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  =  ( A  -h  C )  <->  ( A  +h  ( -u 1  .h  B ) )  =  ( A  +h  ( -u 1  .h  C ) ) ) )
6 neg1cn 9809 . . . 4  |-  -u 1  e.  CC
7 hvmulcl 21589 . . . 4  |-  ( (
-u 1  e.  CC  /\  B  e.  ~H )  ->  ( -u 1  .h  B )  e.  ~H )
86, 7mpan 651 . . 3  |-  ( B  e.  ~H  ->  ( -u 1  .h  B )  e.  ~H )
9 hvmulcl 21589 . . . . 5  |-  ( (
-u 1  e.  CC  /\  C  e.  ~H )  ->  ( -u 1  .h  C )  e.  ~H )
106, 9mpan 651 . . . 4  |-  ( C  e.  ~H  ->  ( -u 1  .h  C )  e.  ~H )
11 hvaddcan 21645 . . . 4  |-  ( ( A  e.  ~H  /\  ( -u 1  .h  B
)  e.  ~H  /\  ( -u 1  .h  C
)  e.  ~H )  ->  ( ( A  +h  ( -u 1  .h  B
) )  =  ( A  +h  ( -u
1  .h  C ) )  <->  ( -u 1  .h  B )  =  (
-u 1  .h  C
) ) )
1210, 11syl3an3 1217 . . 3  |-  ( ( A  e.  ~H  /\  ( -u 1  .h  B
)  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  ( -u 1  .h  B
) )  =  ( A  +h  ( -u
1  .h  C ) )  <->  ( -u 1  .h  B )  =  (
-u 1  .h  C
) ) )
138, 12syl3an2 1216 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  ( -u 1  .h  B ) )  =  ( A  +h  ( -u 1  .h  C ) )  <->  ( -u 1  .h  B )  =  (
-u 1  .h  C
) ) )
14 ax-1cn 8791 . . . . . 6  |-  1  e.  CC
15 ax-1ne0 8802 . . . . . 6  |-  1  =/=  0
1614, 15negne0i 9117 . . . . 5  |-  -u 1  =/=  0
176, 16pm3.2i 441 . . . 4  |-  ( -u
1  e.  CC  /\  -u 1  =/=  0 )
18 hvmulcan 21647 . . . 4  |-  ( ( ( -u 1  e.  CC  /\  -u 1  =/=  0 )  /\  B  e.  ~H  /\  C  e. 
~H )  ->  (
( -u 1  .h  B
)  =  ( -u
1  .h  C )  <-> 
B  =  C ) )
1917, 18mp3an1 1264 . . 3  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( ( -u 1  .h  B )  =  (
-u 1  .h  C
)  <->  B  =  C
) )
20193adant1 973 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( -u 1  .h  B
)  =  ( -u
1  .h  C )  <-> 
B  =  C ) )
215, 13, 203bitrd 270 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  =  ( A  -h  C )  <->  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1685    =/= wne 2447  (class class class)co 5820   CCcc 8731   0cc0 8733   1c1 8734   -ucneg 9034   ~Hchil 21495    +h cva 21496    .h csm 21497    -h cmv 21501
This theorem is referenced by:  hvsubcan2  21650
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-hfvadd 21576  ax-hvcom 21577  ax-hvass 21578  ax-hv0cl 21579  ax-hvaddid 21580  ax-hfvmul 21581  ax-hvmulid 21582  ax-hvmulass 21583  ax-hvdistr1 21584  ax-hvdistr2 21585  ax-hvmul0 21586
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-hvsub 21547
  Copyright terms: Public domain W3C validator