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Theorem hvsubcan 21655
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 21598 . . . 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 21598 . . . 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 2299 . 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 9815 . . . 4  |-  -u 1  e.  CC
7 hvmulcl 21595 . . . 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 21595 . . . . 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 21651 . . . 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 8797 . . . . . 6  |-  1  e.  CC
15 ax-1ne0 8808 . . . . . 6  |-  1  =/=  0
1614, 15negne0i 9123 . . . . 5  |-  -u 1  =/=  0
176, 16pm3.2i 441 . . . 4  |-  ( -u
1  e.  CC  /\  -u 1  =/=  0 )
18 hvmulcan 21653 . . . 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 1625    e. wcel 1686    =/= wne 2448  (class class class)co 5860   CCcc 8737   0cc0 8739   1c1 8740   -ucneg 9040   ~Hchil 21501    +h cva 21502    .h csm 21503    -h cmv 21507
This theorem is referenced by:  hvsubcan2  21656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-hfvadd 21582  ax-hvcom 21583  ax-hvass 21584  ax-hv0cl 21585  ax-hvaddid 21586  ax-hfvmul 21587  ax-hvmulid 21588  ax-hvmulass 21589  ax-hvdistr1 21590  ax-hvdistr2 21591  ax-hvmul0 21592
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-riota 6306  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-hvsub 21553
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