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

Theorem hvaddcani 22598
Description: Cancellation law for vector addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvnegdi.1  |-  A  e. 
~H
hvnegdi.2  |-  B  e. 
~H
hvaddcan.3  |-  C  e. 
~H
Assertion
Ref Expression
hvaddcani  |-  ( ( A  +h  B )  =  ( A  +h  C )  <->  B  =  C )

Proof of Theorem hvaddcani
StepHypRef Expression
1 oveq1 6117 . . 3  |-  ( ( A  +h  B )  =  ( A  +h  C )  ->  (
( A  +h  B
)  +h  ( -u
1  .h  A ) )  =  ( ( A  +h  C )  +h  ( -u 1  .h  A ) ) )
2 hvnegdi.1 . . . . 5  |-  A  e. 
~H
3 hvnegdi.2 . . . . 5  |-  B  e. 
~H
4 neg1cn 10098 . . . . . 6  |-  -u 1  e.  CC
54, 2hvmulcli 22548 . . . . 5  |-  ( -u
1  .h  A )  e.  ~H
62, 3, 5hvadd32i 22587 . . . 4  |-  ( ( A  +h  B )  +h  ( -u 1  .h  A ) )  =  ( ( A  +h  ( -u 1  .h  A
) )  +h  B
)
72hvnegidi 22563 . . . . 5  |-  ( A  +h  ( -u 1  .h  A ) )  =  0h
87oveq1i 6120 . . . 4  |-  ( ( A  +h  ( -u
1  .h  A ) )  +h  B )  =  ( 0h  +h  B )
93hvaddid2i 22562 . . . 4  |-  ( 0h 
+h  B )  =  B
106, 8, 93eqtri 2466 . . 3  |-  ( ( A  +h  B )  +h  ( -u 1  .h  A ) )  =  B
11 hvaddcan.3 . . . . 5  |-  C  e. 
~H
122, 11, 5hvadd32i 22587 . . . 4  |-  ( ( A  +h  C )  +h  ( -u 1  .h  A ) )  =  ( ( A  +h  ( -u 1  .h  A
) )  +h  C
)
137oveq1i 6120 . . . 4  |-  ( ( A  +h  ( -u
1  .h  A ) )  +h  C )  =  ( 0h  +h  C )
1411hvaddid2i 22562 . . . 4  |-  ( 0h 
+h  C )  =  C
1512, 13, 143eqtri 2466 . . 3  |-  ( ( A  +h  C )  +h  ( -u 1  .h  A ) )  =  C
161, 10, 153eqtr3g 2497 . 2  |-  ( ( A  +h  B )  =  ( A  +h  C )  ->  B  =  C )
17 oveq2 6118 . 2  |-  ( B  =  C  ->  ( A  +h  B )  =  ( A  +h  C
) )
1816, 17impbii 182 1  |-  ( ( A  +h  B )  =  ( A  +h  C )  <->  B  =  C )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1727  (class class class)co 6110   1c1 9022   -ucneg 9323   ~Hchil 22453    +h cva 22454    .h csm 22455   0hc0v 22458
This theorem is referenced by:  hvsubaddi  22599  hvaddcan  22603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-hvcom 22535  ax-hvass 22536  ax-hv0cl 22537  ax-hvaddid 22538  ax-hfvmul 22539  ax-hvmulid 22540  ax-hvdistr2 22543  ax-hvmul0 22544
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-riota 6578  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-ltxr 9156  df-sub 9324  df-neg 9325  df-hvsub 22505
  Copyright terms: Public domain W3C validator