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

Theorem hvpncan 24360
Description: Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
hvpncan  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  -h  B
)  =  A )

Proof of Theorem hvpncan
StepHypRef Expression
1 hvaddcl 24333 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  e.  ~H )
2 hvsubval 24337 . . 3  |-  ( ( ( A  +h  B
)  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  -h  B
)  =  ( ( A  +h  B )  +h  ( -u 1  .h  B ) ) )
31, 2sylancom 662 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  -h  B
)  =  ( ( A  +h  B )  +h  ( -u 1  .h  B ) ) )
4 neg1cn 10421 . . . . 5  |-  -u 1  e.  CC
5 hvmulcl 24334 . . . . 5  |-  ( (
-u 1  e.  CC  /\  B  e.  ~H )  ->  ( -u 1  .h  B )  e.  ~H )
64, 5mpan 665 . . . 4  |-  ( B  e.  ~H  ->  ( -u 1  .h  B )  e.  ~H )
76ancli 548 . . 3  |-  ( B  e.  ~H  ->  ( B  e.  ~H  /\  ( -u 1  .h  B )  e.  ~H ) )
8 ax-hvass 24323 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  ( -u 1  .h  B )  e.  ~H )  -> 
( ( A  +h  B )  +h  ( -u 1  .h  B ) )  =  ( A  +h  ( B  +h  ( -u 1  .h  B
) ) ) )
983expb 1183 . . 3  |-  ( ( A  e.  ~H  /\  ( B  e.  ~H  /\  ( -u 1  .h  B )  e.  ~H ) )  ->  (
( A  +h  B
)  +h  ( -u
1  .h  B ) )  =  ( A  +h  ( B  +h  ( -u 1  .h  B
) ) ) )
107, 9sylan2 471 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  +h  ( -u 1  .h  B ) )  =  ( A  +h  ( B  +h  ( -u 1  .h  B
) ) ) )
11 hvnegid 24348 . . . 4  |-  ( B  e.  ~H  ->  ( B  +h  ( -u 1  .h  B ) )  =  0h )
1211oveq2d 6106 . . 3  |-  ( B  e.  ~H  ->  ( A  +h  ( B  +h  ( -u 1  .h  B
) ) )  =  ( A  +h  0h ) )
13 ax-hvaddid 24325 . . 3  |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )
1412, 13sylan9eqr 2495 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  ( B  +h  ( -u 1  .h  B ) ) )  =  A )
153, 10, 143eqtrd 2477 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  -h  B
)  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761  (class class class)co 6090   CCcc 9276   1c1 9279   -ucneg 9592   ~Hchil 24240    +h cva 24241    .h csm 24242   0hc0v 24245    -h cmv 24246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-hfvadd 24321  ax-hvass 24323  ax-hvaddid 24325  ax-hfvmul 24326  ax-hvmulid 24327  ax-hvdistr2 24330  ax-hvmul0 24331
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-ltxr 9419  df-sub 9593  df-neg 9594  df-hvsub 24292
This theorem is referenced by:  hvpncan2  24361  mayete3i  25050  mayete3iOLD  25051  lnop0  25289
  Copyright terms: Public domain W3C validator