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Theorem hvpncan 22494
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 22468 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  B
)  e.  ~H )
2 hvsubval 22472 . . 3  |-  ( ( ( A  +h  B
)  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  -h  B
)  =  ( ( A  +h  B )  +h  ( -u 1  .h  B ) ) )
31, 2sylancom 649 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  -h  B
)  =  ( ( A  +h  B )  +h  ( -u 1  .h  B ) ) )
4 neg1cn 10023 . . . . 5  |-  -u 1  e.  CC
5 hvmulcl 22469 . . . . 5  |-  ( (
-u 1  e.  CC  /\  B  e.  ~H )  ->  ( -u 1  .h  B )  e.  ~H )
64, 5mpan 652 . . . 4  |-  ( B  e.  ~H  ->  ( -u 1  .h  B )  e.  ~H )
76ancli 535 . . 3  |-  ( B  e.  ~H  ->  ( B  e.  ~H  /\  ( -u 1  .h  B )  e.  ~H ) )
8 ax-hvass 22458 . . . 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 1154 . . 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 461 . 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 22482 . . . 4  |-  ( B  e.  ~H  ->  ( B  +h  ( -u 1  .h  B ) )  =  0h )
1211oveq2d 6056 . . 3  |-  ( B  e.  ~H  ->  ( A  +h  ( B  +h  ( -u 1  .h  B
) ) )  =  ( A  +h  0h ) )
13 ax-hvaddid 22460 . . 3  |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )
1412, 13sylan9eqr 2458 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  +h  ( B  +h  ( -u 1  .h  B ) ) )  =  A )
153, 10, 143eqtrd 2440 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  -h  B
)  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721  (class class class)co 6040   CCcc 8944   1c1 8947   -ucneg 9248   ~Hchil 22375    +h cva 22376    .h csm 22377   0hc0v 22380    -h cmv 22381
This theorem is referenced by:  hvpncan2  22495  mayete3i  23183  mayete3iOLD  23184  lnop0  23422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-hfvadd 22456  ax-hvass 22458  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvdistr2 22465  ax-hvmul0 22466
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-neg 9250  df-hvsub 22427
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