HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  his7 Structured version   Unicode version

Theorem his7 22584
Description: Distributive law for inner product. Lemma 3.1(S7) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
Assertion
Ref Expression
his7  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  ( B  +h  C ) )  =  ( ( A  .ih  B )  +  ( A 
.ih  C ) ) )

Proof of Theorem his7
StepHypRef Expression
1 ax-his2 22577 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
( B  +h  C
)  .ih  A )  =  ( ( B 
.ih  A )  +  ( C  .ih  A
) ) )
21fveq2d 5724 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
* `  ( ( B  +h  C )  .ih  A ) )  =  ( * `  ( ( B  .ih  A )  +  ( C  .ih  A ) ) ) )
3 hicl 22574 . . . . . 6  |-  ( ( B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  A
)  e.  CC )
4 hicl 22574 . . . . . 6  |-  ( ( C  e.  ~H  /\  A  e.  ~H )  ->  ( C  .ih  A
)  e.  CC )
5 cjadd 11938 . . . . . 6  |-  ( ( ( B  .ih  A
)  e.  CC  /\  ( C  .ih  A )  e.  CC )  -> 
( * `  (
( B  .ih  A
)  +  ( C 
.ih  A ) ) )  =  ( ( * `  ( B 
.ih  A ) )  +  ( * `  ( C  .ih  A ) ) ) )
63, 4, 5syl2an 464 . . . . 5  |-  ( ( ( B  e.  ~H  /\  A  e.  ~H )  /\  ( C  e.  ~H  /\  A  e.  ~H )
)  ->  ( * `  ( ( B  .ih  A )  +  ( C 
.ih  A ) ) )  =  ( ( * `  ( B 
.ih  A ) )  +  ( * `  ( C  .ih  A ) ) ) )
763impdir 1240 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
* `  ( ( B  .ih  A )  +  ( C  .ih  A
) ) )  =  ( ( * `  ( B  .ih  A ) )  +  ( * `
 ( C  .ih  A ) ) ) )
82, 7eqtrd 2467 . . 3  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
* `  ( ( B  +h  C )  .ih  A ) )  =  ( ( * `  ( B  .ih  A ) )  +  ( * `  ( C  .ih  A ) ) ) )
983comr 1161 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( ( B  +h  C )  .ih  A ) )  =  ( ( * `  ( B  .ih  A ) )  +  ( * `  ( C  .ih  A ) ) ) )
10 hvaddcl 22507 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C
)  e.  ~H )
11 ax-his1 22576 . . . 4  |-  ( ( A  e.  ~H  /\  ( B  +h  C
)  e.  ~H )  ->  ( A  .ih  ( B  +h  C ) )  =  ( * `  ( ( B  +h  C )  .ih  A
) ) )
1210, 11sylan2 461 . . 3  |-  ( ( A  e.  ~H  /\  ( B  e.  ~H  /\  C  e.  ~H )
)  ->  ( A  .ih  ( B  +h  C
) )  =  ( * `  ( ( B  +h  C ) 
.ih  A ) ) )
13123impb 1149 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  ( B  +h  C ) )  =  ( * `  (
( B  +h  C
)  .ih  A )
) )
14 ax-his1 22576 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B
)  =  ( * `
 ( B  .ih  A ) ) )
15143adant3 977 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  B )  =  ( * `  ( B  .ih  A ) ) )
16 ax-his1 22576 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  C
)  =  ( * `
 ( C  .ih  A ) ) )
17163adant2 976 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  C )  =  ( * `  ( C  .ih  A ) ) )
1815, 17oveq12d 6091 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  .ih  B
)  +  ( A 
.ih  C ) )  =  ( ( * `
 ( B  .ih  A ) )  +  ( * `  ( C 
.ih  A ) ) ) )
199, 13, 183eqtr4d 2477 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  ( B  +h  C ) )  =  ( ( A  .ih  B )  +  ( A 
.ih  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980    + caddc 8985   *ccj 11893   ~Hchil 22414    +h cva 22415    .ih csp 22417
This theorem is referenced by:  normlem0  22603  normlem8  22611  pjadjii  23168  lnopunilem1  23505  hmops  23515  cnlnadjlem6  23567  adjlnop  23581  adjadd  23588  hstoh  23727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-hfvadd 22495  ax-hfi 22573  ax-his1 22576  ax-his2 22577
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-2 10050  df-cj 11896  df-re 11897  df-im 11898
  Copyright terms: Public domain W3C validator