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Theorem his7 22545
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 22538 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
( B  +h  C
)  .ih  A )  =  ( ( B 
.ih  A )  +  ( C  .ih  A
) ) )
21fveq2d 5691 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
* `  ( ( B  +h  C )  .ih  A ) )  =  ( * `  ( ( B  .ih  A )  +  ( C  .ih  A ) ) ) )
3 hicl 22535 . . . . . 6  |-  ( ( B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  A
)  e.  CC )
4 hicl 22535 . . . . . 6  |-  ( ( C  e.  ~H  /\  A  e.  ~H )  ->  ( C  .ih  A
)  e.  CC )
5 cjadd 11901 . . . . . 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 2436 . . 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 22468 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C
)  e.  ~H )
11 ax-his1 22537 . . . 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 22537 . . . 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 22537 . . . 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 6058 . 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 2446 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 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   CCcc 8944    + caddc 8949   *ccj 11856   ~Hchil 22375    +h cva 22376    .ih csp 22378
This theorem is referenced by:  normlem0  22564  normlem8  22572  pjadjii  23129  lnopunilem1  23466  hmops  23476  cnlnadjlem6  23528  adjlnop  23542  adjadd  23549  hstoh  23688
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-pre-mulgt0 9023  ax-hfvadd 22456  ax-hfi 22534  ax-his1 22537  ax-his2 22538
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-rmo 2674  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-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-cj 11859  df-re 11860  df-im 11861
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