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Theorem his7 21663
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 21656 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
( B  +h  C
)  .ih  A )  =  ( ( B 
.ih  A )  +  ( C  .ih  A
) ) )
21fveq2d 5491 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
* `  ( ( B  +h  C )  .ih  A ) )  =  ( * `  ( ( B  .ih  A )  +  ( C  .ih  A ) ) ) )
3 hicl 21653 . . . . . 6  |-  ( ( B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  A
)  e.  CC )
4 hicl 21653 . . . . . 6  |-  ( ( C  e.  ~H  /\  A  e.  ~H )  ->  ( C  .ih  A
)  e.  CC )
5 cjadd 11622 . . . . . 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 465 . . . . 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 2318 . . 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 21586 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C
)  e.  ~H )
11 ax-his1 21655 . . . 4  |-  ( ( A  e.  ~H  /\  ( B  +h  C
)  e.  ~H )  ->  ( A  .ih  ( B  +h  C ) )  =  ( * `  ( ( B  +h  C )  .ih  A
) ) )
1210, 11sylan2 462 . . 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 21655 . . . 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 21655 . . . 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 5839 . 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 2328 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 6    /\ wa 360    /\ w3a 936    = wceq 1625    e. wcel 1687   ` cfv 5223  (class class class)co 5821   CCcc 8732    + caddc 8737   *ccj 11577   ~Hchil 21493    +h cva 21494    .ih csp 21496
This theorem is referenced by:  normlem0  21682  normlem8  21690  pjadjii  22247  lnopunilem1  22584  hmops  22594  cnlnadjlem6  22646  adjlnop  22660  adjadd  22667  hstoh  22806
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-hfvadd 21574  ax-hfi 21652  ax-his1 21655  ax-his2 21656
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3831  df-iun 3910  df-br 4027  df-opab 4081  df-mpt 4082  df-id 4310  df-po 4315  df-so 4316  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-iota 6254  df-riota 6301  df-er 6657  df-en 6861  df-dom 6862  df-sdom 6863  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-div 9421  df-2 9801  df-cj 11580  df-re 11581  df-im 11582
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