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Theorem his7 21499
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 21492 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
( B  +h  C
)  .ih  A )  =  ( ( B 
.ih  A )  +  ( C  .ih  A
) ) )
21fveq2d 5381 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
* `  ( ( B  +h  C )  .ih  A ) )  =  ( * `  ( ( B  .ih  A )  +  ( C  .ih  A ) ) ) )
3 hicl 21489 . . . . . 6  |-  ( ( B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  A
)  e.  CC )
4 hicl 21489 . . . . . 6  |-  ( ( C  e.  ~H  /\  A  e.  ~H )  ->  ( C  .ih  A
)  e.  CC )
5 cjadd 11503 . . . . . 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 1243 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
* `  ( ( B  .ih  A )  +  ( C  .ih  A
) ) )  =  ( ( * `  ( B  .ih  A ) )  +  ( * `
 ( C  .ih  A ) ) ) )
82, 7eqtrd 2285 . . 3  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
* `  ( ( B  +h  C )  .ih  A ) )  =  ( ( * `  ( B  .ih  A ) )  +  ( * `  ( C  .ih  A ) ) ) )
983comr 1164 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( ( B  +h  C )  .ih  A ) )  =  ( ( * `  ( B  .ih  A ) )  +  ( * `  ( C  .ih  A ) ) ) )
10 hvaddcl 21422 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C
)  e.  ~H )
11 ax-his1 21491 . . . 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 1152 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  ( B  +h  C ) )  =  ( * `  (
( B  +h  C
)  .ih  A )
) )
14 ax-his1 21491 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B
)  =  ( * `
 ( B  .ih  A ) ) )
15143adant3 980 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  B )  =  ( * `  ( B  .ih  A ) ) )
16 ax-his1 21491 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  C
)  =  ( * `
 ( C  .ih  A ) ) )
17163adant2 979 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  C )  =  ( * `  ( C  .ih  A ) ) )
1815, 17oveq12d 5728 . 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 2295 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 939    = wceq 1619    e. wcel 1621   ` cfv 4592  (class class class)co 5710   CCcc 8615    + caddc 8620   *ccj 11458   ~Hchil 21329    +h cva 21330    .ih csp 21332
This theorem is referenced by:  normlem0  21518  normlem8  21526  pjadjii  22101  lnopunilem1  22420  hmops  22430  cnlnadjlem6  22482  adjlnop  22496  adjadd  22503  hstoh  22642
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-hfvadd 21410  ax-hfi 21488  ax-his1 21491  ax-his2 21492
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-2 9684  df-cj 11461  df-re 11462  df-im 11463
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