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Theorem his7 21629
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 21622 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
( B  +h  C
)  .ih  A )  =  ( ( B 
.ih  A )  +  ( C  .ih  A
) ) )
21fveq2d 5462 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
* `  ( ( B  +h  C )  .ih  A ) )  =  ( * `  ( ( B  .ih  A )  +  ( C  .ih  A ) ) ) )
3 hicl 21619 . . . . . 6  |-  ( ( B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  A
)  e.  CC )
4 hicl 21619 . . . . . 6  |-  ( ( C  e.  ~H  /\  A  e.  ~H )  ->  ( C  .ih  A
)  e.  CC )
5 cjadd 11591 . . . . . 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 2290 . . 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 21552 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C
)  e.  ~H )
11 ax-his1 21621 . . . 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 21621 . . . 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 21621 . . . 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 5810 . 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 2300 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 4673  (class class class)co 5792   CCcc 8703    + caddc 8708   *ccj 11546   ~Hchil 21459    +h cva 21460    .ih csp 21462
This theorem is referenced by:  normlem0  21648  normlem8  21656  pjadjii  22213  lnopunilem1  22550  hmops  22560  cnlnadjlem6  22612  adjlnop  22626  adjadd  22633  hstoh  22772
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-hfvadd 21540  ax-hfi 21618  ax-his1 21621  ax-his2 21622
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-2 9772  df-cj 11549  df-re 11550  df-im 11551
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