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Theorem his5 21657
Description: Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
Assertion
Ref Expression
his5  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  ( A  .h  C ) )  =  ( ( * `  A )  x.  ( B  .ih  C ) ) )

Proof of Theorem his5
StepHypRef Expression
1 hvmulcl 21585 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  C
)  e.  ~H )
2 ax-his1 21653 . . . . 5  |-  ( ( B  e.  ~H  /\  ( A  .h  C
)  e.  ~H )  ->  ( B  .ih  ( A  .h  C )
)  =  ( * `
 ( ( A  .h  C )  .ih  B ) ) )
31, 2sylan2 462 . . . 4  |-  ( ( B  e.  ~H  /\  ( A  e.  CC  /\  C  e.  ~H )
)  ->  ( B  .ih  ( A  .h  C
) )  =  ( * `  ( ( A  .h  C ) 
.ih  B ) ) )
433impb 1152 . . 3  |-  ( ( B  e.  ~H  /\  A  e.  CC  /\  C  e.  ~H )  ->  ( B  .ih  ( A  .h  C ) )  =  ( * `  (
( A  .h  C
)  .ih  B )
) )
543com12 1160 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  ( A  .h  C ) )  =  ( * `  (
( A  .h  C
)  .ih  B )
) )
6 ax-his3 21655 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  ~H  /\  B  e.  ~H )  ->  (
( A  .h  C
)  .ih  B )  =  ( A  x.  ( C  .ih  B ) ) )
763com23 1162 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  .h  C
)  .ih  B )  =  ( A  x.  ( C  .ih  B ) ) )
87fveq2d 5489 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( ( A  .h  C )  .ih  B ) )  =  ( * `  ( A  x.  ( C  .ih  B ) ) ) )
9 hicl 21651 . . . . . 6  |-  ( ( C  e.  ~H  /\  B  e.  ~H )  ->  ( C  .ih  B
)  e.  CC )
10 cjmul 11621 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  .ih  B )  e.  CC )  -> 
( * `  ( A  x.  ( C  .ih  B ) ) )  =  ( ( * `
 A )  x.  ( * `  ( C  .ih  B ) ) ) )
119, 10sylan2 462 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  ~H  /\  B  e.  ~H )
)  ->  ( * `  ( A  x.  ( C  .ih  B ) ) )  =  ( ( * `  A )  x.  ( * `  ( C  .ih  B ) ) ) )
12113impb 1152 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  ~H  /\  B  e.  ~H )  ->  (
* `  ( A  x.  ( C  .ih  B
) ) )  =  ( ( * `  A )  x.  (
* `  ( C  .ih  B ) ) ) )
13123com23 1162 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( A  x.  ( C  .ih  B
) ) )  =  ( ( * `  A )  x.  (
* `  ( C  .ih  B ) ) ) )
14 ax-his1 21653 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  C
)  =  ( * `
 ( C  .ih  B ) ) )
15143adant1 978 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  C )  =  ( * `  ( C  .ih  B ) ) )
1615oveq2d 5835 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( * `  A
)  x.  ( B 
.ih  C ) )  =  ( ( * `
 A )  x.  ( * `  ( C  .ih  B ) ) ) )
1713, 16eqtr4d 2319 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( A  x.  ( C  .ih  B
) ) )  =  ( ( * `  A )  x.  ( B  .ih  C ) ) )
185, 8, 173eqtrd 2320 1  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  ( A  .h  C ) )  =  ( ( * `  A )  x.  ( B  .ih  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688   ` cfv 5221  (class class class)co 5819   CCcc 8730    x. cmul 8737   *ccj 11575   ~Hchil 21491    .h csm 21493    .ih csp 21494
This theorem is referenced by:  his52  21658  his35  21659  normlem0  21680  normlem9  21689  bcseqi  21691  polid2i  21728  pjhthlem1  21962  eigrei  22406  eigposi  22408  eigorthi  22409  brafnmul  22523  lnopunilem1  22582  hmopm  22593  cnlnadjlem6  22644  adjlnop  22658
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-hfvmul 21577  ax-hfi 21650  ax-his1 21653  ax-his3 21655
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-iota 6252  df-riota 6299  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-2 9799  df-cj 11578  df-re 11579  df-im 11580
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