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Theorem his5 21611
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 21539 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  C
)  e.  ~H )
2 ax-his1 21607 . . . . 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 21609 . . . 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 5448 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( ( A  .h  C )  .ih  B ) )  =  ( * `  ( A  x.  ( C  .ih  B ) ) ) )
9 hicl 21605 . . . . . 6  |-  ( ( C  e.  ~H  /\  B  e.  ~H )  ->  ( C  .ih  B
)  e.  CC )
10 cjmul 11578 . . . . . 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 21607 . . . . 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 5794 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( * `  A
)  x.  ( B 
.ih  C ) )  =  ( ( * `
 A )  x.  ( * `  ( C  .ih  B ) ) ) )
1713, 16eqtr4d 2291 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( A  x.  ( C  .ih  B
) ) )  =  ( ( * `  A )  x.  ( B  .ih  C ) ) )
185, 8, 173eqtrd 2292 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 1619    e. wcel 1621   ` cfv 4659  (class class class)co 5778   CCcc 8689    x. cmul 8696   *ccj 11532   ~Hchil 21445    .h csm 21447    .ih csp 21448
This theorem is referenced by:  his52  21612  his35  21613  normlem0  21634  normlem9  21643  bcseqi  21645  polid2i  21682  pjhthlem1  21916  eigrei  22360  eigposi  22362  eigorthi  22363  brafnmul  22477  lnopunilem1  22536  hmopm  22547  cnlnadjlem6  22598  adjlnop  22612
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 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-hfvmul 21531  ax-hfi 21604  ax-his1 21607  ax-his3 21609
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-po 4272  df-so 4273  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-iota 6211  df-riota 6258  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-2 9758  df-cj 11535  df-re 11536  df-im 11537
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