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Theorem his5 21779
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 21707 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  C
)  e.  ~H )
2 ax-his1 21775 . . . . 5  |-  ( ( B  e.  ~H  /\  ( A  .h  C
)  e.  ~H )  ->  ( B  .ih  ( A  .h  C )
)  =  ( * `
 ( ( A  .h  C )  .ih  B ) ) )
31, 2sylan2 460 . . . 4  |-  ( ( B  e.  ~H  /\  ( A  e.  CC  /\  C  e.  ~H )
)  ->  ( B  .ih  ( A  .h  C
) )  =  ( * `  ( ( A  .h  C ) 
.ih  B ) ) )
433impb 1147 . . 3  |-  ( ( B  e.  ~H  /\  A  e.  CC  /\  C  e.  ~H )  ->  ( B  .ih  ( A  .h  C ) )  =  ( * `  (
( A  .h  C
)  .ih  B )
) )
543com12 1155 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  ( A  .h  C ) )  =  ( * `  (
( A  .h  C
)  .ih  B )
) )
6 ax-his3 21777 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  ~H  /\  B  e.  ~H )  ->  (
( A  .h  C
)  .ih  B )  =  ( A  x.  ( C  .ih  B ) ) )
763com23 1157 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  .h  C
)  .ih  B )  =  ( A  x.  ( C  .ih  B ) ) )
87fveq2d 5612 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( ( A  .h  C )  .ih  B ) )  =  ( * `  ( A  x.  ( C  .ih  B ) ) ) )
9 hicl 21773 . . . . . 6  |-  ( ( C  e.  ~H  /\  B  e.  ~H )  ->  ( C  .ih  B
)  e.  CC )
10 cjmul 11723 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  .ih  B )  e.  CC )  -> 
( * `  ( A  x.  ( C  .ih  B ) ) )  =  ( ( * `
 A )  x.  ( * `  ( C  .ih  B ) ) ) )
119, 10sylan2 460 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  ~H  /\  B  e.  ~H )
)  ->  ( * `  ( A  x.  ( C  .ih  B ) ) )  =  ( ( * `  A )  x.  ( * `  ( C  .ih  B ) ) ) )
12113impb 1147 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  ~H  /\  B  e.  ~H )  ->  (
* `  ( A  x.  ( C  .ih  B
) ) )  =  ( ( * `  A )  x.  (
* `  ( C  .ih  B ) ) ) )
13123com23 1157 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( A  x.  ( C  .ih  B
) ) )  =  ( ( * `  A )  x.  (
* `  ( C  .ih  B ) ) ) )
14 ax-his1 21775 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  C
)  =  ( * `
 ( C  .ih  B ) ) )
15143adant1 973 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  C )  =  ( * `  ( C  .ih  B ) ) )
1615oveq2d 5961 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( * `  A
)  x.  ( B 
.ih  C ) )  =  ( ( * `
 A )  x.  ( * `  ( C  .ih  B ) ) ) )
1713, 16eqtr4d 2393 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( A  x.  ( C  .ih  B
) ) )  =  ( ( * `  A )  x.  ( B  .ih  C ) ) )
185, 8, 173eqtrd 2394 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 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   ` cfv 5337  (class class class)co 5945   CCcc 8825    x. cmul 8832   *ccj 11677   ~Hchil 21613    .h csm 21615    .ih csp 21616
This theorem is referenced by:  his52  21780  his35  21781  normlem0  21802  normlem9  21811  bcseqi  21813  polid2i  21850  pjhthlem1  22084  eigrei  22528  eigposi  22530  eigorthi  22531  brafnmul  22645  lnopunilem1  22704  hmopm  22715  cnlnadjlem6  22766  adjlnop  22780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-hfvmul 21699  ax-hfi 21772  ax-his1 21775  ax-his3 21777
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-2 9894  df-cj 11680  df-re 11681  df-im 11682
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