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Theorem his5 26129
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 26056 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  C
)  e.  ~H )
2 ax-his1 26125 . . . . 5  |-  ( ( B  e.  ~H  /\  ( A  .h  C
)  e.  ~H )  ->  ( B  .ih  ( A  .h  C )
)  =  ( * `
 ( ( A  .h  C )  .ih  B ) ) )
31, 2sylan2 474 . . . 4  |-  ( ( B  e.  ~H  /\  ( A  e.  CC  /\  C  e.  ~H )
)  ->  ( B  .ih  ( A  .h  C
) )  =  ( * `  ( ( A  .h  C ) 
.ih  B ) ) )
433impb 1192 . . 3  |-  ( ( B  e.  ~H  /\  A  e.  CC  /\  C  e.  ~H )  ->  ( B  .ih  ( A  .h  C ) )  =  ( * `  (
( A  .h  C
)  .ih  B )
) )
543com12 1200 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  ( A  .h  C ) )  =  ( * `  (
( A  .h  C
)  .ih  B )
) )
6 ax-his3 26127 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  ~H  /\  B  e.  ~H )  ->  (
( A  .h  C
)  .ih  B )  =  ( A  x.  ( C  .ih  B ) ) )
763com23 1202 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  .h  C
)  .ih  B )  =  ( A  x.  ( C  .ih  B ) ) )
87fveq2d 5876 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( ( A  .h  C )  .ih  B ) )  =  ( * `  ( A  x.  ( C  .ih  B ) ) ) )
9 hicl 26123 . . . . . 6  |-  ( ( C  e.  ~H  /\  B  e.  ~H )  ->  ( C  .ih  B
)  e.  CC )
10 cjmul 12986 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  .ih  B )  e.  CC )  -> 
( * `  ( A  x.  ( C  .ih  B ) ) )  =  ( ( * `
 A )  x.  ( * `  ( C  .ih  B ) ) ) )
119, 10sylan2 474 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  ~H  /\  B  e.  ~H )
)  ->  ( * `  ( A  x.  ( C  .ih  B ) ) )  =  ( ( * `  A )  x.  ( * `  ( C  .ih  B ) ) ) )
12113impb 1192 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  ~H  /\  B  e.  ~H )  ->  (
* `  ( A  x.  ( C  .ih  B
) ) )  =  ( ( * `  A )  x.  (
* `  ( C  .ih  B ) ) ) )
13123com23 1202 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( A  x.  ( C  .ih  B
) ) )  =  ( ( * `  A )  x.  (
* `  ( C  .ih  B ) ) ) )
14 ax-his1 26125 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  C
)  =  ( * `
 ( C  .ih  B ) ) )
15143adant1 1014 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  C )  =  ( * `  ( C  .ih  B ) ) )
1615oveq2d 6312 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( * `  A
)  x.  ( B 
.ih  C ) )  =  ( ( * `
 A )  x.  ( * `  ( C  .ih  B ) ) ) )
1713, 16eqtr4d 2501 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( A  x.  ( C  .ih  B
) ) )  =  ( ( * `  A )  x.  ( B  .ih  C ) ) )
185, 8, 173eqtrd 2502 1  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  ( A  .h  C ) )  =  ( ( * `  A )  x.  ( B  .ih  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   CCcc 9507    x. cmul 9514   *ccj 12940   ~Hchil 25962    .h csm 25964    .ih csp 25965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-hfvmul 26048  ax-hfi 26122  ax-his1 26125  ax-his3 26127
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-2 10615  df-cj 12943  df-re 12944  df-im 12945
This theorem is referenced by:  his52  26130  his35  26131  normlem0  26152  normlem9  26161  bcseqi  26163  polid2i  26200  pjhthlem1  26435  eigrei  26879  eigposi  26881  eigorthi  26882  brafnmul  26996  lnopunilem1  27055  hmopm  27066  cnlnadjlem6  27117  adjlnop  27131
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