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Theorem his5 26417
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 26344 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  C
)  e.  ~H )
2 ax-his1 26413 . . . . 5  |-  ( ( B  e.  ~H  /\  ( A  .h  C
)  e.  ~H )  ->  ( B  .ih  ( A  .h  C )
)  =  ( * `
 ( ( A  .h  C )  .ih  B ) ) )
31, 2sylan2 472 . . . 4  |-  ( ( B  e.  ~H  /\  ( A  e.  CC  /\  C  e.  ~H )
)  ->  ( B  .ih  ( A  .h  C
) )  =  ( * `  ( ( A  .h  C ) 
.ih  B ) ) )
433impb 1193 . . 3  |-  ( ( B  e.  ~H  /\  A  e.  CC  /\  C  e.  ~H )  ->  ( B  .ih  ( A  .h  C ) )  =  ( * `  (
( A  .h  C
)  .ih  B )
) )
543com12 1201 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  ( A  .h  C ) )  =  ( * `  (
( A  .h  C
)  .ih  B )
) )
6 ax-his3 26415 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  ~H  /\  B  e.  ~H )  ->  (
( A  .h  C
)  .ih  B )  =  ( A  x.  ( C  .ih  B ) ) )
763com23 1203 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  .h  C
)  .ih  B )  =  ( A  x.  ( C  .ih  B ) ) )
87fveq2d 5853 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( ( A  .h  C )  .ih  B ) )  =  ( * `  ( A  x.  ( C  .ih  B ) ) ) )
9 hicl 26411 . . . . . 6  |-  ( ( C  e.  ~H  /\  B  e.  ~H )  ->  ( C  .ih  B
)  e.  CC )
10 cjmul 13124 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  .ih  B )  e.  CC )  -> 
( * `  ( A  x.  ( C  .ih  B ) ) )  =  ( ( * `
 A )  x.  ( * `  ( C  .ih  B ) ) ) )
119, 10sylan2 472 . . . . 5  |-  ( ( A  e.  CC  /\  ( C  e.  ~H  /\  B  e.  ~H )
)  ->  ( * `  ( A  x.  ( C  .ih  B ) ) )  =  ( ( * `  A )  x.  ( * `  ( C  .ih  B ) ) ) )
12113impb 1193 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  ~H  /\  B  e.  ~H )  ->  (
* `  ( A  x.  ( C  .ih  B
) ) )  =  ( ( * `  A )  x.  (
* `  ( C  .ih  B ) ) ) )
13123com23 1203 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( A  x.  ( C  .ih  B
) ) )  =  ( ( * `  A )  x.  (
* `  ( C  .ih  B ) ) ) )
14 ax-his1 26413 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  C
)  =  ( * `
 ( C  .ih  B ) ) )
15143adant1 1015 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  C )  =  ( * `  ( C  .ih  B ) ) )
1615oveq2d 6294 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( * `  A
)  x.  ( B 
.ih  C ) )  =  ( ( * `
 A )  x.  ( * `  ( C  .ih  B ) ) ) )
1713, 16eqtr4d 2446 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  ( A  x.  ( C  .ih  B
) ) )  =  ( ( * `  A )  x.  ( B  .ih  C ) ) )
185, 8, 173eqtrd 2447 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5569  (class class class)co 6278   CCcc 9520    x. cmul 9527   *ccj 13078   ~Hchil 26250    .h csm 26252    .ih csp 26253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-hfvmul 26336  ax-hfi 26410  ax-his1 26413  ax-his3 26415
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-2 10635  df-cj 13081  df-re 13082  df-im 13083
This theorem is referenced by:  his52  26418  his35  26419  normlem0  26440  normlem9  26449  bcseqi  26451  polid2i  26488  pjhthlem1  26723  eigrei  27166  eigposi  27168  eigorthi  27169  brafnmul  27283  lnopunilem1  27342  hmopm  27353  cnlnadjlem6  27404  adjlnop  27418
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