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Theorem kbmul 28814
Description: Multiplication property of outer product. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
kbmul |- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( ( A .h B ) ketbra C ) = ( B ketbra ( ( * ` A ) .h C ) ) )
Proof of Theorem kbmul
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 hvmulcl 27870 . . 3 |- ( ( A e. CC /\ B e. ~H ) -> ( A .h B ) e. ~H )
2 kbfval 28811 . . 3 |- ( ( ( A .h B ) e. ~H /\ C e. ~H ) -> ( ( A .h B ) ketbra C ) = ( x e. ~H |-> ( ( x .ih C ) .h ( A .h B ) ) ) )
31, 2stoic3 1701 . 2 |- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( ( A .h B ) ketbra C ) = ( x e. ~H |-> ( ( x .ih C ) .h ( A .h B ) ) ) )
4 simp2 1062 . . . 4 |- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> B e. ~H )
5 cjcl 13845 . . . . . 6 |- ( A e. CC -> ( * ` A ) e. CC )
653ad2ant1 1082 . . . . 5 |- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( * ` A ) e. CC )
7 simp3 1063 . . . . 5 |- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> C e. ~H )
8 hvmulcl 27870 . . . . 5 |- ( ( ( * ` A ) e. CC /\ C e. ~H ) -> ( ( * ` A ) .h C ) e. ~H )
96, 7, 8syl2anc 693 . . . 4 |- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( ( * ` A ) .h C ) e. ~H )
10 kbfval 28811 . . . 4 |- ( ( B e. ~H /\ ( ( * ` A ) .h C ) e. ~H ) -> ( B ketbra ( ( * ` A ) .h C ) ) = ( x e. ~H |-> ( ( x .ih ( ( * ` A ) .h C ) ) .h B ) ) )
114, 9, 10syl2anc 693 . . 3 |- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( B ketbra ( ( * ` A ) .h C ) ) = ( x e. ~H |-> ( ( x .ih ( ( * ` A ) .h C ) ) .h B ) ) )
12 simpr 477 . . . . . . 7 |- ( ( ( A e. CC /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> x e. ~H )
13 simpl3 1066 . . . . . . 7 |- ( ( ( A e. CC /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> C e. ~H )
14 hicl 27937 . . . . . . 7 |- ( ( x e. ~H /\ C e. ~H ) -> ( x .ih C ) e. CC )
1512, 13, 14syl2anc 693 . . . . . 6 |- ( ( ( A e. CC /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( x .ih C ) e. CC )
16 simpl1 1064 . . . . . 6 |- ( ( ( A e. CC /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> A e. CC )
17 simpl2 1065 . . . . . 6 |- ( ( ( A e. CC /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> B e. ~H )
18 ax-hvmulass 27864 . . . . . 6 |- ( ( ( x .ih C ) e. CC /\ A e. CC /\ B e. ~H ) -> ( ( ( x .ih C ) x. A ) .h B ) = ( ( x .ih C ) .h ( A .h B ) ) )
1915, 16, 17, 18syl3anc 1326 . . . . 5 |- ( ( ( A e. CC /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( ( x .ih C ) x. A ) .h B ) = ( ( x .ih C ) .h ( A .h B ) ) )
2015, 16mulcomd 10061 . . . . . . 7 |- ( ( ( A e. CC /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( x .ih C ) x. A ) = ( A x. ( x .ih C ) ) )
21 his52 27944 . . . . . . . 8 |- ( ( A e. CC /\ x e. ~H /\ C e. ~H ) -> ( x .ih ( ( * ` A ) .h C ) ) = ( A x. ( x .ih C ) ) )
2216, 12, 13, 21syl3anc 1326 . . . . . . 7 |- ( ( ( A e. CC /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( x .ih ( ( * ` A ) .h C ) ) = ( A x. ( x .ih C ) ) )
2320, 22eqtr4d 2659 . . . . . 6 |- ( ( ( A e. CC /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( x .ih C ) x. A ) = ( x .ih ( ( * ` A ) .h C ) ) )
2423oveq1d 6665 . . . . 5 |- ( ( ( A e. CC /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( ( x .ih C ) x. A ) .h B ) = ( ( x .ih ( ( * ` A ) .h C ) ) .h B ) )
2519, 24eqtr3d 2658 . . . 4 |- ( ( ( A e. CC /\ B e. ~H /\ C e. ~H ) /\ x e. ~H ) -> ( ( x .ih C ) .h ( A .h B ) ) = ( ( x .ih ( ( * ` A ) .h C ) ) .h B ) )
2625mpteq2dva 4744 . . 3 |- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( x e. ~H |-> ( ( x .ih C ) .h ( A .h B ) ) ) = ( x e. ~H |-> ( ( x .ih ( ( * ` A ) .h C ) ) .h B ) ) )
2711, 26eqtr4d 2659 . 2 |- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( B ketbra ( ( * ` A ) .h C ) ) = ( x e. ~H |-> ( ( x .ih C ) .h ( A .h B ) ) ) )
283, 27eqtr4d 2659 1 |- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( ( A .h B ) ketbra C ) = ( B ketbra ( ( * ` A ) .h C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   x. cmul 9941   *ccj 13836   ~Hchil 27776   .h csm 27778   .ih csp 27779   ketbra ck 27814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-hilex 27856  ax-hfvmul 27862  ax-hvmulass 27864  ax-hfi 27936  ax-his1 27939  ax-his3 27941
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-cj 13839  df-re 13840  df-im 13841  df-kb 28710
This theorem is referenced by:  kbass6  28980
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