Theorem hiassdi 27948

Description: Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)

Assertion

Ref	Expression
hiassdi	|- ( ( ( A e. CC /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( ( A .h B ) +h C ) .ih D ) = ( ( A x. ( B .ih D ) ) + ( C .ih D ) ) )

Proof of Theorem hiassdi

Step	Hyp	Ref	Expression
1		hvmulcl 27870	. . 3 |- ( ( A e. CC /\ B e. ~H ) -> ( A .h B ) e. ~H )
2		ax-his2 27940	. . . 4 |- ( ( ( A .h B ) e. ~H /\ C e. ~H /\ D e. ~H ) -> ( ( ( A .h B ) +h C ) .ih D ) = ( ( ( A .h B ) .ih D ) + ( C .ih D ) ) )
3	2	3expb 1266	. . 3 |- ( ( ( A .h B ) e. ~H /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( ( A .h B ) +h C ) .ih D ) = ( ( ( A .h B ) .ih D ) + ( C .ih D ) ) )
4	1, 3	sylan 488	. 2 |- ( ( ( A e. CC /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( ( A .h B ) +h C ) .ih D ) = ( ( ( A .h B ) .ih D ) + ( C .ih D ) ) )
5		ax-his3 27941	. . . . 5 |- ( ( A e. CC /\ B e. ~H /\ D e. ~H ) -> ( ( A .h B ) .ih D ) = ( A x. ( B .ih D ) ) )
6	5	3expa 1265	. . . 4 |- ( ( ( A e. CC /\ B e. ~H ) /\ D e. ~H ) -> ( ( A .h B ) .ih D ) = ( A x. ( B .ih D ) ) )
7	6	adantrl 752	. . 3 |- ( ( ( A e. CC /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( A .h B ) .ih D ) = ( A x. ( B .ih D ) ) )
8	7	oveq1d 6665	. 2 |- ( ( ( A e. CC /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( ( A .h B ) .ih D ) + ( C .ih D ) ) = ( ( A x. ( B .ih D ) ) + ( C .ih D ) ) )
9	4, 8	eqtrd 2656	1 |- ( ( ( A e. CC /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( ( A .h B ) +h C ) .ih D ) = ( ( A x. ( B .ih D ) ) + ( C .ih D ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   + caddc 9939   x. cmul 9941   ~Hchil 27776   +h cva 27777   .h csm 27778   .ih csp 27779

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-hfvmul 27862  ax-his2 27940  ax-his3 27941

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-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-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653

This theorem is referenced by:  unoplin  28779  hmoplin  28801