Theorem his2sub2 27950

Description: Distributive law for inner product of vector subtraction. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
his2sub2	|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A .ih ( B -h C ) ) = ( ( A .ih B ) - ( A .ih C ) ) )

Proof of Theorem his2sub2

Step	Hyp	Ref	Expression
1		his2sub 27949	. . . . 5 |- ( ( B e. ~H /\ C e. ~H /\ A e. ~H ) -> ( ( B -h C ) .ih A ) = ( ( B .ih A ) - ( C .ih A ) ) )
2	1	fveq2d 6195	. . . 4 |- ( ( B e. ~H /\ C e. ~H /\ A e. ~H ) -> ( * ` ( ( B -h C ) .ih A ) ) = ( * ` ( ( B .ih A ) - ( C .ih A ) ) ) )
3		hicl 27937	. . . . . 6 |- ( ( B e. ~H /\ A e. ~H ) -> ( B .ih A ) e. CC )
4		hicl 27937	. . . . . 6 |- ( ( C e. ~H /\ A e. ~H ) -> ( C .ih A ) e. CC )
5		cjsub 13889	. . . . . 6 |- ( ( ( B .ih A ) e. CC /\ ( C .ih A ) e. CC ) -> ( * ` ( ( B .ih A ) - ( C .ih A ) ) ) = ( ( * ` ( B .ih A ) ) - ( * ` ( C .ih A ) ) ) )
6	3, 4, 5	syl2an 494	. . . . 5 |- ( ( ( B e. ~H /\ A e. ~H ) /\ ( C e. ~H /\ A e. ~H ) ) -> ( * ` ( ( B .ih A ) - ( C .ih A ) ) ) = ( ( * ` ( B .ih A ) ) - ( * ` ( C .ih A ) ) ) )
7	6	3impdir 1382	. . . 4 |- ( ( B e. ~H /\ C e. ~H /\ A e. ~H ) -> ( * ` ( ( B .ih A ) - ( C .ih A ) ) ) = ( ( * ` ( B .ih A ) ) - ( * ` ( C .ih A ) ) ) )
8	2, 7	eqtrd 2656	. . 3 |- ( ( B e. ~H /\ C e. ~H /\ A e. ~H ) -> ( * ` ( ( B -h C ) .ih A ) ) = ( ( * ` ( B .ih A ) ) - ( * ` ( C .ih A ) ) ) )
9	8	3comr 1273	. 2 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( * ` ( ( B -h C ) .ih A ) ) = ( ( * ` ( B .ih A ) ) - ( * ` ( C .ih A ) ) ) )
10		hvsubcl 27874	. . . 4 |- ( ( B e. ~H /\ C e. ~H ) -> ( B -h C ) e. ~H )
11		ax-his1 27939	. . . 4 |- ( ( A e. ~H /\ ( B -h C ) e. ~H ) -> ( A .ih ( B -h C ) ) = ( * ` ( ( B -h C ) .ih A ) ) )
12	10, 11	sylan2 491	. . 3 |- ( ( A e. ~H /\ ( B e. ~H /\ C e. ~H ) ) -> ( A .ih ( B -h C ) ) = ( * ` ( ( B -h C ) .ih A ) ) )
13	12	3impb 1260	. 2 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A .ih ( B -h C ) ) = ( * ` ( ( B -h C ) .ih A ) ) )
14		ax-his1 27939	. . . 4 |- ( ( A e. ~H /\ B e. ~H ) -> ( A .ih B ) = ( * ` ( B .ih A ) ) )
15	14	3adant3 1081	. . 3 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A .ih B ) = ( * ` ( B .ih A ) ) )
16		ax-his1 27939	. . . 4 |- ( ( A e. ~H /\ C e. ~H ) -> ( A .ih C ) = ( * ` ( C .ih A ) ) )
17	16	3adant2 1080	. . 3 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A .ih C ) = ( * ` ( C .ih A ) ) )
18	15, 17	oveq12d 6668	. 2 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A .ih B ) - ( A .ih C ) ) = ( ( * ` ( B .ih A ) ) - ( * ` ( C .ih A ) ) ) )
19	9, 13, 18	3eqtr4d 2666	1 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A .ih ( B -h C ) ) = ( ( A .ih B ) - ( A .ih C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   - cmin 10266   *ccj 13836   ~Hchil 27776   .ih csp 27779   -h cmv 27782

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-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-hfvadd 27857  ax-hfvmul 27862  ax-hfi 27936  ax-his1 27939  ax-his2 27940  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-hvsub 27828

This theorem is referenced by:  pjhthlem1  28250  riesz4i  28922