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Theorem his2sub 27949
Description: Distributive law for inner product of vector subtraction. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
Assertion
Ref Expression
his2sub |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A -h B ) .ih C ) = ( ( A .ih C ) - ( B .ih C ) ) )
Proof of Theorem his2sub
StepHypRef Expression
1 hvsubval 27873 . . . 4 |- ( ( A e. ~H /\ B e. ~H ) -> ( A -h B ) = ( A +h ( -u 1 .h B ) ) )
21oveq1d 6665 . . 3 |- ( ( A e. ~H /\ B e. ~H ) -> ( ( A -h B ) .ih C ) = ( ( A +h ( -u 1 .h B ) ) .ih C ) )
323adant3 1081 . 2 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A -h B ) .ih C ) = ( ( A +h ( -u 1 .h B ) ) .ih C ) )
4 neg1cn 11124 . . . . 5 |- -u 1 e. CC
5 hvmulcl 27870 . . . . 5 |- ( ( -u 1 e. CC /\ B e. ~H ) -> ( -u 1 .h B ) e. ~H )
64, 5mpan 706 . . . 4 |- ( B e. ~H -> ( -u 1 .h B ) e. ~H )
7 ax-his2 27940 . . . 4 |- ( ( A e. ~H /\ ( -u 1 .h B ) e. ~H /\ C e. ~H ) -> ( ( A +h ( -u 1 .h B ) ) .ih C ) = ( ( A .ih C ) + ( ( -u 1 .h B ) .ih C ) ) )
86, 7syl3an2 1360 . . 3 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h ( -u 1 .h B ) ) .ih C ) = ( ( A .ih C ) + ( ( -u 1 .h B ) .ih C ) ) )
9 ax-his3 27941 . . . . . . 7 |- ( ( -u 1 e. CC /\ B e. ~H /\ C e. ~H ) -> ( ( -u 1 .h B ) .ih C ) = ( -u 1 x. ( B .ih C ) ) )
104, 9mp3an1 1411 . . . . . 6 |- ( ( B e. ~H /\ C e. ~H ) -> ( ( -u 1 .h B ) .ih C ) = ( -u 1 x. ( B .ih C ) ) )
11 hicl 27937 . . . . . . 7 |- ( ( B e. ~H /\ C e. ~H ) -> ( B .ih C ) e. CC )
1211mulm1d 10482 . . . . . 6 |- ( ( B e. ~H /\ C e. ~H ) -> ( -u 1 x. ( B .ih C ) ) = -u ( B .ih C ) )
1310, 12eqtrd 2656 . . . . 5 |- ( ( B e. ~H /\ C e. ~H ) -> ( ( -u 1 .h B ) .ih C ) = -u ( B .ih C ) )
1413oveq2d 6666 . . . 4 |- ( ( B e. ~H /\ C e. ~H ) -> ( ( A .ih C ) + ( ( -u 1 .h B ) .ih C ) ) = ( ( A .ih C ) + -u ( B .ih C ) ) )
15143adant1 1079 . . 3 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A .ih C ) + ( ( -u 1 .h B ) .ih C ) ) = ( ( A .ih C ) + -u ( B .ih C ) ) )
168, 15eqtrd 2656 . 2 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h ( -u 1 .h B ) ) .ih C ) = ( ( A .ih C ) + -u ( B .ih C ) ) )
17 hicl 27937 . . . 4 |- ( ( A e. ~H /\ C e. ~H ) -> ( A .ih C ) e. CC )
18173adant2 1080 . . 3 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A .ih C ) e. CC )
19113adant1 1079 . . 3 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( B .ih C ) e. CC )
2018, 19negsubd 10398 . 2 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A .ih C ) + -u ( B .ih C ) ) = ( ( A .ih C ) - ( B .ih C ) ) )
213, 16, 203eqtrd 2660 1 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A -h B ) .ih C ) = ( ( A .ih C ) - ( B .ih C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   ~Hchil 27776   +h cva 27777   .h csm 27778   .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-hfvmul 27862  ax-hfi 27936  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-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-ltxr 10079  df-sub 10268  df-neg 10269  df-hvsub 27828
This theorem is referenced by:  his2sub2  27950  hi2eq  27962  pjhthlem1  28250  h1de2i  28412  pjdifnormii  28542  lnopeqi  28867  riesz3i  28921  leop2  28983  hmopidmpji  29011  pjssposi  29031  pjclem4  29058  pj3si  29066
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