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Theorem hvsubass 27901
Description: Hilbert vector space associative law for subtraction. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hvsubass |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A -h B ) -h C ) = ( A -h ( B +h C ) ) )
Proof of Theorem hvsubass
StepHypRef Expression
1 neg1cn 11124 . . . 4 |- -u 1 e. CC
2 hvmulcl 27870 . . . 4 |- ( ( -u 1 e. CC /\ B e. ~H ) -> ( -u 1 .h B ) e. ~H )
31, 2mpan 706 . . 3 |- ( B e. ~H -> ( -u 1 .h B ) e. ~H )
4 hvaddsubass 27898 . . 3 |- ( ( A e. ~H /\ ( -u 1 .h B ) e. ~H /\ C e. ~H ) -> ( ( A +h ( -u 1 .h B ) ) -h C ) = ( A +h ( ( -u 1 .h B ) -h C ) ) )
53, 4syl3an2 1360 . 2 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h ( -u 1 .h B ) ) -h C ) = ( A +h ( ( -u 1 .h B ) -h C ) ) )
6 hvsubval 27873 . . . 4 |- ( ( A e. ~H /\ B e. ~H ) -> ( A -h B ) = ( A +h ( -u 1 .h B ) ) )
763adant3 1081 . . 3 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A -h B ) = ( A +h ( -u 1 .h B ) ) )
87oveq1d 6665 . 2 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A -h B ) -h C ) = ( ( A +h ( -u 1 .h B ) ) -h C ) )
9 simp1 1061 . . . 4 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> A e. ~H )
10 hvaddcl 27869 . . . . 5 |- ( ( B e. ~H /\ C e. ~H ) -> ( B +h C ) e. ~H )
11103adant1 1079 . . . 4 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( B +h C ) e. ~H )
12 hvsubval 27873 . . . 4 |- ( ( A e. ~H /\ ( B +h C ) e. ~H ) -> ( A -h ( B +h C ) ) = ( A +h ( -u 1 .h ( B +h C ) ) ) )
139, 11, 12syl2anc 693 . . 3 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A -h ( B +h C ) ) = ( A +h ( -u 1 .h ( B +h C ) ) ) )
14 hvsubval 27873 . . . . . . 7 |- ( ( ( -u 1 .h B ) e. ~H /\ C e. ~H ) -> ( ( -u 1 .h B ) -h C ) = ( ( -u 1 .h B ) +h ( -u 1 .h C ) ) )
153, 14sylan 488 . . . . . 6 |- ( ( B e. ~H /\ C e. ~H ) -> ( ( -u 1 .h B ) -h C ) = ( ( -u 1 .h B ) +h ( -u 1 .h C ) ) )
16153adant1 1079 . . . . 5 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( -u 1 .h B ) -h C ) = ( ( -u 1 .h B ) +h ( -u 1 .h C ) ) )
17 ax-hvdistr1 27865 . . . . . . 7 |- ( ( -u 1 e. CC /\ B e. ~H /\ C e. ~H ) -> ( -u 1 .h ( B +h C ) ) = ( ( -u 1 .h B ) +h ( -u 1 .h C ) ) )
181, 17mp3an1 1411 . . . . . 6 |- ( ( B e. ~H /\ C e. ~H ) -> ( -u 1 .h ( B +h C ) ) = ( ( -u 1 .h B ) +h ( -u 1 .h C ) ) )
19183adant1 1079 . . . . 5 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( -u 1 .h ( B +h C ) ) = ( ( -u 1 .h B ) +h ( -u 1 .h C ) ) )
2016, 19eqtr4d 2659 . . . 4 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( -u 1 .h B ) -h C ) = ( -u 1 .h ( B +h C ) ) )
2120oveq2d 6666 . . 3 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A +h ( ( -u 1 .h B ) -h C ) ) = ( A +h ( -u 1 .h ( B +h C ) ) ) )
2213, 21eqtr4d 2659 . 2 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( A -h ( B +h C ) ) = ( A +h ( ( -u 1 .h B ) -h C ) ) )
235, 8, 223eqtr4d 2666 1 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A -h B ) -h C ) = ( A -h ( B +h C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   1c1 9937   -ucneg 10267   ~Hchil 27776   +h cva 27777   .h csm 27778   -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-hfvadd 27857  ax-hvass 27859  ax-hfvmul 27862  ax-hvdistr1 27865
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:  hvsub32  27902  hvsubassi  27912  pjhthlem1  28250
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