Theorem hvsubsub4 27917

Description: Hilbert vector space addition/subtraction law. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)

Assertion

Ref	Expression
hvsubsub4	|- ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( A -h B ) -h ( C -h D ) ) = ( ( A -h C ) -h ( B -h D ) ) )

Proof of Theorem hvsubsub4

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . 4 |- ( A = if ( A e. ~H , A , 0h ) -> ( A -h B ) = ( if ( A e. ~H , A , 0h ) -h B ) )
2	1	oveq1d 6665	. . 3 |- ( A = if ( A e. ~H , A , 0h ) -> ( ( A -h B ) -h ( C -h D ) ) = ( ( if ( A e. ~H , A , 0h ) -h B ) -h ( C -h D ) ) )
3		oveq1 6657	. . . 4 |- ( A = if ( A e. ~H , A , 0h ) -> ( A -h C ) = ( if ( A e. ~H , A , 0h ) -h C ) )
4	3	oveq1d 6665	. . 3 |- ( A = if ( A e. ~H , A , 0h ) -> ( ( A -h C ) -h ( B -h D ) ) = ( ( if ( A e. ~H , A , 0h ) -h C ) -h ( B -h D ) ) )
5	2, 4	eqeq12d 2637	. 2 |- ( A = if ( A e. ~H , A , 0h ) -> ( ( ( A -h B ) -h ( C -h D ) ) = ( ( A -h C ) -h ( B -h D ) ) <-> ( ( if ( A e. ~H , A , 0h ) -h B ) -h ( C -h D ) ) = ( ( if ( A e. ~H , A , 0h ) -h C ) -h ( B -h D ) ) ) )
6		oveq2 6658	. . . 4 |- ( B = if ( B e. ~H , B , 0h ) -> ( if ( A e. ~H , A , 0h ) -h B ) = ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) )
7	6	oveq1d 6665	. . 3 |- ( B = if ( B e. ~H , B , 0h ) -> ( ( if ( A e. ~H , A , 0h ) -h B ) -h ( C -h D ) ) = ( ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) -h ( C -h D ) ) )
8		oveq1 6657	. . . 4 |- ( B = if ( B e. ~H , B , 0h ) -> ( B -h D ) = ( if ( B e. ~H , B , 0h ) -h D ) )
9	8	oveq2d 6666	. . 3 |- ( B = if ( B e. ~H , B , 0h ) -> ( ( if ( A e. ~H , A , 0h ) -h C ) -h ( B -h D ) ) = ( ( if ( A e. ~H , A , 0h ) -h C ) -h ( if ( B e. ~H , B , 0h ) -h D ) ) )
10	7, 9	eqeq12d 2637	. 2 |- ( B = if ( B e. ~H , B , 0h ) -> ( ( ( if ( A e. ~H , A , 0h ) -h B ) -h ( C -h D ) ) = ( ( if ( A e. ~H , A , 0h ) -h C ) -h ( B -h D ) ) <-> ( ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) -h ( C -h D ) ) = ( ( if ( A e. ~H , A , 0h ) -h C ) -h ( if ( B e. ~H , B , 0h ) -h D ) ) ) )
11		oveq1 6657	. . . 4 |- ( C = if ( C e. ~H , C , 0h ) -> ( C -h D ) = ( if ( C e. ~H , C , 0h ) -h D ) )
12	11	oveq2d 6666	. . 3 |- ( C = if ( C e. ~H , C , 0h ) -> ( ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) -h ( C -h D ) ) = ( ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) -h ( if ( C e. ~H , C , 0h ) -h D ) ) )
13		oveq2 6658	. . . 4 |- ( C = if ( C e. ~H , C , 0h ) -> ( if ( A e. ~H , A , 0h ) -h C ) = ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) )
14	13	oveq1d 6665	. . 3 |- ( C = if ( C e. ~H , C , 0h ) -> ( ( if ( A e. ~H , A , 0h ) -h C ) -h ( if ( B e. ~H , B , 0h ) -h D ) ) = ( ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) -h ( if ( B e. ~H , B , 0h ) -h D ) ) )
15	12, 14	eqeq12d 2637	. 2 |- ( C = if ( C e. ~H , C , 0h ) -> ( ( ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) -h ( C -h D ) ) = ( ( if ( A e. ~H , A , 0h ) -h C ) -h ( if ( B e. ~H , B , 0h ) -h D ) ) <-> ( ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) -h ( if ( C e. ~H , C , 0h ) -h D ) ) = ( ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) -h ( if ( B e. ~H , B , 0h ) -h D ) ) ) )
16		oveq2 6658	. . . 4 |- ( D = if ( D e. ~H , D , 0h ) -> ( if ( C e. ~H , C , 0h ) -h D ) = ( if ( C e. ~H , C , 0h ) -h if ( D e. ~H , D , 0h ) ) )
17	16	oveq2d 6666	. . 3 |- ( D = if ( D e. ~H , D , 0h ) -> ( ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) -h ( if ( C e. ~H , C , 0h ) -h D ) ) = ( ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) -h ( if ( C e. ~H , C , 0h ) -h if ( D e. ~H , D , 0h ) ) ) )
18		oveq2 6658	. . . 4 |- ( D = if ( D e. ~H , D , 0h ) -> ( if ( B e. ~H , B , 0h ) -h D ) = ( if ( B e. ~H , B , 0h ) -h if ( D e. ~H , D , 0h ) ) )
19	18	oveq2d 6666	. . 3 |- ( D = if ( D e. ~H , D , 0h ) -> ( ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) -h ( if ( B e. ~H , B , 0h ) -h D ) ) = ( ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) -h ( if ( B e. ~H , B , 0h ) -h if ( D e. ~H , D , 0h ) ) ) )
20	17, 19	eqeq12d 2637	. 2 |- ( D = if ( D e. ~H , D , 0h ) -> ( ( ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) -h ( if ( C e. ~H , C , 0h ) -h D ) ) = ( ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) -h ( if ( B e. ~H , B , 0h ) -h D ) ) <-> ( ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) -h ( if ( C e. ~H , C , 0h ) -h if ( D e. ~H , D , 0h ) ) ) = ( ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) -h ( if ( B e. ~H , B , 0h ) -h if ( D e. ~H , D , 0h ) ) ) ) )
21		ifhvhv0 27879	. . 3 |- if ( A e. ~H , A , 0h ) e. ~H
22		ifhvhv0 27879	. . 3 |- if ( B e. ~H , B , 0h ) e. ~H
23		ifhvhv0 27879	. . 3 |- if ( C e. ~H , C , 0h ) e. ~H
24		ifhvhv0 27879	. . 3 |- if ( D e. ~H , D , 0h ) e. ~H
25	21, 22, 23, 24	hvsubsub4i 27916	. 2 |- ( ( if ( A e. ~H , A , 0h ) -h if ( B e. ~H , B , 0h ) ) -h ( if ( C e. ~H , C , 0h ) -h if ( D e. ~H , D , 0h ) ) ) = ( ( if ( A e. ~H , A , 0h ) -h if ( C e. ~H , C , 0h ) ) -h ( if ( B e. ~H , B , 0h ) -h if ( D e. ~H , D , 0h ) ) )
26	5, 10, 15, 20, 25	dedth4h 4142	1 |- ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( A -h B ) -h ( C -h D ) ) = ( ( A -h C ) -h ( B -h D ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086  (class class class)co 6650   ~Hchil 27776   0hc0v 27781   -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-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  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:  chscllem2  28497  5oalem3  28515  5oalem5  28517