Metamath Proof Explorer

Theorem hdmapsub

Description: Part of proof of part 12 in Baer p. 49 line 5, (a-b)S = aS-bS in their notation (S = sigma). (Contributed by NM, 26-May-2015)

Ref Expression
Hypotheses hdmap12c.h 
|- H = ( LHyp ` K )
hdmap12c.u 
|- U = ( ( DVecH ` K ) ` W )
hdmap12c.v 
|- V = ( Base ` U )
hdmap12c.m 
|- .- = ( -g ` U )
hdmap12c.c 
|- C = ( ( LCDual ` K ) ` W )
hdmap12c.n 
|- N = ( -g ` C )
hdmap12c.s 
|- S = ( ( HDMap ` K ) ` W )
hdmap12c.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
hdmap12c.x 
|- ( ph -> X e. V )
hdmap12c.y 
|- ( ph -> Y e. V )
Assertion hdmapsub 
|- ( ph -> ( S ` ( X .- Y ) ) = ( ( S ` X ) N ( S ` Y ) ) )

Proof

Step Hyp Ref Expression
1 hdmap12c.h 
 |-  H = ( LHyp ` K )
2 hdmap12c.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 hdmap12c.v 
 |-  V = ( Base ` U )
4 hdmap12c.m 
 |-  .- = ( -g ` U )
5 hdmap12c.c 
 |-  C = ( ( LCDual ` K ) ` W )
6 hdmap12c.n 
 |-  N = ( -g ` C )
7 hdmap12c.s 
 |-  S = ( ( HDMap ` K ) ` W )
8 hdmap12c.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
9 hdmap12c.x 
 |-  ( ph -> X e. V )
10 hdmap12c.y 
 |-  ( ph -> Y e. V )
11 eqid 
 |-  ( +g ` U ) = ( +g ` U )
12 eqid 
 |-  ( invg ` U ) = ( invg ` U )
13 3 11 12 4 grpsubval 
 |-  ( ( X e. V /\ Y e. V ) -> ( X .- Y ) = ( X ( +g ` U ) ( ( invg ` U ) ` Y ) ) )
14 9 10 13 syl2anc 
 |-  ( ph -> ( X .- Y ) = ( X ( +g ` U ) ( ( invg ` U ) ` Y ) ) )
15 14 fveq2d 
 |-  ( ph -> ( S ` ( X .- Y ) ) = ( S ` ( X ( +g ` U ) ( ( invg ` U ) ` Y ) ) ) )
16 eqid 
 |-  ( +g ` C ) = ( +g ` C )
17 1 2 8 dvhlmod 
 |-  ( ph -> U e. LMod )
18 3 12 lmodvnegcl 
 |-  ( ( U e. LMod /\ Y e. V ) -> ( ( invg ` U ) ` Y ) e. V )
19 17 10 18 syl2anc 
 |-  ( ph -> ( ( invg ` U ) ` Y ) e. V )
20 1 2 3 11 5 16 7 8 9 19 hdmapadd 
 |-  ( ph -> ( S ` ( X ( +g ` U ) ( ( invg ` U ) ` Y ) ) ) = ( ( S ` X ) ( +g ` C ) ( S ` ( ( invg ` U ) ` Y ) ) ) )
21 eqid 
 |-  ( invg ` C ) = ( invg ` C )
22 1 2 3 12 5 21 7 8 10 hdmapneg 
 |-  ( ph -> ( S ` ( ( invg ` U ) ` Y ) ) = ( ( invg ` C ) ` ( S ` Y ) ) )
23 22 oveq2d 
 |-  ( ph -> ( ( S ` X ) ( +g ` C ) ( S ` ( ( invg ` U ) ` Y ) ) ) = ( ( S ` X ) ( +g ` C ) ( ( invg ` C ) ` ( S ` Y ) ) ) )
24 15 20 23 3eqtrd 
 |-  ( ph -> ( S ` ( X .- Y ) ) = ( ( S ` X ) ( +g ` C ) ( ( invg ` C ) ` ( S ` Y ) ) ) )
25 eqid 
 |-  ( Base ` C ) = ( Base ` C )
26 1 2 3 5 25 7 8 9 hdmapcl 
 |-  ( ph -> ( S ` X ) e. ( Base ` C ) )
27 1 2 3 5 25 7 8 10 hdmapcl 
 |-  ( ph -> ( S ` Y ) e. ( Base ` C ) )
28 25 16 21 6 grpsubval 
 |-  ( ( ( S ` X ) e. ( Base ` C ) /\ ( S ` Y ) e. ( Base ` C ) ) -> ( ( S ` X ) N ( S ` Y ) ) = ( ( S ` X ) ( +g ` C ) ( ( invg ` C ) ` ( S ` Y ) ) ) )
29 26 27 28 syl2anc 
 |-  ( ph -> ( ( S ` X ) N ( S ` Y ) ) = ( ( S ` X ) ( +g ` C ) ( ( invg ` C ) ` ( S ` Y ) ) ) )
30 24 29 eqtr4d 
 |-  ( ph -> ( S ` ( X .- Y ) ) = ( ( S ` X ) N ( S ` Y ) ) )