Metamath Proof Explorer

Theorem mapdpglem2

Description: Lemma for mapdpg . Baer p. 45, lines 1 and 2: "we have (F(x-y))* = Gt where t necessarily belongs to (Fx)*+(Fy)*." (We scope $d t ph locally to avoid clashes with later substitutions into ph .) (Contributed by NM, 15-Mar-2015)

Ref Expression
Hypotheses mapdpglem.h 
|- H = ( LHyp ` K )
mapdpglem.m 
|- M = ( ( mapd ` K ) ` W )
mapdpglem.u 
|- U = ( ( DVecH ` K ) ` W )
mapdpglem.v 
|- V = ( Base ` U )
mapdpglem.s 
|- .- = ( -g ` U )
mapdpglem.n 
|- N = ( LSpan ` U )
mapdpglem.c 
|- C = ( ( LCDual ` K ) ` W )
mapdpglem.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdpglem.x 
|- ( ph -> X e. V )
mapdpglem.y 
|- ( ph -> Y e. V )
mapdpglem1.p 
|- .(+) = ( LSSum ` C )
mapdpglem2.j 
|- J = ( LSpan ` C )
Assertion mapdpglem2 
|- ( ph -> E. t e. ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) )

Proof

Step Hyp Ref Expression
1 mapdpglem.h 
 |-  H = ( LHyp ` K )
2 mapdpglem.m 
 |-  M = ( ( mapd ` K ) ` W )
3 mapdpglem.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 mapdpglem.v 
 |-  V = ( Base ` U )
5 mapdpglem.s 
 |-  .- = ( -g ` U )
6 mapdpglem.n 
 |-  N = ( LSpan ` U )
7 mapdpglem.c 
 |-  C = ( ( LCDual ` K ) ` W )
8 mapdpglem.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
9 mapdpglem.x 
 |-  ( ph -> X e. V )
10 mapdpglem.y 
 |-  ( ph -> Y e. V )
11 mapdpglem1.p 
 |-  .(+) = ( LSSum ` C )
12 mapdpglem2.j 
 |-  J = ( LSpan ` C )
13 eqid 
 |-  ( Base ` C ) = ( Base ` C )
14 1 3 8 dvhlmod 
 |-  ( ph -> U e. LMod )
15 4 5 lmodvsubcl 
 |-  ( ( U e. LMod /\ X e. V /\ Y e. V ) -> ( X .- Y ) e. V )
16 14 9 10 15 syl3anc 
 |-  ( ph -> ( X .- Y ) e. V )
17 1 2 3 4 6 7 13 12 8 16 mapdspex 
 |-  ( ph -> E. t e. ( Base ` C ) ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) )
18 1 7 8 lcdlmod 
 |-  ( ph -> C e. LMod )
19 13 12 lspsnid 
 |-  ( ( C e. LMod /\ t e. ( Base ` C ) ) -> t e. ( J ` { t } ) )
20 18 19 sylan 
 |-  ( ( ph /\ t e. ( Base ` C ) ) -> t e. ( J ` { t } ) )
21 20 adantrr 
 |-  ( ( ph /\ ( t e. ( Base ` C ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) ) ) -> t e. ( J ` { t } ) )
22 simprr 
 |-  ( ( ph /\ ( t e. ( Base ` C ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) ) ) -> ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) )
23 21 22 eleqtrrd 
 |-  ( ( ph /\ ( t e. ( Base ` C ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) ) ) -> t e. ( M ` ( N ` { ( X .- Y ) } ) ) )
24 17 23 22 reximssdv 
 |-  ( ph -> E. t e. ( M ` ( N ` { ( X .- Y ) } ) ) ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) )
25 1 2 3 4 5 6 7 8 9 10 11 mapdpglem1 
 |-  ( ph -> ( M ` ( N ` { ( X .- Y ) } ) ) C_ ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) )
26 25 sseld 
 |-  ( ph -> ( t e. ( M ` ( N ` { ( X .- Y ) } ) ) -> t e. ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) ) )
27 26 anim1d 
 |-  ( ph -> ( ( t e. ( M ` ( N ` { ( X .- Y ) } ) ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) ) -> ( t e. ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) ) ) )
28 27 reximdv2 
 |-  ( ph -> ( E. t e. ( M ` ( N ` { ( X .- Y ) } ) ) ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) -> E. t e. ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) ) )
29 24 28 mpd 
 |-  ( ph -> E. t e. ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { t } ) )