Metamath Proof Explorer

Theorem mapdpglem1

Description: Lemma for mapdpg . Baer p. 44, last line: "(F(x-y))* <= (Fx)*+(Fy)*." (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 )
Assertion mapdpglem1 
|- ( ph -> ( M ` ( N ` { ( X .- Y ) } ) ) C_ ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) )

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 1 3 8 dvhlmod 
 |-  ( ph -> U e. LMod )
13 eqid 
 |-  ( LSSum ` U ) = ( LSSum ` U )
14 4 5 13 6 lspsntrim 
 |-  ( ( U e. LMod /\ X e. V /\ Y e. V ) -> ( N ` { ( X .- Y ) } ) C_ ( ( N ` { X } ) ( LSSum ` U ) ( N ` { Y } ) ) )
15 12 9 10 14 syl3anc 
 |-  ( ph -> ( N ` { ( X .- Y ) } ) C_ ( ( N ` { X } ) ( LSSum ` U ) ( N ` { Y } ) ) )
16 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
17 4 5 lmodvsubcl 
 |-  ( ( U e. LMod /\ X e. V /\ Y e. V ) -> ( X .- Y ) e. V )
18 12 9 10 17 syl3anc 
 |-  ( ph -> ( X .- Y ) e. V )
19 4 16 6 lspsncl 
 |-  ( ( U e. LMod /\ ( X .- Y ) e. V ) -> ( N ` { ( X .- Y ) } ) e. ( LSubSp ` U ) )
20 12 18 19 syl2anc 
 |-  ( ph -> ( N ` { ( X .- Y ) } ) e. ( LSubSp ` U ) )
21 4 16 6 lspsncl 
 |-  ( ( U e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` U ) )
22 12 9 21 syl2anc 
 |-  ( ph -> ( N ` { X } ) e. ( LSubSp ` U ) )
23 4 16 6 lspsncl 
 |-  ( ( U e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` U ) )
24 12 10 23 syl2anc 
 |-  ( ph -> ( N ` { Y } ) e. ( LSubSp ` U ) )
25 16 13 lsmcl 
 |-  ( ( U e. LMod /\ ( N ` { X } ) e. ( LSubSp ` U ) /\ ( N ` { Y } ) e. ( LSubSp ` U ) ) -> ( ( N ` { X } ) ( LSSum ` U ) ( N ` { Y } ) ) e. ( LSubSp ` U ) )
26 12 22 24 25 syl3anc 
 |-  ( ph -> ( ( N ` { X } ) ( LSSum ` U ) ( N ` { Y } ) ) e. ( LSubSp ` U ) )
27 1 3 16 2 8 20 26 mapdord 
 |-  ( ph -> ( ( M ` ( N ` { ( X .- Y ) } ) ) C_ ( M ` ( ( N ` { X } ) ( LSSum ` U ) ( N ` { Y } ) ) ) <-> ( N ` { ( X .- Y ) } ) C_ ( ( N ` { X } ) ( LSSum ` U ) ( N ` { Y } ) ) ) )
28 15 27 mpbird 
 |-  ( ph -> ( M ` ( N ` { ( X .- Y ) } ) ) C_ ( M ` ( ( N ` { X } ) ( LSSum ` U ) ( N ` { Y } ) ) ) )
29 1 2 3 16 13 7 11 8 22 24 mapdlsm 
 |-  ( ph -> ( M ` ( ( N ` { X } ) ( LSSum ` U ) ( N ` { Y } ) ) ) = ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) )
30 28 29 sseqtrd 
 |-  ( ph -> ( M ` ( N ` { ( X .- Y ) } ) ) C_ ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) )