Metamath Proof Explorer

Theorem mapdpglem2a

Description: Lemma for mapdpg . (Contributed by NM, 20-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 )
mapdpglem3.f 
|- F = ( Base ` C )
mapdpglem3.te 
|- ( ph -> t e. ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) )
Assertion mapdpglem2a 
|- ( ph -> t e. F )

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 mapdpglem3.f 
 |-  F = ( Base ` C )
14 mapdpglem3.te 
 |-  ( ph -> t e. ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) )
15 1 7 8 lcdlmod 
 |-  ( ph -> C e. LMod )
16 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
17 eqid 
 |-  ( LSubSp ` C ) = ( LSubSp ` C )
18 1 3 8 dvhlmod 
 |-  ( ph -> U e. LMod )
19 4 16 6 lspsncl 
 |-  ( ( U e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` U ) )
20 18 9 19 syl2anc 
 |-  ( ph -> ( N ` { X } ) e. ( LSubSp ` U ) )
21 1 2 3 16 7 17 8 20 mapdcl2 
 |-  ( ph -> ( M ` ( N ` { X } ) ) e. ( LSubSp ` C ) )
22 4 16 6 lspsncl 
 |-  ( ( U e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` U ) )
23 18 10 22 syl2anc 
 |-  ( ph -> ( N ` { Y } ) e. ( LSubSp ` U ) )
24 1 2 3 16 7 17 8 23 mapdcl2 
 |-  ( ph -> ( M ` ( N ` { Y } ) ) e. ( LSubSp ` C ) )
25 17 11 lsmcl 
 |-  ( ( C e. LMod /\ ( M ` ( N ` { X } ) ) e. ( LSubSp ` C ) /\ ( M ` ( N ` { Y } ) ) e. ( LSubSp ` C ) ) -> ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) e. ( LSubSp ` C ) )
26 15 21 24 25 syl3anc 
 |-  ( ph -> ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) e. ( LSubSp ` C ) )
27 13 17 lssel 
 |-  ( ( ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) e. ( LSubSp ` C ) /\ t e. ( ( M ` ( N ` { X } ) ) .(+) ( M ` ( N ` { Y } ) ) ) ) -> t e. F )
28 26 14 27 syl2anc 
 |-  ( ph -> t e. F )