Metamath Proof Explorer

Theorem lcfrlem39

Description: Lemma for lcfr . Eliminate J . (Contributed by NM, 11-Mar-2015)

Ref Expression
Hypotheses lcfrlem38.h 
|- H = ( LHyp ` K )
lcfrlem38.o 
|- ._|_ = ( ( ocH ` K ) ` W )
lcfrlem38.u 
|- U = ( ( DVecH ` K ) ` W )
lcfrlem38.p 
|- .+ = ( +g ` U )
lcfrlem38.f 
|- F = ( LFnl ` U )
lcfrlem38.l 
|- L = ( LKer ` U )
lcfrlem38.d 
|- D = ( LDual ` U )
lcfrlem38.q 
|- Q = ( LSubSp ` D )
lcfrlem38.c 
|- C = { f e. ( LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
lcfrlem38.e 
|- E = U_ g e. G ( ._|_ ` ( L ` g ) )
lcfrlem38.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
lcfrlem38.g 
|- ( ph -> G e. Q )
lcfrlem38.gs 
|- ( ph -> G C_ C )
lcfrlem38.xe 
|- ( ph -> X e. E )
lcfrlem38.ye 
|- ( ph -> Y e. E )
lcfrlem38.z 
|- .0. = ( 0g ` U )
lcfrlem38.x 
|- ( ph -> X =/= .0. )
lcfrlem38.y 
|- ( ph -> Y =/= .0. )
lcfrlem38.sp 
|- N = ( LSpan ` U )
lcfrlem38.ne 
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
lcfrlem38.b 
|- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )
lcfrlem38.i 
|- ( ph -> I e. B )
lcfrlem38.n 
|- ( ph -> I =/= .0. )
Assertion lcfrlem39 
|- ( ph -> ( X .+ Y ) e. E )

Proof

Step Hyp Ref Expression
1 lcfrlem38.h 
 |-  H = ( LHyp ` K )
2 lcfrlem38.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
3 lcfrlem38.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 lcfrlem38.p 
 |-  .+ = ( +g ` U )
5 lcfrlem38.f 
 |-  F = ( LFnl ` U )
6 lcfrlem38.l 
 |-  L = ( LKer ` U )
7 lcfrlem38.d 
 |-  D = ( LDual ` U )
8 lcfrlem38.q 
 |-  Q = ( LSubSp ` D )
9 lcfrlem38.c 
 |-  C = { f e. ( LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
10 lcfrlem38.e 
 |-  E = U_ g e. G ( ._|_ ` ( L ` g ) )
11 lcfrlem38.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
12 lcfrlem38.g 
 |-  ( ph -> G e. Q )
13 lcfrlem38.gs 
 |-  ( ph -> G C_ C )
14 lcfrlem38.xe 
 |-  ( ph -> X e. E )
15 lcfrlem38.ye 
 |-  ( ph -> Y e. E )
16 lcfrlem38.z 
 |-  .0. = ( 0g ` U )
17 lcfrlem38.x 
 |-  ( ph -> X =/= .0. )
18 lcfrlem38.y 
 |-  ( ph -> Y =/= .0. )
19 lcfrlem38.sp 
 |-  N = ( LSpan ` U )
20 lcfrlem38.ne 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
21 lcfrlem38.b 
 |-  B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )
22 lcfrlem38.i 
 |-  ( ph -> I e. B )
23 lcfrlem38.n 
 |-  ( ph -> I =/= .0. )
24 eqid 
 |-  ( Base ` U ) = ( Base ` U )
25 eqid 
 |-  ( .s ` U ) = ( .s ` U )
26 eqid 
 |-  ( Scalar ` U ) = ( Scalar ` U )
27 eqid 
 |-  ( Base ` ( Scalar ` U ) ) = ( Base ` ( Scalar ` U ) )
28 oveq1 
 |-  ( j = k -> ( j ( .s ` U ) x ) = ( k ( .s ` U ) x ) )
29 28 oveq2d 
 |-  ( j = k -> ( w .+ ( j ( .s ` U ) x ) ) = ( w .+ ( k ( .s ` U ) x ) ) )
30 29 eqeq2d 
 |-  ( j = k -> ( v = ( w .+ ( j ( .s ` U ) x ) ) <-> v = ( w .+ ( k ( .s ` U ) x ) ) ) )
31 30 rexbidv 
 |-  ( j = k -> ( E. w e. ( ._|_ ` { x } ) v = ( w .+ ( j ( .s ` U ) x ) ) <-> E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k ( .s ` U ) x ) ) ) )
32 31 cbvriotavw 
 |-  ( iota_ j e. ( Base ` ( Scalar ` U ) ) E. w e. ( ._|_ ` { x } ) v = ( w .+ ( j ( .s ` U ) x ) ) ) = ( iota_ k e. ( Base ` ( Scalar ` U ) ) E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k ( .s ` U ) x ) ) )
33 32 mpteq2i 
 |-  ( v e. ( Base ` U ) |-> ( iota_ j e. ( Base ` ( Scalar ` U ) ) E. w e. ( ._|_ ` { x } ) v = ( w .+ ( j ( .s ` U ) x ) ) ) ) = ( v e. ( Base ` U ) |-> ( iota_ k e. ( Base ` ( Scalar ` U ) ) E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k ( .s ` U ) x ) ) ) )
34 33 mpteq2i 
 |-  ( x e. ( ( Base ` U ) \ { .0. } ) |-> ( v e. ( Base ` U ) |-> ( iota_ j e. ( Base ` ( Scalar ` U ) ) E. w e. ( ._|_ ` { x } ) v = ( w .+ ( j ( .s ` U ) x ) ) ) ) ) = ( x e. ( ( Base ` U ) \ { .0. } ) |-> ( v e. ( Base ` U ) |-> ( iota_ k e. ( Base ` ( Scalar ` U ) ) E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k ( .s ` U ) x ) ) ) ) )
35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 34 lcfrlem38 
 |-  ( ph -> ( X .+ Y ) e. E )