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Metamath Proof Explorer

Theorem dochexmidat

Description: Special case of excluded middle for the singleton of a vector. (Contributed by NM, 27-Oct-2014)

Ref Expression
Hypotheses dochexmidat.h 
|- H = ( LHyp ` K )
dochexmidat.o 
|- ._|_ = ( ( ocH ` K ) ` W )
dochexmidat.u 
|- U = ( ( DVecH ` K ) ` W )
dochexmidat.v 
|- V = ( Base ` U )
dochexmidat.z 
|- .0. = ( 0g ` U )
dochexmidat.r 
|- N = ( LSpan ` U )
dochexmidat.p 
|- .(+) = ( LSSum ` U )
dochexmidat.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dochexmidat.x 
|- ( ph -> X e. ( V \ { .0. } ) )
Assertion dochexmidat 
|- ( ph -> ( ( ._|_ ` { X } ) .(+) ( N ` { X } ) ) = V )

Proof

Step Hyp Ref Expression
1 dochexmidat.h 
 |-  H = ( LHyp ` K )
2 dochexmidat.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
3 dochexmidat.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 dochexmidat.v 
 |-  V = ( Base ` U )
5 dochexmidat.z 
 |-  .0. = ( 0g ` U )
6 dochexmidat.r 
 |-  N = ( LSpan ` U )
7 dochexmidat.p 
 |-  .(+) = ( LSSum ` U )
8 dochexmidat.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
9 dochexmidat.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
10 1 2 3 4 5 8 9 dochnel 
 |-  ( ph -> -. X e. ( ._|_ ` { X } ) )
11 eqid 
 |-  ( LSHyp ` U ) = ( LSHyp ` U )
12 1 3 8 dvhlvec 
 |-  ( ph -> U e. LVec )
13 1 2 3 4 5 11 8 9 dochsnshp 
 |-  ( ph -> ( ._|_ ` { X } ) e. ( LSHyp ` U ) )
14 9 eldifad 
 |-  ( ph -> X e. V )
15 4 6 7 11 12 13 14 lshpnelb 
 |-  ( ph -> ( -. X e. ( ._|_ ` { X } ) <-> ( ( ._|_ ` { X } ) .(+) ( N ` { X } ) ) = V ) )
16 10 15 mpbid 
 |-  ( ph -> ( ( ._|_ ` { X } ) .(+) ( N ` { X } ) ) = V )