Metamath Proof Explorer


Theorem mapdh75cN

Description: Part (7) of Baer p. 48 line 10 (3 of 6 cases). (Contributed by NM, 2-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses mapdh75.h
|- H = ( LHyp ` K )
mapdh75.u
|- U = ( ( DVecH ` K ) ` W )
mapdh75.v
|- V = ( Base ` U )
mapdh75.s
|- .- = ( -g ` U )
mapdh75.o
|- .0. = ( 0g ` U )
mapdh75.n
|- N = ( LSpan ` U )
mapdh75.c
|- C = ( ( LCDual ` K ) ` W )
mapdh75.d
|- D = ( Base ` C )
mapdh75.r
|- R = ( -g ` C )
mapdh75.q
|- Q = ( 0g ` C )
mapdh75.j
|- J = ( LSpan ` C )
mapdh75.m
|- M = ( ( mapd ` K ) ` W )
mapdh75.i
|- I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
mapdh75.k
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdh75.f
|- ( ph -> F e. D )
mapdh75.mn
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
mapdh75a
|- ( ph -> ( I ` <. X , F , Y >. ) = G )
mapdh75c.ne
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
mapdh75c.x
|- ( ph -> X e. ( V \ { .0. } ) )
mapdh75c.y
|- ( ph -> Y e. ( V \ { .0. } ) )
Assertion mapdh75cN
|- ( ph -> ( I ` <. Y , G , X >. ) = F )

Proof

Step Hyp Ref Expression
1 mapdh75.h
 |-  H = ( LHyp ` K )
2 mapdh75.u
 |-  U = ( ( DVecH ` K ) ` W )
3 mapdh75.v
 |-  V = ( Base ` U )
4 mapdh75.s
 |-  .- = ( -g ` U )
5 mapdh75.o
 |-  .0. = ( 0g ` U )
6 mapdh75.n
 |-  N = ( LSpan ` U )
7 mapdh75.c
 |-  C = ( ( LCDual ` K ) ` W )
8 mapdh75.d
 |-  D = ( Base ` C )
9 mapdh75.r
 |-  R = ( -g ` C )
10 mapdh75.q
 |-  Q = ( 0g ` C )
11 mapdh75.j
 |-  J = ( LSpan ` C )
12 mapdh75.m
 |-  M = ( ( mapd ` K ) ` W )
13 mapdh75.i
 |-  I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
14 mapdh75.k
 |-  ( ph -> ( K e. HL /\ W e. H ) )
15 mapdh75.f
 |-  ( ph -> F e. D )
16 mapdh75.mn
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
17 mapdh75a
 |-  ( ph -> ( I ` <. X , F , Y >. ) = G )
18 mapdh75c.ne
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
19 mapdh75c.x
 |-  ( ph -> X e. ( V \ { .0. } ) )
20 mapdh75c.y
 |-  ( ph -> Y e. ( V \ { .0. } ) )
21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 mapdh75e
 |-  ( ph -> ( I ` <. Y , G , X >. ) = F )