Metamath Proof Explorer

Theorem mapdhcl

Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015)

Ref Expression
Hypotheses mapdh.q 
|- Q = ( 0g ` C )
mapdh.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 ) } ) ) ) ) )
mapdh.h 
|- H = ( LHyp ` K )
mapdh.m 
|- M = ( ( mapd ` K ) ` W )
mapdh.u 
|- U = ( ( DVecH ` K ) ` W )
mapdh.v 
|- V = ( Base ` U )
mapdh.s 
|- .- = ( -g ` U )
mapdhc.o 
|- .0. = ( 0g ` U )
mapdh.n 
|- N = ( LSpan ` U )
mapdh.c 
|- C = ( ( LCDual ` K ) ` W )
mapdh.d 
|- D = ( Base ` C )
mapdh.r 
|- R = ( -g ` C )
mapdh.j 
|- J = ( LSpan ` C )
mapdh.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdhc.f 
|- ( ph -> F e. D )
mapdh.mn 
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
mapdhcl.x 
|- ( ph -> X e. ( V \ { .0. } ) )
mapdhc.y 
|- ( ph -> Y e. V )
mapdh.ne 
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
Assertion mapdhcl 
|- ( ph -> ( I ` <. X , F , Y >. ) e. D )

Proof

Step Hyp Ref Expression
1 mapdh.q 
 |-  Q = ( 0g ` C )
2 mapdh.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 ) } ) ) ) ) )
3 mapdh.h 
 |-  H = ( LHyp ` K )
4 mapdh.m 
 |-  M = ( ( mapd ` K ) ` W )
5 mapdh.u 
 |-  U = ( ( DVecH ` K ) ` W )
6 mapdh.v 
 |-  V = ( Base ` U )
7 mapdh.s 
 |-  .- = ( -g ` U )
8 mapdhc.o 
 |-  .0. = ( 0g ` U )
9 mapdh.n 
 |-  N = ( LSpan ` U )
10 mapdh.c 
 |-  C = ( ( LCDual ` K ) ` W )
11 mapdh.d 
 |-  D = ( Base ` C )
12 mapdh.r 
 |-  R = ( -g ` C )
13 mapdh.j 
 |-  J = ( LSpan ` C )
14 mapdh.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
15 mapdhc.f 
 |-  ( ph -> F e. D )
16 mapdh.mn 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
17 mapdhcl.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
18 mapdhc.y 
 |-  ( ph -> Y e. V )
19 mapdh.ne 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
20 oteq3 
 |-  ( Y = .0. -> <. X , F , Y >. = <. X , F , .0. >. )
21 20 fveq2d 
 |-  ( Y = .0. -> ( I ` <. X , F , Y >. ) = ( I ` <. X , F , .0. >. ) )
22 21 eleq1d 
 |-  ( Y = .0. -> ( ( I ` <. X , F , Y >. ) e. D <-> ( I ` <. X , F , .0. >. ) e. D ) )
23 17 adantr 
 |-  ( ( ph /\ Y =/= .0. ) -> X e. ( V \ { .0. } ) )
24 15 adantr 
 |-  ( ( ph /\ Y =/= .0. ) -> F e. D )
25 18 anim1i 
 |-  ( ( ph /\ Y =/= .0. ) -> ( Y e. V /\ Y =/= .0. ) )
26 eldifsn 
 |-  ( Y e. ( V \ { .0. } ) <-> ( Y e. V /\ Y =/= .0. ) )
27 25 26 sylibr 
 |-  ( ( ph /\ Y =/= .0. ) -> Y e. ( V \ { .0. } ) )
28 1 2 23 24 27 mapdhval2 
 |-  ( ( ph /\ Y =/= .0. ) -> ( I ` <. X , F , Y >. ) = ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) )
29 14 adantr 
 |-  ( ( ph /\ Y =/= .0. ) -> ( K e. HL /\ W e. H ) )
30 19 adantr 
 |-  ( ( ph /\ Y =/= .0. ) -> ( N ` { X } ) =/= ( N ` { Y } ) )
31 16 adantr 
 |-  ( ( ph /\ Y =/= .0. ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
32 3 4 5 6 7 8 9 10 11 12 13 29 23 27 24 30 31 mapdpg 
 |-  ( ( ph /\ Y =/= .0. ) -> E! h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) )
33 riotacl 
 |-  ( E! h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) -> ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) e. D )
34 32 33 syl 
 |-  ( ( ph /\ Y =/= .0. ) -> ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) e. D )
35 28 34 eqeltrd 
 |-  ( ( ph /\ Y =/= .0. ) -> ( I ` <. X , F , Y >. ) e. D )
36 1 2 8 17 15 mapdhval0 
 |-  ( ph -> ( I ` <. X , F , .0. >. ) = Q )
37 3 10 11 1 14 lcd0vcl 
 |-  ( ph -> Q e. D )
38 36 37 eqeltrd 
 |-  ( ph -> ( I ` <. X , F , .0. >. ) e. D )
39 22 35 38 pm2.61ne 
 |-  ( ph -> ( I ` <. X , F , Y >. ) e. D )