Metamath Proof Explorer

Theorem mapdheq

Description: Lemmma for ~? mapdh . The defining equation for h(x,x',y)=y' in part (2) in Baer p. 45 line 24. (Contributed by NM, 4-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. } ) )
mapdhe.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
mapdhe.g 
|- ( ph -> G e. D )
mapdh.ne2 
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
Assertion mapdheq 
|- ( ph -> ( ( I ` <. X , F , Y >. ) = G <-> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) )

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 mapdhe.y 
 |-  ( ph -> Y e. ( V \ { .0. } ) )
19 mapdhe.g 
 |-  ( ph -> G e. D )
20 mapdh.ne2 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
21 1 2 17 15 18 mapdhval2 
 |-  ( ph -> ( I ` <. X , F , Y >. ) = ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) )
22 21 eqeq1d 
 |-  ( ph -> ( ( I ` <. X , F , Y >. ) = G <-> ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) = G ) )
23 3 4 5 6 7 8 9 10 11 12 13 14 17 18 15 20 16 mapdpg 
 |-  ( ph -> E! h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) )
24 nfv 
 |-  F/ h ph
25 nfcvd 
 |-  ( ph -> F/_ h G )
26 nfvd 
 |-  ( ph -> F/ h ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) )
27 sneq 
 |-  ( h = G -> { h } = { G } )
28 27 fveq2d 
 |-  ( h = G -> ( J ` { h } ) = ( J ` { G } ) )
29 28 eqeq2d 
 |-  ( h = G -> ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) <-> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) ) )
30 oveq2 
 |-  ( h = G -> ( F R h ) = ( F R G ) )
31 30 sneqd 
 |-  ( h = G -> { ( F R h ) } = { ( F R G ) } )
32 31 fveq2d 
 |-  ( h = G -> ( J ` { ( F R h ) } ) = ( J ` { ( F R G ) } ) )
33 32 eqeq2d 
 |-  ( h = G -> ( ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) <-> ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) )
34 29 33 anbi12d 
 |-  ( h = G -> ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) <-> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) )
35 34 adantl 
 |-  ( ( ph /\ h = G ) -> ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) <-> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) )
36 24 25 26 19 35 riota2df 
 |-  ( ( ph /\ E! h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) -> ( ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) <-> ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) = G ) )
37 23 36 mpdan 
 |-  ( ph -> ( ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) <-> ( iota_ h e. D ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R h ) } ) ) ) = G ) )
38 22 37 bitr4d 
 |-  ( ph -> ( ( I ` <. X , F , Y >. ) = G <-> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) )