Metamath Proof Explorer

Theorem mapdheq2

Description: Lemmma for ~? mapdh . One direction of part (2) in Baer p. 45. (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 mapdheq2 
|- ( ph -> ( ( I ` <. X , F , Y >. ) = G -> ( I ` <. Y , G , X >. ) = F ) )

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 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 mapdheq 
 |-  ( ph -> ( ( I ` <. X , F , Y >. ) = G <-> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) )
22 16 adantr 
 |-  ( ( ph /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
23 3 5 14 dvhlmod 
 |-  ( ph -> U e. LMod )
24 17 eldifad 
 |-  ( ph -> X e. V )
25 18 eldifad 
 |-  ( ph -> Y e. V )
26 6 7 9 23 24 25 lspsnsub 
 |-  ( ph -> ( N ` { ( X .- Y ) } ) = ( N ` { ( Y .- X ) } ) )
27 26 fveq2d 
 |-  ( ph -> ( M ` ( N ` { ( X .- Y ) } ) ) = ( M ` ( N ` { ( Y .- X ) } ) ) )
28 3 10 14 lcdlmod 
 |-  ( ph -> C e. LMod )
29 11 12 13 28 15 19 lspsnsub 
 |-  ( ph -> ( J ` { ( F R G ) } ) = ( J ` { ( G R F ) } ) )
30 27 29 eqeq12d 
 |-  ( ph -> ( ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) <-> ( M ` ( N ` { ( Y .- X ) } ) ) = ( J ` { ( G R F ) } ) ) )
31 30 biimpa 
 |-  ( ( ph /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) -> ( M ` ( N ` { ( Y .- X ) } ) ) = ( J ` { ( G R F ) } ) )
32 31 adantrl 
 |-  ( ( ph /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) -> ( M ` ( N ` { ( Y .- X ) } ) ) = ( J ` { ( G R F ) } ) )
33 14 adantr 
 |-  ( ( ph /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) -> ( K e. HL /\ W e. H ) )
34 19 adantr 
 |-  ( ( ph /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) -> G e. D )
35 simprl 
 |-  ( ( ph /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) -> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) )
36 18 adantr 
 |-  ( ( ph /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) -> Y e. ( V \ { .0. } ) )
37 17 adantr 
 |-  ( ( ph /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) -> X e. ( V \ { .0. } ) )
38 15 adantr 
 |-  ( ( ph /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) -> F e. D )
39 20 necomd 
 |-  ( ph -> ( N ` { Y } ) =/= ( N ` { X } ) )
40 39 adantr 
 |-  ( ( ph /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) -> ( N ` { Y } ) =/= ( N ` { X } ) )
41 1 2 3 4 5 6 7 8 9 10 11 12 13 33 34 35 36 37 38 40 mapdheq 
 |-  ( ( ph /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) -> ( ( I ` <. Y , G , X >. ) = F <-> ( ( M ` ( N ` { X } ) ) = ( J ` { F } ) /\ ( M ` ( N ` { ( Y .- X ) } ) ) = ( J ` { ( G R F ) } ) ) ) )
42 22 32 41 mpbir2and 
 |-  ( ( ph /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) -> ( I ` <. Y , G , X >. ) = F )
43 42 ex 
 |-  ( ph -> ( ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) -> ( I ` <. Y , G , X >. ) = F ) )
44 21 43 sylbid 
 |-  ( ph -> ( ( I ` <. X , F , Y >. ) = G -> ( I ` <. Y , G , X >. ) = F ) )