Metamath Proof Explorer

Theorem mapdpglem32

Description: Lemma for mapdpg . Uniqueness part - consolidate hypotheses in mapdpglem31 . (Contributed by NM, 23-Mar-2015)

Ref Expression
Hypotheses mapdpg.h 
|- H = ( LHyp ` K )
mapdpg.m 
|- M = ( ( mapd ` K ) ` W )
mapdpg.u 
|- U = ( ( DVecH ` K ) ` W )
mapdpg.v 
|- V = ( Base ` U )
mapdpg.s 
|- .- = ( -g ` U )
mapdpg.z 
|- .0. = ( 0g ` U )
mapdpg.n 
|- N = ( LSpan ` U )
mapdpg.c 
|- C = ( ( LCDual ` K ) ` W )
mapdpg.f 
|- F = ( Base ` C )
mapdpg.r 
|- R = ( -g ` C )
mapdpg.j 
|- J = ( LSpan ` C )
mapdpg.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdpg.x 
|- ( ph -> X e. ( V \ { .0. } ) )
mapdpg.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
mapdpg.g 
|- ( ph -> G e. F )
mapdpg.ne 
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
mapdpg.e 
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { G } ) )
Assertion mapdpglem32 
|- ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> h = i )

Proof

Step Hyp Ref Expression
1 mapdpg.h 
 |-  H = ( LHyp ` K )
2 mapdpg.m 
 |-  M = ( ( mapd ` K ) ` W )
3 mapdpg.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 mapdpg.v 
 |-  V = ( Base ` U )
5 mapdpg.s 
 |-  .- = ( -g ` U )
6 mapdpg.z 
 |-  .0. = ( 0g ` U )
7 mapdpg.n 
 |-  N = ( LSpan ` U )
8 mapdpg.c 
 |-  C = ( ( LCDual ` K ) ` W )
9 mapdpg.f 
 |-  F = ( Base ` C )
10 mapdpg.r 
 |-  R = ( -g ` C )
11 mapdpg.j 
 |-  J = ( LSpan ` C )
12 mapdpg.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
13 mapdpg.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
14 mapdpg.y 
 |-  ( ph -> Y e. ( V \ { .0. } ) )
15 mapdpg.g 
 |-  ( ph -> G e. F )
16 mapdpg.ne 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
17 mapdpg.e 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { G } ) )
18 12 3ad2ant1 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> ( K e. HL /\ W e. H ) )
19 13 3ad2ant1 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> X e. ( V \ { .0. } ) )
20 14 3ad2ant1 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> Y e. ( V \ { .0. } ) )
21 15 3ad2ant1 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> G e. F )
22 16 3ad2ant1 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> ( N ` { X } ) =/= ( N ` { Y } ) )
23 17 3ad2ant1 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { G } ) )
24 simp2l 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> h e. F )
25 simp3l 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) )
26 24 25 jca 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> ( h e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) ) )
27 simp2r 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> i e. F )
28 simp3r 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) )
29 27 28 jca 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> ( i e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) )
30 eqid 
 |-  ( Scalar ` U ) = ( Scalar ` U )
31 eqid 
 |-  ( Base ` ( Scalar ` U ) ) = ( Base ` ( Scalar ` U ) )
32 eqid 
 |-  ( .s ` C ) = ( .s ` C )
33 eqid 
 |-  ( 0g ` ( Scalar ` U ) ) = ( 0g ` ( Scalar ` U ) )
34 1 2 3 4 5 6 7 8 9 10 11 18 19 20 21 22 23 26 29 30 31 32 33 mapdpglem26 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> E. u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) h = ( u ( .s ` C ) i ) )
35 1 2 3 4 5 6 7 8 9 10 11 18 19 20 21 22 23 26 29 30 31 32 33 mapdpglem27 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> E. v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ( G R h ) = ( v ( .s ` C ) ( G R i ) ) )
36 reeanv 
 |-  ( E. u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) E. v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) <-> ( E. u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) h = ( u ( .s ` C ) i ) /\ E. v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) )
37 34 35 36 sylanbrc 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> E. u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) E. v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) )
38 18 3ad2ant1 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> ( K e. HL /\ W e. H ) )
39 19 3ad2ant1 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> X e. ( V \ { .0. } ) )
40 20 3ad2ant1 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> Y e. ( V \ { .0. } ) )
41 21 3ad2ant1 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> G e. F )
42 22 3ad2ant1 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> ( N ` { X } ) =/= ( N ` { Y } ) )
43 23 3ad2ant1 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { G } ) )
44 simp12l 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> h e. F )
45 simp13l 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) )
46 44 45 jca 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> ( h e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) ) )
47 simp12r 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> i e. F )
48 simp13r 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) )
49 47 48 jca 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> ( i e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) )
50 eldifi 
 |-  ( v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) -> v e. ( Base ` ( Scalar ` U ) ) )
51 50 adantl 
 |-  ( ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) -> v e. ( Base ` ( Scalar ` U ) ) )
52 51 3ad2ant2 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> v e. ( Base ` ( Scalar ` U ) ) )
53 simp3l 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> h = ( u ( .s ` C ) i ) )
54 simp3r 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> ( G R h ) = ( v ( .s ` C ) ( G R i ) ) )
55 eldifi 
 |-  ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) -> u e. ( Base ` ( Scalar ` U ) ) )
56 55 adantr 
 |-  ( ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) -> u e. ( Base ` ( Scalar ` U ) ) )
57 56 3ad2ant2 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> u e. ( Base ` ( Scalar ` U ) ) )
58 1 2 3 4 5 6 7 8 9 10 11 38 39 40 41 42 43 46 49 30 31 32 33 52 53 54 57 mapdpglem31 
 |-  ( ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) /\ ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) /\ ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) ) -> h = i )
59 58 3exp 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> ( ( u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) /\ v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ) -> ( ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) -> h = i ) ) )
60 59 rexlimdvv 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> ( E. u e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) E. v e. ( ( Base ` ( Scalar ` U ) ) \ { ( 0g ` ( Scalar ` U ) ) } ) ( h = ( u ( .s ` C ) i ) /\ ( G R h ) = ( v ( .s ` C ) ( G R i ) ) ) -> h = i ) )
61 37 60 mpd 
 |-  ( ( ph /\ ( h e. F /\ i e. F ) /\ ( ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) ) -> h = i )