Metamath Proof Explorer

Theorem mapdpg

Description: Part 1 of proof of the first fundamental theorem of projective geometry. Part (1) in Baer p. 44. Our notation corresponds to Baer's as follows: M for *, N{ } for F(), J{ } for G(), X for x, G for x', Y for y, h for y'. TODO: Rename variables per mapdhval . (Contributed by NM, 22-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 mapdpg 
|- ( ph -> E! h e. F ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) )

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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 mapdpglem24 
 |-  ( ph -> E. h e. F ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) )
19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 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 )
20 19 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 ) } ) ) ) -> h = i ) ) )
21 20 ralrimivv 
 |-  ( ph -> A. h e. F A. 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 ) )
22 sneq 
 |-  ( h = i -> { h } = { i } )
23 22 fveq2d 
 |-  ( h = i -> ( J ` { h } ) = ( J ` { i } ) )
24 23 eqeq2d 
 |-  ( h = i -> ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) <-> ( M ` ( N ` { Y } ) ) = ( J ` { i } ) ) )
25 oveq2 
 |-  ( h = i -> ( G R h ) = ( G R i ) )
26 25 sneqd 
 |-  ( h = i -> { ( G R h ) } = { ( G R i ) } )
27 26 fveq2d 
 |-  ( h = i -> ( J ` { ( G R h ) } ) = ( J ` { ( G R i ) } ) )
28 27 eqeq2d 
 |-  ( h = i -> ( ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) <-> ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) )
29 24 28 anbi12d 
 |-  ( h = i -> ( ( ( 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 ) } ) ) ) )
30 29 reu4 
 |-  ( E! h e. F ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) <-> ( E. h e. F ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) /\ A. h e. F A. 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 ) ) )
31 18 21 30 sylanbrc 
 |-  ( ph -> E! h e. F ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) )