Metamath Proof Explorer

Theorem mapdh8b

Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 6-May-2015)

Ref Expression
Hypotheses mapdh8a.h 
|- H = ( LHyp ` K )
mapdh8a.u 
|- U = ( ( DVecH ` K ) ` W )
mapdh8a.v 
|- V = ( Base ` U )
mapdh8a.s 
|- .- = ( -g ` U )
mapdh8a.o 
|- .0. = ( 0g ` U )
mapdh8a.n 
|- N = ( LSpan ` U )
mapdh8a.c 
|- C = ( ( LCDual ` K ) ` W )
mapdh8a.d 
|- D = ( Base ` C )
mapdh8a.r 
|- R = ( -g ` C )
mapdh8a.q 
|- Q = ( 0g ` C )
mapdh8a.j 
|- J = ( LSpan ` C )
mapdh8a.m 
|- M = ( ( mapd ` K ) ` W )
mapdh8a.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 ) } ) ) ) ) )
mapdh8a.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdh8b.f 
|- ( ph -> G e. D )
mapdh8b.mn 
|- ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) )
mapdh8b.a 
|- ( ph -> ( I ` <. Y , G , w >. ) = E )
mapdh8b.x 
|- ( ph -> Y e. ( V \ { .0. } ) )
mapdh8b.y 
|- ( ph -> w e. ( V \ { .0. } ) )
mapdh8b.yz 
|- ( ph -> ( N ` { w } ) =/= ( N ` { T } ) )
mapdh8b.xt 
|- ( ph -> T e. ( V \ { .0. } ) )
mapdh8b.vw 
|- ( ph -> ( N ` { Y } ) =/= ( N ` { w } ) )
mapdh8b.e 
|- ( ph -> X e. ( N ` { Y , T } ) )
mapdh8b.xn 
|- ( ph -> -. X e. ( N ` { Y , w } ) )
Assertion mapdh8b 
|- ( ph -> ( I ` <. w , E , T >. ) = ( I ` <. Y , G , T >. ) )

Proof

Step Hyp Ref Expression
1 mapdh8a.h 
 |-  H = ( LHyp ` K )
2 mapdh8a.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 mapdh8a.v 
 |-  V = ( Base ` U )
4 mapdh8a.s 
 |-  .- = ( -g ` U )
5 mapdh8a.o 
 |-  .0. = ( 0g ` U )
6 mapdh8a.n 
 |-  N = ( LSpan ` U )
7 mapdh8a.c 
 |-  C = ( ( LCDual ` K ) ` W )
8 mapdh8a.d 
 |-  D = ( Base ` C )
9 mapdh8a.r 
 |-  R = ( -g ` C )
10 mapdh8a.q 
 |-  Q = ( 0g ` C )
11 mapdh8a.j 
 |-  J = ( LSpan ` C )
12 mapdh8a.m 
 |-  M = ( ( mapd ` K ) ` W )
13 mapdh8a.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 ) } ) ) ) ) )
14 mapdh8a.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
15 mapdh8b.f 
 |-  ( ph -> G e. D )
16 mapdh8b.mn 
 |-  ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) )
17 mapdh8b.a 
 |-  ( ph -> ( I ` <. Y , G , w >. ) = E )
18 mapdh8b.x 
 |-  ( ph -> Y e. ( V \ { .0. } ) )
19 mapdh8b.y 
 |-  ( ph -> w e. ( V \ { .0. } ) )
20 mapdh8b.yz 
 |-  ( ph -> ( N ` { w } ) =/= ( N ` { T } ) )
21 mapdh8b.xt 
 |-  ( ph -> T e. ( V \ { .0. } ) )
22 mapdh8b.vw 
 |-  ( ph -> ( N ` { Y } ) =/= ( N ` { w } ) )
23 mapdh8b.e 
 |-  ( ph -> X e. ( N ` { Y , T } ) )
24 mapdh8b.xn 
 |-  ( ph -> -. X e. ( N ` { Y , w } ) )
25 1 2 14 dvhlvec 
 |-  ( ph -> U e. LVec )
26 18 eldifad 
 |-  ( ph -> Y e. V )
27 19 eldifad 
 |-  ( ph -> w e. V )
28 21 eldifad 
 |-  ( ph -> T e. V )
29 3 6 25 26 27 28 23 24 lspindp5 
 |-  ( ph -> -. T e. ( N ` { Y , w } ) )
30 prcom 
 |-  { w , T } = { T , w }
31 30 fveq2i 
 |-  ( N ` { w , T } ) = ( N ` { T , w } )
32 31 eleq2i 
 |-  ( Y e. ( N ` { w , T } ) <-> Y e. ( N ` { T , w } ) )
33 25 adantr 
 |-  ( ( ph /\ Y e. ( N ` { T , w } ) ) -> U e. LVec )
34 18 adantr 
 |-  ( ( ph /\ Y e. ( N ` { T , w } ) ) -> Y e. ( V \ { .0. } ) )
35 28 adantr 
 |-  ( ( ph /\ Y e. ( N ` { T , w } ) ) -> T e. V )
36 27 adantr 
 |-  ( ( ph /\ Y e. ( N ` { T , w } ) ) -> w e. V )
37 22 adantr 
 |-  ( ( ph /\ Y e. ( N ` { T , w } ) ) -> ( N ` { Y } ) =/= ( N ` { w } ) )
38 simpr 
 |-  ( ( ph /\ Y e. ( N ` { T , w } ) ) -> Y e. ( N ` { T , w } ) )
39 3 5 6 33 34 35 36 37 38 lspexch 
 |-  ( ( ph /\ Y e. ( N ` { T , w } ) ) -> T e. ( N ` { Y , w } ) )
40 39 ex 
 |-  ( ph -> ( Y e. ( N ` { T , w } ) -> T e. ( N ` { Y , w } ) ) )
41 32 40 syl5bi 
 |-  ( ph -> ( Y e. ( N ` { w , T } ) -> T e. ( N ` { Y , w } ) ) )
42 29 41 mtod 
 |-  ( ph -> -. Y e. ( N ` { w , T } ) )
43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 42 mapdh8a 
 |-  ( ph -> ( I ` <. w , E , T >. ) = ( I ` <. Y , G , T >. ) )