Metamath Proof Explorer

Theorem hdmaprnlem10N

Description: Lemma for hdmaprnN . Show s is in the range of S . (Contributed by NM, 29-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses hdmaprnlem1.h 
|- H = ( LHyp ` K )
hdmaprnlem1.u 
|- U = ( ( DVecH ` K ) ` W )
hdmaprnlem1.v 
|- V = ( Base ` U )
hdmaprnlem1.n 
|- N = ( LSpan ` U )
hdmaprnlem1.c 
|- C = ( ( LCDual ` K ) ` W )
hdmaprnlem1.l 
|- L = ( LSpan ` C )
hdmaprnlem1.m 
|- M = ( ( mapd ` K ) ` W )
hdmaprnlem1.s 
|- S = ( ( HDMap ` K ) ` W )
hdmaprnlem1.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
hdmaprnlem1.se 
|- ( ph -> s e. ( D \ { Q } ) )
hdmaprnlem1.ve 
|- ( ph -> v e. V )
hdmaprnlem1.e 
|- ( ph -> ( M ` ( N ` { v } ) ) = ( L ` { s } ) )
hdmaprnlem1.ue 
|- ( ph -> u e. V )
hdmaprnlem1.un 
|- ( ph -> -. u e. ( N ` { v } ) )
hdmaprnlem1.d 
|- D = ( Base ` C )
hdmaprnlem1.q 
|- Q = ( 0g ` C )
hdmaprnlem1.o 
|- .0. = ( 0g ` U )
hdmaprnlem1.a 
|- .+b = ( +g ` C )
hdmaprnlem3e.p 
|- .+ = ( +g ` U )
Assertion hdmaprnlem10N 
|- ( ph -> E. t e. V ( S ` t ) = s )

Proof

Step Hyp Ref Expression
1 hdmaprnlem1.h 
 |-  H = ( LHyp ` K )
2 hdmaprnlem1.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 hdmaprnlem1.v 
 |-  V = ( Base ` U )
4 hdmaprnlem1.n 
 |-  N = ( LSpan ` U )
5 hdmaprnlem1.c 
 |-  C = ( ( LCDual ` K ) ` W )
6 hdmaprnlem1.l 
 |-  L = ( LSpan ` C )
7 hdmaprnlem1.m 
 |-  M = ( ( mapd ` K ) ` W )
8 hdmaprnlem1.s 
 |-  S = ( ( HDMap ` K ) ` W )
9 hdmaprnlem1.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
10 hdmaprnlem1.se 
 |-  ( ph -> s e. ( D \ { Q } ) )
11 hdmaprnlem1.ve 
 |-  ( ph -> v e. V )
12 hdmaprnlem1.e 
 |-  ( ph -> ( M ` ( N ` { v } ) ) = ( L ` { s } ) )
13 hdmaprnlem1.ue 
 |-  ( ph -> u e. V )
14 hdmaprnlem1.un 
 |-  ( ph -> -. u e. ( N ` { v } ) )
15 hdmaprnlem1.d 
 |-  D = ( Base ` C )
16 hdmaprnlem1.q 
 |-  Q = ( 0g ` C )
17 hdmaprnlem1.o 
 |-  .0. = ( 0g ` U )
18 hdmaprnlem1.a 
 |-  .+b = ( +g ` C )
19 hdmaprnlem3e.p 
 |-  .+ = ( +g ` U )
20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 hdmaprnlem3eN 
 |-  ( ph -> E. t e. ( ( N ` { v } ) \ { .0. } ) ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) )
21 9 adantr 
 |-  ( ( ph /\ ( t e. ( ( N ` { v } ) \ { .0. } ) /\ ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) ) ) -> ( K e. HL /\ W e. H ) )
22 10 adantr 
 |-  ( ( ph /\ ( t e. ( ( N ` { v } ) \ { .0. } ) /\ ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) ) ) -> s e. ( D \ { Q } ) )
23 11 adantr 
 |-  ( ( ph /\ ( t e. ( ( N ` { v } ) \ { .0. } ) /\ ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) ) ) -> v e. V )
24 12 adantr 
 |-  ( ( ph /\ ( t e. ( ( N ` { v } ) \ { .0. } ) /\ ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) ) ) -> ( M ` ( N ` { v } ) ) = ( L ` { s } ) )
25 13 adantr 
 |-  ( ( ph /\ ( t e. ( ( N ` { v } ) \ { .0. } ) /\ ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) ) ) -> u e. V )
26 14 adantr 
 |-  ( ( ph /\ ( t e. ( ( N ` { v } ) \ { .0. } ) /\ ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) ) ) -> -. u e. ( N ` { v } ) )
27 simprl 
 |-  ( ( ph /\ ( t e. ( ( N ` { v } ) \ { .0. } ) /\ ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) ) ) -> t e. ( ( N ` { v } ) \ { .0. } ) )
28 1 2 3 4 5 6 7 8 21 22 23 24 25 26 15 16 17 18 27 hdmaprnlem4tN 
 |-  ( ( ph /\ ( t e. ( ( N ` { v } ) \ { .0. } ) /\ ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) ) ) -> t e. V )
29 simprr 
 |-  ( ( ph /\ ( t e. ( ( N ` { v } ) \ { .0. } ) /\ ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) ) ) -> ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) )
30 1 2 3 4 5 6 7 8 21 22 23 24 25 26 15 16 17 18 27 19 29 hdmaprnlem9N 
 |-  ( ( ph /\ ( t e. ( ( N ` { v } ) \ { .0. } ) /\ ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) ) ) -> s = ( S ` t ) )
31 30 eqcomd 
 |-  ( ( ph /\ ( t e. ( ( N ` { v } ) \ { .0. } ) /\ ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) ) ) -> ( S ` t ) = s )
32 20 28 31 reximssdv 
 |-  ( ph -> E. t e. V ( S ` t ) = s )