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Theorem hdmaprnlem3eN 37689
Description: Lemma for hdmaprnN 37695. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
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
Ref Expression
hdmaprnlem3eN  |-  ( ph  ->  E. t  e.  ( ( N `  {
v } )  \  {  .0.  } ) ( L `  { ( ( S `  u
)  .+b  s ) } )  =  ( M `  ( N `
 { ( u 
.+  t ) } ) ) )
Distinct variable groups:    t,  .+b    t, L    t, M    t, N    t,  .0.    t,  .+    t, S   
t, U    t, V    ph, t    t, s, u, v
Allowed substitution hints:    ph( v, u, s)    C( v, u, t, s)    D( v, u, t, s)    .+ ( v, u, s)    .+b ( v, u, s)    Q( v, u, t, s)    S( v, u, s)    U( v, u, s)    H( v, u, t, s)    K( v, u, t, s)    L( v, u, s)    M( v, u, s)    N( v, u, s)    V( v, u, s)    W( v, u, t, s)    .0. ( v, u, s)

Proof of Theorem hdmaprnlem3eN
StepHypRef Expression
1 hdmaprnlem1.v . . 3  |-  V  =  ( Base `  U
)
2 hdmaprnlem3e.p . . 3  |-  .+  =  ( +g  `  U )
3 hdmaprnlem1.o . . 3  |-  .0.  =  ( 0g `  U )
4 hdmaprnlem1.n . . 3  |-  N  =  ( LSpan `  U )
5 eqid 2457 . . 3  |-  (LSAtoms `  U
)  =  (LSAtoms `  U
)
6 hdmaprnlem1.h . . . 4  |-  H  =  ( LHyp `  K
)
7 hdmaprnlem1.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
8 hdmaprnlem1.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
96, 7, 8dvhlvec 36937 . . 3  |-  ( ph  ->  U  e.  LVec )
10 hdmaprnlem1.m . . . 4  |-  M  =  ( (mapd `  K
) `  W )
11 hdmaprnlem1.c . . . 4  |-  C  =  ( (LCDual `  K
) `  W )
12 eqid 2457 . . . 4  |-  (LSAtoms `  C
)  =  (LSAtoms `  C
)
13 hdmaprnlem1.d . . . . 5  |-  D  =  ( Base `  C
)
14 hdmaprnlem1.l . . . . 5  |-  L  =  ( LSpan `  C )
15 hdmaprnlem1.q . . . . 5  |-  Q  =  ( 0g `  C
)
166, 11, 8lcdlmod 37420 . . . . 5  |-  ( ph  ->  C  e.  LMod )
17 hdmaprnlem1.s . . . . . . . 8  |-  S  =  ( (HDMap `  K
) `  W )
18 hdmaprnlem1.ue . . . . . . . 8  |-  ( ph  ->  u  e.  V )
196, 7, 1, 11, 13, 17, 8, 18hdmapcl 37661 . . . . . . 7  |-  ( ph  ->  ( S `  u
)  e.  D )
20 hdmaprnlem1.se . . . . . . . 8  |-  ( ph  ->  s  e.  ( D 
\  { Q }
) )
2120eldifad 3483 . . . . . . 7  |-  ( ph  ->  s  e.  D )
22 hdmaprnlem1.a . . . . . . . 8  |-  .+b  =  ( +g  `  C )
2313, 22lmodvacl 17652 . . . . . . 7  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  s  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  D )
2416, 19, 21, 23syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  D )
25 hdmaprnlem1.ve . . . . . . . 8  |-  ( ph  ->  v  e.  V )
26 hdmaprnlem1.e . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =  ( L `  {
s } ) )
27 hdmaprnlem1.un . . . . . . . 8  |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )
286, 7, 1, 4, 11, 14, 10, 17, 8, 20, 25, 26, 18, 27hdmaprnlem1N 37680 . . . . . . 7  |-  ( ph  ->  ( L `  {
( S `  u
) } )  =/=  ( L `  {
s } ) )
2913, 22, 15, 14, 16, 19, 21, 28lmodindp1 17786 . . . . . 6  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  =/=  Q )
30 eldifsn 4157 . . . . . 6  |-  ( ( ( S `  u
)  .+b  s )  e.  ( D  \  { Q } )  <->  ( (
( S `  u
)  .+b  s )  e.  D  /\  (
( S `  u
)  .+b  s )  =/=  Q ) )
3124, 29, 30sylanbrc 664 . . . . 5  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  ( D 
\  { Q }
) )
3213, 14, 15, 12, 16, 31lsatlspsn 34819 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  (LSAtoms `  C )
)
336, 10, 7, 5, 11, 12, 8, 32mapdcnvatN 37494 . . 3  |-  ( ph  ->  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  e.  (LSAtoms `  U )
)
346, 7, 1, 4, 11, 14, 10, 17, 8, 20, 25, 26, 18, 27, 13, 15, 3, 22hdmaprnlem3uN 37682 . . . 4  |-  ( ph  ->  ( N `  {
u } )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
3534necomd 2728 . . 3  |-  ( ph  ->  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  =/=  ( N `  {
u } ) )
366, 7, 1, 4, 11, 14, 10, 17, 8, 20, 25, 26, 18, 27, 13, 15, 3, 22hdmaprnlem3N 37681 . . . 4  |-  ( ph  ->  ( N `  {
v } )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
3736necomd 2728 . . 3  |-  ( ph  ->  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  =/=  ( N `  {
v } ) )
38 eqid 2457 . . . . . . 7  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
39 eqid 2457 . . . . . . . . 9  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
406, 7, 8dvhlmod 36938 . . . . . . . . . 10  |-  ( ph  ->  U  e.  LMod )
411, 39, 4lspsncl 17749 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  u  e.  V )  ->  ( N `  { u } )  e.  (
LSubSp `  U ) )
4240, 18, 41syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( N `  {
u } )  e.  ( LSubSp `  U )
)
436, 10, 7, 39, 11, 38, 8, 42mapdcl2 37484 . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { u } ) )  e.  ( LSubSp `  C )
)
441, 39, 4lspsncl 17749 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  v  e.  V )  ->  ( N `  { v } )  e.  (
LSubSp `  U ) )
4540, 25, 44syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( N `  {
v } )  e.  ( LSubSp `  U )
)
466, 10, 7, 39, 11, 38, 8, 45mapdcl2 37484 . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { v } ) )  e.  ( LSubSp `  C )
)
47 eqid 2457 . . . . . . . . 9  |-  ( LSSum `  C )  =  (
LSSum `  C )
4838, 47lsmcl 17855 . . . . . . . 8  |-  ( ( C  e.  LMod  /\  ( M `  ( N `  { u } ) )  e.  ( LSubSp `  C )  /\  ( M `  ( N `  { v } ) )  e.  ( LSubSp `  C ) )  -> 
( ( M `  ( N `  { u } ) ) (
LSSum `  C ) ( M `  ( N `
 { v } ) ) )  e.  ( LSubSp `  C )
)
4916, 43, 46, 48syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { u } ) ) (
LSSum `  C ) ( M `  ( N `
 { v } ) ) )  e.  ( LSubSp `  C )
)
5038lsssssubg 17730 . . . . . . . . . 10  |-  ( C  e.  LMod  ->  ( LSubSp `  C )  C_  (SubGrp `  C ) )
5116, 50syl 16 . . . . . . . . 9  |-  ( ph  ->  ( LSubSp `  C )  C_  (SubGrp `  C )
)
5251, 43sseldd 3500 . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { u } ) )  e.  (SubGrp `  C )
)
5351, 46sseldd 3500 . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { v } ) )  e.  (SubGrp `  C )
)
5413, 14lspsnid 17765 . . . . . . . . . 10  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D )  ->  ( S `  u )  e.  ( L `  {
( S `  u
) } ) )
5516, 19, 54syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( S `  u
)  e.  ( L `
 { ( S `
 u ) } ) )
566, 7, 1, 4, 11, 14, 10, 17, 8, 18hdmap10 37671 . . . . . . . . 9  |-  ( ph  ->  ( M `  ( N `  { u } ) )  =  ( L `  {
( S `  u
) } ) )
5755, 56eleqtrrd 2548 . . . . . . . 8  |-  ( ph  ->  ( S `  u
)  e.  ( M `
 ( N `  { u } ) ) )
58 eqimss2 3552 . . . . . . . . . 10  |-  ( ( M `  ( N `
 { v } ) )  =  ( L `  { s } )  ->  ( L `  { s } )  C_  ( M `  ( N `  { v } ) ) )
5926, 58syl 16 . . . . . . . . 9  |-  ( ph  ->  ( L `  {
s } )  C_  ( M `  ( N `
 { v } ) ) )
6013, 38, 14, 16, 46, 21lspsnel5 17767 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( M `  ( N `
 { v } ) )  <->  ( L `  { s } ) 
C_  ( M `  ( N `  { v } ) ) ) )
6159, 60mpbird 232 . . . . . . . 8  |-  ( ph  ->  s  e.  ( M `
 ( N `  { v } ) ) )
6222, 47lsmelvali 16796 . . . . . . . 8  |-  ( ( ( ( M `  ( N `  { u } ) )  e.  (SubGrp `  C )  /\  ( M `  ( N `  { v } ) )  e.  (SubGrp `  C )
)  /\  ( ( S `  u )  e.  ( M `  ( N `  { u } ) )  /\  s  e.  ( M `  ( N `  {
v } ) ) ) )  ->  (
( S `  u
)  .+b  s )  e.  ( ( M `  ( N `  { u } ) ) (
LSSum `  C ) ( M `  ( N `
 { v } ) ) ) )
6352, 53, 57, 61, 62syl22anc 1229 . . . . . . 7  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  ( ( M `  ( N `
 { u }
) ) ( LSSum `  C ) ( M `
 ( N `  { v } ) ) ) )
6438, 14, 16, 49, 63lspsnel5a 17768 . . . . . 6  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  C_  ( ( M `  ( N `  { u } ) ) (
LSSum `  C ) ( M `  ( N `
 { v } ) ) ) )
65 eqid 2457 . . . . . . 7  |-  ( LSSum `  U )  =  (
LSSum `  U )
666, 10, 7, 39, 65, 11, 47, 8, 42, 45mapdlsm 37492 . . . . . 6  |-  ( ph  ->  ( M `  (
( N `  {
u } ) (
LSSum `  U ) ( N `  { v } ) ) )  =  ( ( M `
 ( N `  { u } ) ) ( LSSum `  C
) ( M `  ( N `  { v } ) ) ) )
6764, 66sseqtr4d 3536 . . . . 5  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  C_  ( M `  ( ( N `  { u } ) ( LSSum `  U ) ( N `
 { v } ) ) ) )
6813, 38, 14lspsncl 17749 . . . . . . . 8  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  ( L `  { (
( S `  u
)  .+b  s ) } )  e.  (
LSubSp `  C ) )
6916, 24, 68syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  ( LSubSp `  C )
)
706, 10, 11, 38, 8mapdrn2 37479 . . . . . . 7  |-  ( ph  ->  ran  M  =  (
LSubSp `  C ) )
7169, 70eleqtrrd 2548 . . . . . 6  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e. 
ran  M )
7239, 65lsmcl 17855 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  ( N `  { u } )  e.  (
LSubSp `  U )  /\  ( N `  { v } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { u }
) ( LSSum `  U
) ( N `  { v } ) )  e.  ( LSubSp `  U ) )
7340, 42, 45, 72syl3anc 1228 . . . . . . 7  |-  ( ph  ->  ( ( N `  { u } ) ( LSSum `  U )
( N `  {
v } ) )  e.  ( LSubSp `  U
) )
746, 10, 7, 39, 8, 73mapdcl 37481 . . . . . 6  |-  ( ph  ->  ( M `  (
( N `  {
u } ) (
LSSum `  U ) ( N `  { v } ) ) )  e.  ran  M )
756, 10, 8, 71, 74mapdcnvordN 37486 . . . . 5  |-  ( ph  ->  ( ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) ) 
C_  ( `' M `  ( M `  (
( N `  {
u } ) (
LSSum `  U ) ( N `  { v } ) ) ) )  <->  ( L `  { ( ( S `
 u )  .+b  s ) } ) 
C_  ( M `  ( ( N `  { u } ) ( LSSum `  U )
( N `  {
v } ) ) ) ) )
7667, 75mpbird 232 . . . 4  |-  ( ph  ->  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  C_  ( `' M `  ( M `
 ( ( N `
 { u }
) ( LSSum `  U
) ( N `  { v } ) ) ) ) )
771, 4, 65, 40, 18, 25lsmpr 17861 . . . . 5  |-  ( ph  ->  ( N `  {
u ,  v } )  =  ( ( N `  { u } ) ( LSSum `  U ) ( N `
 { v } ) ) )
786, 10, 7, 39, 8, 73mapdcnvid1N 37482 . . . . 5  |-  ( ph  ->  ( `' M `  ( M `  ( ( N `  { u } ) ( LSSum `  U ) ( N `
 { v } ) ) ) )  =  ( ( N `
 { u }
) ( LSSum `  U
) ( N `  { v } ) ) )
7977, 78eqtr4d 2501 . . . 4  |-  ( ph  ->  ( N `  {
u ,  v } )  =  ( `' M `  ( M `
 ( ( N `
 { u }
) ( LSSum `  U
) ( N `  { v } ) ) ) ) )
8076, 79sseqtr4d 3536 . . 3  |-  ( ph  ->  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  C_  ( N `  { u ,  v } ) )
811, 2, 3, 4, 5, 9, 33, 18, 25, 35, 37, 80lsatfixedN 34835 . 2  |-  ( ph  ->  E. t  e.  ( ( N `  {
v } )  \  {  .0.  } ) ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )
82 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  =  ( N `  {
( u  .+  t
) } ) )
838ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
8440ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  ->  U  e.  LMod )
8518ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  ->  u  e.  V )
8620ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
s  e.  ( D 
\  { Q }
) )
8725ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
v  e.  V )
8826ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( M `  ( N `  { v } ) )  =  ( L `  {
s } ) )
8927ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  ->  -.  u  e.  ( N `  { v } ) )
90 simplr 755 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
t  e.  ( ( N `  { v } )  \  {  .0.  } ) )
916, 7, 1, 4, 11, 14, 10, 17, 83, 86, 87, 88, 85, 89, 13, 15, 3, 22, 90hdmaprnlem4tN 37683 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
t  e.  V )
921, 2lmodvacl 17652 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  u  e.  V  /\  t  e.  V )  ->  (
u  .+  t )  e.  V )
9384, 85, 91, 92syl3anc 1228 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( u  .+  t
)  e.  V )
941, 39, 4lspsncl 17749 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  (
u  .+  t )  e.  V )  ->  ( N `  { (
u  .+  t ) } )  e.  (
LSubSp `  U ) )
9584, 93, 94syl2anc 661 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( N `  {
( u  .+  t
) } )  e.  ( LSubSp `  U )
)
966, 10, 7, 39, 83, 95mapdcnvid1N 37482 . . . . . 6  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( `' M `  ( M `  ( N `
 { ( u 
.+  t ) } ) ) )  =  ( N `  {
( u  .+  t
) } ) )
9782, 96eqtr4d 2501 . . . . 5  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  =  ( `' M `  ( M `  ( N `
 { ( u 
.+  t ) } ) ) ) )
9871ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( L `  {
( ( S `  u )  .+b  s
) } )  e. 
ran  M )
996, 10, 7, 39, 83, 95mapdcl 37481 . . . . . 6  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( M `  ( N `  { (
u  .+  t ) } ) )  e. 
ran  M )
1006, 10, 83, 98, 99mapdcnv11N 37487 . . . . 5  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) )  =  ( `' M `  ( M `  ( N `  { (
u  .+  t ) } ) ) )  <-> 
( L `  {
( ( S `  u )  .+b  s
) } )  =  ( M `  ( N `  { (
u  .+  t ) } ) ) ) )
10197, 100mpbid 210 . . . 4  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( L `  {
( ( S `  u )  .+b  s
) } )  =  ( M `  ( N `  { (
u  .+  t ) } ) ) )
102101ex 434 . . 3  |-  ( (
ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  ->  (
( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  =  ( N `  {
( u  .+  t
) } )  -> 
( L `  {
( ( S `  u )  .+b  s
) } )  =  ( M `  ( N `  { (
u  .+  t ) } ) ) ) )
103102reximdva 2932 . 2  |-  ( ph  ->  ( E. t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) )  =  ( N `  { ( u  .+  t ) } )  ->  E. t  e.  ( ( N `  {
v } )  \  {  .0.  } ) ( L `  { ( ( S `  u
)  .+b  s ) } )  =  ( M `  ( N `
 { ( u 
.+  t ) } ) ) ) )
10481, 103mpd 15 1  |-  ( ph  ->  E. t  e.  ( ( N `  {
v } )  \  {  .0.  } ) ( L `  { ( ( S `  u
)  .+b  s ) } )  =  ( M `  ( N `
 { ( u 
.+  t ) } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808    \ cdif 3468    C_ wss 3471   {csn 4032   {cpr 4034   `'ccnv 5007   ran crn 5009   ` cfv 5594  (class class class)co 6296   Basecbs 14643   +g cplusg 14711   0gc0g 14856  SubGrpcsubg 16321   LSSumclsm 16780   LModclmod 17638   LSubSpclss 17704   LSpanclspn 17743  LSAtomsclsa 34800   HLchlt 35176   LHypclh 35809   DVecHcdvh 36906  LCDualclcd 37414  mapdcmpd 37452  HDMapchdma 37621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-riotaBAD 34785
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-ot 4041  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-undef 7020  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-sca 14727  df-vsca 14728  df-0g 14858  df-mre 15002  df-mrc 15003  df-acs 15005  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-subg 16324  df-cntz 16481  df-oppg 16507  df-lsm 16782  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-oppr 17398  df-dvdsr 17416  df-unit 17417  df-invr 17447  df-dvr 17458  df-drng 17524  df-lmod 17640  df-lss 17705  df-lsp 17744  df-lvec 17875  df-lsatoms 34802  df-lshyp 34803  df-lcv 34845  df-lfl 34884  df-lkr 34912  df-ldual 34950  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-llines 35323  df-lplanes 35324  df-lvols 35325  df-lines 35326  df-psubsp 35328  df-pmap 35329  df-padd 35621  df-lhyp 35813  df-laut 35814  df-ldil 35929  df-ltrn 35930  df-trl 35985  df-tgrp 36570  df-tendo 36582  df-edring 36584  df-dveca 36830  df-disoa 36857  df-dvech 36907  df-dib 36967  df-dic 37001  df-dih 37057  df-doch 37176  df-djh 37223  df-lcdual 37415  df-mapd 37453  df-hvmap 37585  df-hdmap1 37622  df-hdmap 37623
This theorem is referenced by:  hdmaprnlem10N  37690
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