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Theorem hdmaprnlem3eN 35504
Description: Lemma for hdmaprnN 35510. (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 2442 . . 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 34752 . . 3  |-  ( ph  ->  U  e.  LVec )
10 hdmaprnlem1.m . . . 4  |-  M  =  ( (mapd `  K
) `  W )
11 hdmaprnlem1.c . . . 4  |-  C  =  ( (LCDual `  K
) `  W )
12 eqid 2442 . . . 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 35235 . . . . 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 35476 . . . . . . 7  |-  ( ph  ->  ( S `  u
)  e.  D )
20 hdmaprnlem1.se . . . . . . . 8  |-  ( ph  ->  s  e.  ( D 
\  { Q }
) )
2120eldifad 3339 . . . . . . 7  |-  ( ph  ->  s  e.  D )
22 hdmaprnlem1.a . . . . . . . 8  |-  .+b  =  ( +g  `  C )
2313, 22lmodvacl 16961 . . . . . . 7  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  s  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  D )
2416, 19, 21, 23syl3anc 1218 . . . . . 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 35495 . . . . . . 7  |-  ( ph  ->  ( L `  {
( S `  u
) } )  =/=  ( L `  {
s } ) )
2913, 22, 15, 14, 16, 19, 21, 28lmodindp1 17094 . . . . . 6  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  =/=  Q )
30 eldifsn 3999 . . . . . 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 32636 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  (LSAtoms `  C )
)
336, 10, 7, 5, 11, 12, 8, 32mapdcnvatN 35309 . . 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 35497 . . . 4  |-  ( ph  ->  ( N `  {
u } )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
3534necomd 2694 . . 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 35496 . . . 4  |-  ( ph  ->  ( N `  {
v } )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
3736necomd 2694 . . 3  |-  ( ph  ->  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  =/=  ( N `  {
v } ) )
38 eqid 2442 . . . . . . 7  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
39 eqid 2442 . . . . . . . . 9  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
406, 7, 8dvhlmod 34753 . . . . . . . . . 10  |-  ( ph  ->  U  e.  LMod )
411, 39, 4lspsncl 17057 . . . . . . . . . 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 35299 . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { u } ) )  e.  ( LSubSp `  C )
)
441, 39, 4lspsncl 17057 . . . . . . . . . 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 35299 . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { v } ) )  e.  ( LSubSp `  C )
)
47 eqid 2442 . . . . . . . . 9  |-  ( LSSum `  C )  =  (
LSSum `  C )
4838, 47lsmcl 17163 . . . . . . . 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 1218 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { u } ) ) (
LSSum `  C ) ( M `  ( N `
 { v } ) ) )  e.  ( LSubSp `  C )
)
5038lsssssubg 17038 . . . . . . . . . 10  |-  ( C  e.  LMod  ->  ( LSubSp `  C )  C_  (SubGrp `  C ) )
5116, 50syl 16 . . . . . . . . 9  |-  ( ph  ->  ( LSubSp `  C )  C_  (SubGrp `  C )
)
5251, 43sseldd 3356 . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { u } ) )  e.  (SubGrp `  C )
)
5351, 46sseldd 3356 . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { v } ) )  e.  (SubGrp `  C )
)
5413, 14lspsnid 17073 . . . . . . . . . 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 35486 . . . . . . . . 9  |-  ( ph  ->  ( M `  ( N `  { u } ) )  =  ( L `  {
( S `  u
) } ) )
5755, 56eleqtrrd 2519 . . . . . . . 8  |-  ( ph  ->  ( S `  u
)  e.  ( M `
 ( N `  { u } ) ) )
58 eqimss2 3408 . . . . . . . . . 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 17075 . . . . . . . . 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 16148 . . . . . . . 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 1219 . . . . . . 7  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  ( ( M `  ( N `
 { u }
) ) ( LSSum `  C ) ( M `
 ( N `  { v } ) ) ) )
6438, 14, 16, 49, 63lspsnel5a 17076 . . . . . 6  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  C_  ( ( M `  ( N `  { u } ) ) (
LSSum `  C ) ( M `  ( N `
 { v } ) ) ) )
65 eqid 2442 . . . . . . 7  |-  ( LSSum `  U )  =  (
LSSum `  U )
666, 10, 7, 39, 65, 11, 47, 8, 42, 45mapdlsm 35307 . . . . . 6  |-  ( ph  ->  ( M `  (
( N `  {
u } ) (
LSSum `  U ) ( N `  { v } ) ) )  =  ( ( M `
 ( N `  { u } ) ) ( LSSum `  C
) ( M `  ( N `  { v } ) ) ) )
6764, 66sseqtr4d 3392 . . . . 5  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  C_  ( M `  ( ( N `  { u } ) ( LSSum `  U ) ( N `
 { v } ) ) ) )
6813, 38, 14lspsncl 17057 . . . . . . . 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 35294 . . . . . . 7  |-  ( ph  ->  ran  M  =  (
LSubSp `  C ) )
7169, 70eleqtrrd 2519 . . . . . 6  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e. 
ran  M )
7239, 65lsmcl 17163 . . . . . . . 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 1218 . . . . . . 7  |-  ( ph  ->  ( ( N `  { u } ) ( LSSum `  U )
( N `  {
v } ) )  e.  ( LSubSp `  U
) )
746, 10, 7, 39, 8, 73mapdcl 35296 . . . . . 6  |-  ( ph  ->  ( M `  (
( N `  {
u } ) (
LSSum `  U ) ( N `  { v } ) ) )  e.  ran  M )
756, 10, 8, 71, 74mapdcnvordN 35301 . . . . 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 17169 . . . . 5  |-  ( ph  ->  ( N `  {
u ,  v } )  =  ( ( N `  { u } ) ( LSSum `  U ) ( N `
 { v } ) ) )
786, 10, 7, 39, 8, 73mapdcnvid1N 35297 . . . . 5  |-  ( ph  ->  ( `' M `  ( M `  ( ( N `  { u } ) ( LSSum `  U ) ( N `
 { v } ) ) ) )  =  ( ( N `
 { u }
) ( LSSum `  U
) ( N `  { v } ) ) )
7977, 78eqtr4d 2477 . . . 4  |-  ( ph  ->  ( N `  {
u ,  v } )  =  ( `' M `  ( M `
 ( ( N `
 { u }
) ( LSSum `  U
) ( N `  { v } ) ) ) ) )
8076, 79sseqtr4d 3392 . . 3  |-  ( ph  ->  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  C_  ( N `  { u ,  v } ) )
811, 2, 3, 4, 5, 9, 33, 18, 25, 35, 37, 80lsatfixedN 32652 . 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 754 . . . . . . . . . 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 35498 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
t  e.  V )
921, 2lmodvacl 16961 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  u  e.  V  /\  t  e.  V )  ->  (
u  .+  t )  e.  V )
9384, 85, 91, 92syl3anc 1218 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( u  .+  t
)  e.  V )
941, 39, 4lspsncl 17057 . . . . . . . 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 35297 . . . . . 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 2477 . . . . 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 35296 . . . . . 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 35302 . . . . 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 2827 . 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 1369    e. wcel 1756    =/= wne 2605   E.wrex 2715    \ cdif 3324    C_ wss 3327   {csn 3876   {cpr 3878   `'ccnv 4838   ran crn 4840   ` cfv 5417  (class class class)co 6090   Basecbs 14173   +g cplusg 14237   0gc0g 14377  SubGrpcsubg 15674   LSSumclsm 16132   LModclmod 16947   LSubSpclss 17012   LSpanclspn 17051  LSAtomsclsa 32617   HLchlt 32993   LHypclh 33626   DVecHcdvh 34721  LCDualclcd 35229  mapdcmpd 35267  HDMapchdma 35436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-riotaBAD 32602
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-ot 3885  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-undef 6791  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-sca 14253  df-vsca 14254  df-0g 14379  df-mre 14523  df-mrc 14524  df-acs 14526  df-poset 15115  df-plt 15127  df-lub 15143  df-glb 15144  df-join 15145  df-meet 15146  df-p0 15208  df-p1 15209  df-lat 15215  df-clat 15277  df-mnd 15414  df-submnd 15464  df-grp 15544  df-minusg 15545  df-sbg 15546  df-subg 15677  df-cntz 15834  df-oppg 15860  df-lsm 16134  df-cmn 16278  df-abl 16279  df-mgp 16591  df-ur 16603  df-rng 16646  df-oppr 16714  df-dvdsr 16732  df-unit 16733  df-invr 16763  df-dvr 16774  df-drng 16833  df-lmod 16949  df-lss 17013  df-lsp 17052  df-lvec 17183  df-lsatoms 32619  df-lshyp 32620  df-lcv 32662  df-lfl 32701  df-lkr 32729  df-ldual 32767  df-oposet 32819  df-ol 32821  df-oml 32822  df-covers 32909  df-ats 32910  df-atl 32941  df-cvlat 32965  df-hlat 32994  df-llines 33140  df-lplanes 33141  df-lvols 33142  df-lines 33143  df-psubsp 33145  df-pmap 33146  df-padd 33438  df-lhyp 33630  df-laut 33631  df-ldil 33746  df-ltrn 33747  df-trl 33801  df-tgrp 34385  df-tendo 34397  df-edring 34399  df-dveca 34645  df-disoa 34672  df-dvech 34722  df-dib 34782  df-dic 34816  df-dih 34872  df-doch 34991  df-djh 35038  df-lcdual 35230  df-mapd 35268  df-hvmap 35400  df-hdmap1 35437  df-hdmap 35438
This theorem is referenced by:  hdmaprnlem10N  35505
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