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Theorem hdmaprnlem3eN 32051
Description: Lemma for hdmaprnN 32057. (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 2283 . . 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 31299 . . 3  |-  ( ph  ->  U  e.  LVec )
10 hdmaprnlem1.m . . . 4  |-  M  =  ( (mapd `  K
) `  W )
11 hdmaprnlem1.c . . . 4  |-  C  =  ( (LCDual `  K
) `  W )
12 eqid 2283 . . . 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 31782 . . . . 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 32023 . . . . . . 7  |-  ( ph  ->  ( S `  u
)  e.  D )
20 hdmaprnlem1.se . . . . . . . 8  |-  ( ph  ->  s  e.  ( D 
\  { Q }
) )
21 eldifi 3298 . . . . . . . 8  |-  ( s  e.  ( D  \  { Q } )  -> 
s  e.  D )
2220, 21syl 15 . . . . . . 7  |-  ( ph  ->  s  e.  D )
23 hdmaprnlem1.a . . . . . . . 8  |-  .+b  =  ( +g  `  C )
2413, 23lmodvacl 15641 . . . . . . 7  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  s  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  D )
2516, 19, 22, 24syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  D )
26 hdmaprnlem1.ve . . . . . . . 8  |-  ( ph  ->  v  e.  V )
27 hdmaprnlem1.e . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =  ( L `  {
s } ) )
28 hdmaprnlem1.un . . . . . . . 8  |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )
296, 7, 1, 4, 11, 14, 10, 17, 8, 20, 26, 27, 18, 28hdmaprnlem1N 32042 . . . . . . 7  |-  ( ph  ->  ( L `  {
( S `  u
) } )  =/=  ( L `  {
s } ) )
3013, 23, 15, 14, 16, 19, 22, 29lmodindp1 15771 . . . . . 6  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  =/=  Q )
31 eldifsn 3749 . . . . . 6  |-  ( ( ( S `  u
)  .+b  s )  e.  ( D  \  { Q } )  <->  ( (
( S `  u
)  .+b  s )  e.  D  /\  (
( S `  u
)  .+b  s )  =/=  Q ) )
3225, 30, 31sylanbrc 645 . . . . 5  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  ( D 
\  { Q }
) )
3313, 14, 15, 12, 16, 32lsatlspsn 29183 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  (LSAtoms `  C )
)
346, 10, 7, 5, 11, 12, 8, 33mapdcnvatN 31856 . . 3  |-  ( ph  ->  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  e.  (LSAtoms `  U )
)
356, 7, 1, 4, 11, 14, 10, 17, 8, 20, 26, 27, 18, 28, 13, 15, 3, 23hdmaprnlem3uN 32044 . . . 4  |-  ( ph  ->  ( N `  {
u } )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
3635necomd 2529 . . 3  |-  ( ph  ->  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  =/=  ( N `  {
u } ) )
376, 7, 1, 4, 11, 14, 10, 17, 8, 20, 26, 27, 18, 28, 13, 15, 3, 23hdmaprnlem3N 32043 . . . 4  |-  ( ph  ->  ( N `  {
v } )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
3837necomd 2529 . . 3  |-  ( ph  ->  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  =/=  ( N `  {
v } ) )
39 eqid 2283 . . . . . . 7  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
40 eqid 2283 . . . . . . . . 9  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
416, 7, 8dvhlmod 31300 . . . . . . . . . 10  |-  ( ph  ->  U  e.  LMod )
421, 40, 4lspsncl 15734 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  u  e.  V )  ->  ( N `  { u } )  e.  (
LSubSp `  U ) )
4341, 18, 42syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( N `  {
u } )  e.  ( LSubSp `  U )
)
446, 10, 7, 40, 11, 39, 8, 43mapdcl2 31846 . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { u } ) )  e.  ( LSubSp `  C )
)
451, 40, 4lspsncl 15734 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  v  e.  V )  ->  ( N `  { v } )  e.  (
LSubSp `  U ) )
4641, 26, 45syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( N `  {
v } )  e.  ( LSubSp `  U )
)
476, 10, 7, 40, 11, 39, 8, 46mapdcl2 31846 . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { v } ) )  e.  ( LSubSp `  C )
)
48 eqid 2283 . . . . . . . . 9  |-  ( LSSum `  C )  =  (
LSSum `  C )
4939, 48lsmcl 15836 . . . . . . . 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 )
)
5016, 44, 47, 49syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( N `  { u } ) ) (
LSSum `  C ) ( M `  ( N `
 { v } ) ) )  e.  ( LSubSp `  C )
)
5139lsssssubg 15715 . . . . . . . . . 10  |-  ( C  e.  LMod  ->  ( LSubSp `  C )  C_  (SubGrp `  C ) )
5216, 51syl 15 . . . . . . . . 9  |-  ( ph  ->  ( LSubSp `  C )  C_  (SubGrp `  C )
)
5352, 44sseldd 3181 . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { u } ) )  e.  (SubGrp `  C )
)
5452, 47sseldd 3181 . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { v } ) )  e.  (SubGrp `  C )
)
5513, 14lspsnid 15750 . . . . . . . . . 10  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D )  ->  ( S `  u )  e.  ( L `  {
( S `  u
) } ) )
5616, 19, 55syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( S `  u
)  e.  ( L `
 { ( S `
 u ) } ) )
576, 7, 1, 4, 11, 14, 10, 17, 8, 18hdmap10 32033 . . . . . . . . 9  |-  ( ph  ->  ( M `  ( N `  { u } ) )  =  ( L `  {
( S `  u
) } ) )
5856, 57eleqtrrd 2360 . . . . . . . 8  |-  ( ph  ->  ( S `  u
)  e.  ( M `
 ( N `  { u } ) ) )
59 eqimss2 3231 . . . . . . . . . 10  |-  ( ( M `  ( N `
 { v } ) )  =  ( L `  { s } )  ->  ( L `  { s } )  C_  ( M `  ( N `  { v } ) ) )
6027, 59syl 15 . . . . . . . . 9  |-  ( ph  ->  ( L `  {
s } )  C_  ( M `  ( N `
 { v } ) ) )
6113, 39, 14, 16, 47, 22lspsnel5 15752 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( M `  ( N `
 { v } ) )  <->  ( L `  { s } ) 
C_  ( M `  ( N `  { v } ) ) ) )
6260, 61mpbird 223 . . . . . . . 8  |-  ( ph  ->  s  e.  ( M `
 ( N `  { v } ) ) )
6323, 48lsmelvali 14961 . . . . . . . 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 } ) ) ) )
6453, 54, 58, 62, 63syl22anc 1183 . . . . . . 7  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  ( ( M `  ( N `
 { u }
) ) ( LSSum `  C ) ( M `
 ( N `  { v } ) ) ) )
6539, 14, 16, 50, 64lspsnel5a 15753 . . . . . 6  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  C_  ( ( M `  ( N `  { u } ) ) (
LSSum `  C ) ( M `  ( N `
 { v } ) ) ) )
66 eqid 2283 . . . . . . 7  |-  ( LSSum `  U )  =  (
LSSum `  U )
676, 10, 7, 40, 66, 11, 48, 8, 43, 46mapdlsm 31854 . . . . . 6  |-  ( ph  ->  ( M `  (
( N `  {
u } ) (
LSSum `  U ) ( N `  { v } ) ) )  =  ( ( M `
 ( N `  { u } ) ) ( LSSum `  C
) ( M `  ( N `  { v } ) ) ) )
6865, 67sseqtr4d 3215 . . . . 5  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  C_  ( M `  ( ( N `  { u } ) ( LSSum `  U ) ( N `
 { v } ) ) ) )
6913, 39, 14lspsncl 15734 . . . . . . . 8  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  ( L `  { (
( S `  u
)  .+b  s ) } )  e.  (
LSubSp `  C ) )
7016, 25, 69syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  ( LSubSp `  C )
)
716, 10, 11, 39, 8mapdrn2 31841 . . . . . . 7  |-  ( ph  ->  ran  M  =  (
LSubSp `  C ) )
7270, 71eleqtrrd 2360 . . . . . 6  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e. 
ran  M )
7340, 66lsmcl 15836 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  ( N `  { u } )  e.  (
LSubSp `  U )  /\  ( N `  { v } )  e.  (
LSubSp `  U ) )  ->  ( ( N `
 { u }
) ( LSSum `  U
) ( N `  { v } ) )  e.  ( LSubSp `  U ) )
7441, 43, 46, 73syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( N `  { u } ) ( LSSum `  U )
( N `  {
v } ) )  e.  ( LSubSp `  U
) )
756, 10, 7, 40, 8, 74mapdcl 31843 . . . . . 6  |-  ( ph  ->  ( M `  (
( N `  {
u } ) (
LSSum `  U ) ( N `  { v } ) ) )  e.  ran  M )
766, 10, 8, 72, 75mapdcnvordN 31848 . . . . 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 } ) ) ) ) )
7768, 76mpbird 223 . . . 4  |-  ( ph  ->  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  C_  ( `' M `  ( M `
 ( ( N `
 { u }
) ( LSSum `  U
) ( N `  { v } ) ) ) ) )
781, 4, 66, 41, 18, 26lsmpr 15842 . . . . 5  |-  ( ph  ->  ( N `  {
u ,  v } )  =  ( ( N `  { u } ) ( LSSum `  U ) ( N `
 { v } ) ) )
796, 10, 7, 40, 8, 74mapdcnvid1N 31844 . . . . 5  |-  ( ph  ->  ( `' M `  ( M `  ( ( N `  { u } ) ( LSSum `  U ) ( N `
 { v } ) ) ) )  =  ( ( N `
 { u }
) ( LSSum `  U
) ( N `  { v } ) ) )
8078, 79eqtr4d 2318 . . . 4  |-  ( ph  ->  ( N `  {
u ,  v } )  =  ( `' M `  ( M `
 ( ( N `
 { u }
) ( LSSum `  U
) ( N `  { v } ) ) ) ) )
8177, 80sseqtr4d 3215 . . 3  |-  ( ph  ->  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  C_  ( N `  { u ,  v } ) )
821, 2, 3, 4, 5, 9, 34, 18, 26, 36, 38, 81lsatfixedN 29199 . 2  |-  ( ph  ->  E. t  e.  ( ( N `  {
v } )  \  {  .0.  } ) ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )
83 simpr 447 . . . . . 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
) } ) )
848ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
8541ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  ->  U  e.  LMod )
8618ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  ->  u  e.  V )
8720ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
s  e.  ( D 
\  { Q }
) )
8826ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
v  e.  V )
8927ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( M `  ( N `  { v } ) )  =  ( L `  {
s } ) )
9028ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  ->  -.  u  e.  ( N `  { v } ) )
91 simplr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
t  e.  ( ( N `  { v } )  \  {  .0.  } ) )
926, 7, 1, 4, 11, 14, 10, 17, 84, 87, 88, 89, 86, 90, 13, 15, 3, 23, 91hdmaprnlem4tN 32045 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
t  e.  V )
931, 2lmodvacl 15641 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  u  e.  V  /\  t  e.  V )  ->  (
u  .+  t )  e.  V )
9485, 86, 92, 93syl3anc 1182 . . . . . . . 8  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( u  .+  t
)  e.  V )
951, 40, 4lspsncl 15734 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  (
u  .+  t )  e.  V )  ->  ( N `  { (
u  .+  t ) } )  e.  (
LSubSp `  U ) )
9685, 94, 95syl2anc 642 . . . . . . 7  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( N `  {
( u  .+  t
) } )  e.  ( LSubSp `  U )
)
976, 10, 7, 40, 84, 96mapdcnvid1N 31844 . . . . . 6  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( `' M `  ( M `  ( N `
 { ( u 
.+  t ) } ) ) )  =  ( N `  {
( u  .+  t
) } ) )
9883, 97eqtr4d 2318 . . . . 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 ) } ) ) ) )
9972ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( L `  {
( ( S `  u )  .+b  s
) } )  e. 
ran  M )
1006, 10, 7, 40, 84, 96mapdcl 31843 . . . . . 6  |-  ( ( ( ph  /\  t  e.  ( ( N `  { v } ) 
\  {  .0.  }
) )  /\  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )  =  ( N `  { ( u  .+  t ) } ) )  -> 
( M `  ( N `  { (
u  .+  t ) } ) )  e. 
ran  M )
1016, 10, 84, 99, 100mapdcnv11N 31849 . . . . 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 ) } ) ) ) )
10298, 101mpbid 201 . . . 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 ) } ) ) )
103102ex 423 . . 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 ) } ) ) ) )
104103reximdva 2655 . 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 ) } ) ) ) )
10582, 104mpd 14 1  |-  ( ph  ->  E. t  e.  ( ( N `  {
v } )  \  {  .0.  } ) ( L `  { ( ( S `  u
)  .+b  s ) } )  =  ( M `  ( N `
 { ( u 
.+  t ) } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149    C_ wss 3152   {csn 3640   {cpr 3641   `'ccnv 4688   ran crn 4690   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400  SubGrpcsubg 14615   LSSumclsm 14945   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728  LSAtomsclsa 29164   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268  LCDualclcd 31776  mapdcmpd 31814  HDMapchdma 31983
This theorem is referenced by:  hdmaprnlem10N  32052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-lshyp 29167  df-lcv 29209  df-lfl 29248  df-lkr 29276  df-ldual 29314  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tgrp 30932  df-tendo 30944  df-edring 30946  df-dveca 31192  df-disoa 31219  df-dvech 31269  df-dib 31329  df-dic 31363  df-dih 31419  df-doch 31538  df-djh 31585  df-lcdual 31777  df-mapd 31815  df-hvmap 31947  df-hdmap1 31984  df-hdmap 31985
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