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Theorem islshpat 29500
Description: Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 29463. (Contributed by NM, 11-Jan-2015.)
Hypotheses
Ref Expression
islshpat.v  |-  V  =  ( Base `  W
)
islshpat.s  |-  S  =  ( LSubSp `  W )
islshpat.p  |-  .(+)  =  (
LSSum `  W )
islshpat.h  |-  H  =  (LSHyp `  W )
islshpat.a  |-  A  =  (LSAtoms `  W )
islshpat.w  |-  ( ph  ->  W  e.  LMod )
Assertion
Ref Expression
islshpat  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. q  e.  A  ( U  .(+)  q )  =  V ) ) )
Distinct variable groups:    .(+) , q    S, q    U, q    V, q    W, q    ph, q
Allowed substitution hints:    A( q)    H( q)

Proof of Theorem islshpat
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 islshpat.v . . 3  |-  V  =  ( Base `  W
)
2 eqid 2404 . . 3  |-  ( LSpan `  W )  =  (
LSpan `  W )
3 islshpat.s . . 3  |-  S  =  ( LSubSp `  W )
4 islshpat.p . . 3  |-  .(+)  =  (
LSSum `  W )
5 islshpat.h . . 3  |-  H  =  (LSHyp `  W )
6 islshpat.w . . 3  |-  ( ph  ->  W  e.  LMod )
71, 2, 3, 4, 5, 6islshpsm 29463 . 2  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )
8 df-3an 938 . . . . 5  |-  ( ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V )  <->  ( ( U  e.  S  /\  U  =/=  V )  /\  E. v  e.  V  ( U  .(+)  ( ( LSpan `  W ) `  { v } ) )  =  V ) )
9 r19.42v 2822 . . . . 5  |-  ( E. v  e.  V  ( ( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V )  <->  ( ( U  e.  S  /\  U  =/=  V )  /\  E. v  e.  V  ( U  .(+)  ( ( LSpan `  W ) `  { v } ) )  =  V ) )
108, 9bitr4i 244 . . . 4  |-  ( ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V )  <->  E. v  e.  V  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) )
11 df-rex 2672 . . . . . . . 8  |-  ( E. v  e.  V  ( ( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V )  <->  E. v
( v  e.  V  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )
12 simpr 448 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
v  =  ( 0g
`  W ) )
1312sneqd 3787 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  { v }  =  { ( 0g `  W ) } )
1413fveq2d 5691 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( ( LSpan `  W
) `  { v } )  =  ( ( LSpan `  W ) `  { ( 0g `  W ) } ) )
156ad3antrrr 711 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  W  e.  LMod )
16 eqid 2404 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0g
`  W )  =  ( 0g `  W
)
1716, 2lspsn0 16039 . . . . . . . . . . . . . . . . . . . 20  |-  ( W  e.  LMod  ->  ( (
LSpan `  W ) `  { ( 0g `  W ) } )  =  { ( 0g
`  W ) } )
1815, 17syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( ( LSpan `  W
) `  { ( 0g `  W ) } )  =  { ( 0g `  W ) } )
1914, 18eqtrd 2436 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( ( LSpan `  W
) `  { v } )  =  {
( 0g `  W
) } )
2019oveq2d 6056 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  ( U 
.(+)  { ( 0g `  W ) } ) )
21 simplrl 737 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  U  e.  S )
223lsssubg 15988 . . . . . . . . . . . . . . . . . . 19  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
2315, 21, 22syl2anc 643 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  U  e.  (SubGrp `  W
) )
2416, 4lsm01 15258 . . . . . . . . . . . . . . . . . 18  |-  ( U  e.  (SubGrp `  W
)  ->  ( U  .(+)  { ( 0g `  W ) } )  =  U )
2523, 24syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  { ( 0g `  W ) } )  =  U )
2620, 25eqtrd 2436 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  U )
27 simplrr 738 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  U  =/=  V )
2826, 27eqnetrd 2585 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =/=  V )
2928ex 424 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V ) )  ->  ( v  =  ( 0g `  W
)  ->  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =/= 
V ) )
3029necon2d 2617 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V ) )  ->  ( ( U 
.(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V  ->  v  =/=  ( 0g `  W
) ) )
3130pm4.71rd 617 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V ) )  ->  ( ( U 
.(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V  <->  ( v  =/=  ( 0g `  W
)  /\  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) ) )
3231pm5.32da 623 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  V )  ->  (
( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V )  <->  ( ( U  e.  S  /\  U  =/=  V )  /\  ( v  =/=  ( 0g `  W )  /\  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) ) ) )
3332pm5.32da 623 . . . . . . . . . 10  |-  ( ph  ->  ( ( v  e.  V  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) )  <->  ( v  e.  V  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( v  =/=  ( 0g `  W )  /\  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) ) ) ) )
34 eldifsn 3887 . . . . . . . . . . . 12  |-  ( v  e.  ( V  \  { ( 0g `  W ) } )  <-> 
( v  e.  V  /\  v  =/=  ( 0g `  W ) ) )
3534anbi1i 677 . . . . . . . . . . 11  |-  ( ( v  e.  ( V 
\  { ( 0g
`  W ) } )  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) )  <->  ( ( v  e.  V  /\  v  =/=  ( 0g `  W
) )  /\  (
( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) ) )
36 anass 631 . . . . . . . . . . . 12  |-  ( ( ( v  e.  V  /\  v  =/=  ( 0g `  W ) )  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) )  <-> 
( v  e.  V  /\  ( v  =/=  ( 0g `  W )  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) ) )
37 an12 773 . . . . . . . . . . . . 13  |-  ( ( v  =/=  ( 0g
`  W )  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) )  <-> 
( ( U  e.  S  /\  U  =/= 
V )  /\  (
v  =/=  ( 0g
`  W )  /\  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) ) )
3837anbi2i 676 . . . . . . . . . . . 12  |-  ( ( v  e.  V  /\  ( v  =/=  ( 0g `  W )  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )  <->  ( v  e.  V  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( v  =/=  ( 0g `  W )  /\  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) ) ) )
3936, 38bitri 241 . . . . . . . . . . 11  |-  ( ( ( v  e.  V  /\  v  =/=  ( 0g `  W ) )  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) )  <-> 
( v  e.  V  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  (
v  =/=  ( 0g
`  W )  /\  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) ) ) )
4035, 39bitr2i 242 . . . . . . . . . 10  |-  ( ( v  e.  V  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  (
v  =/=  ( 0g
`  W )  /\  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) ) )  <->  ( v  e.  ( V  \  {
( 0g `  W
) } )  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )
4133, 40syl6bb 253 . . . . . . . . 9  |-  ( ph  ->  ( ( v  e.  V  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) )  <->  ( v  e.  ( V  \  {
( 0g `  W
) } )  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) ) )
4241exbidv 1633 . . . . . . . 8  |-  ( ph  ->  ( E. v ( v  e.  V  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) )  <->  E. v ( v  e.  ( V  \  {
( 0g `  W
) } )  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) ) )
4311, 42syl5bb 249 . . . . . . 7  |-  ( ph  ->  ( E. v  e.  V  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V )  <->  E. v
( v  e.  ( V  \  { ( 0g `  W ) } )  /\  (
( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) ) ) )
44 fvex 5701 . . . . . . . . . 10  |-  ( (
LSpan `  W ) `  { v } )  e.  _V
4544rexcom4b 2937 . . . . . . . . 9  |-  ( E. q E. v  e.  ( V  \  {
( 0g `  W
) } ) ( ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V )  /\  q  =  ( ( LSpan `  W ) `  { v } ) )  <->  E. v  e.  ( V  \  { ( 0g `  W ) } ) ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) )
46 df-rex 2672 . . . . . . . . 9  |-  ( E. v  e.  ( V 
\  { ( 0g
`  W ) } ) ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V )  <->  E. v
( v  e.  ( V  \  { ( 0g `  W ) } )  /\  (
( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) ) )
4745, 46bitr2i 242 . . . . . . . 8  |-  ( E. v ( v  e.  ( V  \  {
( 0g `  W
) } )  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) )  <->  E. q E. v  e.  ( V  \  {
( 0g `  W
) } ) ( ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V )  /\  q  =  ( ( LSpan `  W ) `  { v } ) ) )
48 ancom 438 . . . . . . . . . 10  |-  ( ( ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V )  /\  q  =  ( ( LSpan `  W ) `  { v } ) )  <->  ( q  =  ( ( LSpan `  W
) `  { v } )  /\  (
( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) ) )
4948rexbii 2691 . . . . . . . . 9  |-  ( E. v  e.  ( V 
\  { ( 0g
`  W ) } ) ( ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V )  /\  q  =  ( ( LSpan `  W ) `  { v } ) )  <->  E. v  e.  ( V  \  { ( 0g `  W ) } ) ( q  =  ( ( LSpan `  W ) `  {
v } )  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )
5049exbii 1589 . . . . . . . 8  |-  ( E. q E. v  e.  ( V  \  {
( 0g `  W
) } ) ( ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V )  /\  q  =  ( ( LSpan `  W ) `  { v } ) )  <->  E. q E. v  e.  ( V  \  {
( 0g `  W
) } ) ( q  =  ( (
LSpan `  W ) `  { v } )  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )
5147, 50bitri 241 . . . . . . 7  |-  ( E. v ( v  e.  ( V  \  {
( 0g `  W
) } )  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) )  <->  E. q E. v  e.  ( V  \  {
( 0g `  W
) } ) ( q  =  ( (
LSpan `  W ) `  { v } )  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )
5243, 51syl6bb 253 . . . . . 6  |-  ( ph  ->  ( E. v  e.  V  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V )  <->  E. q E. v  e.  ( V  \  { ( 0g
`  W ) } ) ( q  =  ( ( LSpan `  W
) `  { v } )  /\  (
( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) ) ) )
53 r19.41v 2821 . . . . . . . 8  |-  ( E. v  e.  ( V 
\  { ( 0g
`  W ) } ) ( q  =  ( ( LSpan `  W
) `  { v } )  /\  (
( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
q )  =  V ) )  <->  ( E. v  e.  ( V  \  { ( 0g `  W ) } ) q  =  ( (
LSpan `  W ) `  { v } )  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  q )  =  V ) ) )
54 oveq2 6048 . . . . . . . . . . . 12  |-  ( q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  .(+)  q )  =  ( U  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
5554eqeq1d 2412 . . . . . . . . . . 11  |-  ( q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( U  .(+)  q )  =  V  <->  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) )
5655anbi2d 685 . . . . . . . . . 10  |-  ( q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  q )  =  V )  <->  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  V ) ) )
5756pm5.32i 619 . . . . . . . . 9  |-  ( ( q  =  ( (
LSpan `  W ) `  { v } )  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  q )  =  V ) )  <->  ( q  =  ( ( LSpan `  W ) `  {
v } )  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )
5857rexbii 2691 . . . . . . . 8  |-  ( E. v  e.  ( V 
\  { ( 0g
`  W ) } ) ( q  =  ( ( LSpan `  W
) `  { v } )  /\  (
( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
q )  =  V ) )  <->  E. v  e.  ( V  \  {
( 0g `  W
) } ) ( q  =  ( (
LSpan `  W ) `  { v } )  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )
5953, 58bitr3i 243 . . . . . . 7  |-  ( ( E. v  e.  ( V  \  { ( 0g `  W ) } ) q  =  ( ( LSpan `  W
) `  { v } )  /\  (
( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
q )  =  V ) )  <->  E. v  e.  ( V  \  {
( 0g `  W
) } ) ( q  =  ( (
LSpan `  W ) `  { v } )  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )
6059exbii 1589 . . . . . 6  |-  ( E. q ( E. v  e.  ( V  \  {
( 0g `  W
) } ) q  =  ( ( LSpan `  W ) `  {
v } )  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  q )  =  V ) )  <->  E. q E. v  e.  ( V  \  { ( 0g
`  W ) } ) ( q  =  ( ( LSpan `  W
) `  { v } )  /\  (
( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) ) )
6152, 60syl6bbr 255 . . . . 5  |-  ( ph  ->  ( E. v  e.  V  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V )  <->  E. q
( E. v  e.  ( V  \  {
( 0g `  W
) } ) q  =  ( ( LSpan `  W ) `  {
v } )  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  q )  =  V ) ) ) )
62 islshpat.a . . . . . . . . 9  |-  A  =  (LSAtoms `  W )
631, 2, 16, 62islsat 29474 . . . . . . . 8  |-  ( W  e.  LMod  ->  ( q  e.  A  <->  E. v  e.  ( V  \  {
( 0g `  W
) } ) q  =  ( ( LSpan `  W ) `  {
v } ) ) )
646, 63syl 16 . . . . . . 7  |-  ( ph  ->  ( q  e.  A  <->  E. v  e.  ( V 
\  { ( 0g
`  W ) } ) q  =  ( ( LSpan `  W ) `  { v } ) ) )
6564anbi1d 686 . . . . . 6  |-  ( ph  ->  ( ( q  e.  A  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  q )  =  V ) )  <-> 
( E. v  e.  ( V  \  {
( 0g `  W
) } ) q  =  ( ( LSpan `  W ) `  {
v } )  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  q )  =  V ) ) ) )
6665exbidv 1633 . . . . 5  |-  ( ph  ->  ( E. q ( q  e.  A  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  q )  =  V ) )  <->  E. q
( E. v  e.  ( V  \  {
( 0g `  W
) } ) q  =  ( ( LSpan `  W ) `  {
v } )  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  q )  =  V ) ) ) )
6761, 66bitr4d 248 . . . 4  |-  ( ph  ->  ( E. v  e.  V  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V )  <->  E. q
( q  e.  A  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  q )  =  V ) ) ) )
6810, 67syl5bb 249 . . 3  |-  ( ph  ->  ( ( U  e.  S  /\  U  =/= 
V  /\  E. v  e.  V  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V )  <->  E. q
( q  e.  A  /\  ( ( U  e.  S  /\  U  =/= 
V )  /\  ( U  .(+)  q )  =  V ) ) ) )
69 df-3an 938 . . . 4  |-  ( ( U  e.  S  /\  U  =/=  V  /\  E. q  e.  A  ( U  .(+)  q )  =  V )  <->  ( ( U  e.  S  /\  U  =/=  V )  /\  E. q  e.  A  ( U  .(+)  q )  =  V ) )
70 r19.42v 2822 . . . . 5  |-  ( E. q  e.  A  ( ( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
q )  =  V )  <->  ( ( U  e.  S  /\  U  =/=  V )  /\  E. q  e.  A  ( U  .(+)  q )  =  V ) )
71 df-rex 2672 . . . . 5  |-  ( E. q  e.  A  ( ( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
q )  =  V )  <->  E. q ( q  e.  A  /\  (
( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
q )  =  V ) ) )
7270, 71bitr3i 243 . . . 4  |-  ( ( ( U  e.  S  /\  U  =/=  V
)  /\  E. q  e.  A  ( U  .(+) 
q )  =  V )  <->  E. q ( q  e.  A  /\  (
( U  e.  S  /\  U  =/=  V
)  /\  ( U  .(+) 
q )  =  V ) ) )
7369, 72bitr2i 242 . . 3  |-  ( E. q ( q  e.  A  /\  ( ( U  e.  S  /\  U  =/=  V )  /\  ( U  .(+)  q )  =  V ) )  <-> 
( U  e.  S  /\  U  =/=  V  /\  E. q  e.  A  ( U  .(+)  q )  =  V ) )
7468, 73syl6bb 253 . 2  |-  ( ph  ->  ( ( U  e.  S  /\  U  =/= 
V  /\  E. v  e.  V  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V )  <->  ( U  e.  S  /\  U  =/= 
V  /\  E. q  e.  A  ( U  .(+) 
q )  =  V ) ) )
757, 74bitrd 245 1  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. q  e.  A  ( U  .(+)  q )  =  V ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667    \ cdif 3277   {csn 3774   ` cfv 5413  (class class class)co 6040   Basecbs 13424   0gc0g 13678  SubGrpcsubg 14893   LSSumclsm 15223   LModclmod 15905   LSubSpclss 15963   LSpanclspn 16002  LSAtomsclsa 29457  LSHypclsh 29458
This theorem is referenced by:  islshpcv  29536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lsatoms 29459  df-lshyp 29460
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