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Theorem islshpat 29877
Description: Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 29840. (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 2438 . . 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 29840 . 2  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )
8 df-3an 939 . . . . 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 2864 . . . . 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 245 . . . 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 2713 . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
v  =  ( 0g
`  W ) )
1312sneqd 3829 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  { v }  =  { ( 0g `  W ) } )
1413fveq2d 5734 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( ( LSpan `  W
) `  { v } )  =  ( ( LSpan `  W ) `  { ( 0g `  W ) } ) )
156ad3antrrr 712 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  W  e.  LMod )
16 eqid 2438 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0g
`  W )  =  ( 0g `  W
)
1716, 2lspsn0 16086 . . . . . . . . . . . . . . . . . . . 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 2470 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( ( LSpan `  W
) `  { v } )  =  {
( 0g `  W
) } )
2019oveq2d 6099 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  ( U 
.(+)  { ( 0g `  W ) } ) )
21 simplrl 738 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  U  e.  S )
223lsssubg 16035 . . . . . . . . . . . . . . . . . . 19  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
2315, 21, 22syl2anc 644 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  U  e.  (SubGrp `  W
) )
2416, 4lsm01 15305 . . . . . . . . . . . . . . . . . 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 2470 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  U )
27 simplrr 739 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  U  =/=  V )
2826, 27eqnetrd 2621 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =/=  V )
2928ex 425 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V ) )  ->  ( v  =  ( 0g `  W
)  ->  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =/= 
V ) )
3029necon2d 2656 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V ) )  ->  ( ( U 
.(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V  ->  v  =/=  ( 0g `  W
) ) )
3130pm4.71rd 618 . . . . . . . . . . . 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 624 . . . . . . . . . . 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 624 . . . . . . . . . 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 3929 . . . . . . . . . . . 12  |-  ( v  e.  ( V  \  { ( 0g `  W ) } )  <-> 
( v  e.  V  /\  v  =/=  ( 0g `  W ) ) )
3534anbi1i 678 . . . . . . . . . . 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 632 . . . . . . . . . . . 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 774 . . . . . . . . . . . . 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 677 . . . . . . . . . . . 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 242 . . . . . . . . . . 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 243 . . . . . . . . . 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 254 . . . . . . . . 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 1637 . . . . . . . 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 250 . . . . . . 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 5744 . . . . . . . . . 10  |-  ( (
LSpan `  W ) `  { v } )  e.  _V
4544rexcom4b 2979 . . . . . . . . 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 2713 . . . . . . . . 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 243 . . . . . . . 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 439 . . . . . . . . . 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 2732 . . . . . . . . 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 1593 . . . . . . . 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 242 . . . . . . 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 254 . . . . . 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 2863 . . . . . . . 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 6091 . . . . . . . . . . . 12  |-  ( q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  .(+)  q )  =  ( U  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
5554eqeq1d 2446 . . . . . . . . . . 11  |-  ( q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( U  .(+)  q )  =  V  <->  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) )
5655anbi2d 686 . . . . . . . . . 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 620 . . . . . . . . 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 2732 . . . . . . . 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 244 . . . . . . 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 1593 . . . . . 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 256 . . . . 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 29851 . . . . . . . 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 687 . . . . . 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 1637 . . . . 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 249 . . . 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 250 . . 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 939 . . . 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 2864 . . . . 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 2713 . . . . 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 244 . . . 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 243 . . 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 254 . 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 246 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 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    \ cdif 3319   {csn 3816   ` cfv 5456  (class class class)co 6083   Basecbs 13471   0gc0g 13725  SubGrpcsubg 14940   LSSumclsm 15270   LModclmod 15952   LSubSpclss 16010   LSpanclspn 16049  LSAtomsclsa 29834  LSHypclsh 29835
This theorem is referenced by:  islshpcv  29913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-cntz 15118  df-lsm 15272  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-ur 15667  df-lmod 15954  df-lss 16011  df-lsp 16050  df-lsatoms 29836  df-lshyp 29837
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