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Theorem islshpat 32552
Description: Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 32515. (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 2422 . . 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 32515 . 2  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )
8 df-3an 984 . . . . 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 2980 . . . . 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 255 . . . 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 2777 . . . . . . . 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 462 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
v  =  ( 0g
`  W ) )
1312sneqd 4010 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  { v }  =  { ( 0g `  W ) } )
1413fveq2d 5885 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( ( LSpan `  W
) `  { v } )  =  ( ( LSpan `  W ) `  { ( 0g `  W ) } ) )
156ad3antrrr 734 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  W  e.  LMod )
16 eqid 2422 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0g
`  W )  =  ( 0g `  W
)
1716, 2lspsn0 18230 . . . . . . . . . . . . . . . . . . . 20  |-  ( W  e.  LMod  ->  ( (
LSpan `  W ) `  { ( 0g `  W ) } )  =  { ( 0g
`  W ) } )
1815, 17syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( ( LSpan `  W
) `  { ( 0g `  W ) } )  =  { ( 0g `  W ) } )
1914, 18eqtrd 2463 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( ( LSpan `  W
) `  { v } )  =  {
( 0g `  W
) } )
2019oveq2d 6321 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  ( U 
.(+)  { ( 0g `  W ) } ) )
21 simplrl 768 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  U  e.  S )
223lsssubg 18179 . . . . . . . . . . . . . . . . . . 19  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
2315, 21, 22syl2anc 665 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  U  e.  (SubGrp `  W
) )
2416, 4lsm01 17320 . . . . . . . . . . . . . . . . . 18  |-  ( U  e.  (SubGrp `  W
)  ->  ( U  .(+)  { ( 0g `  W ) } )  =  U )
2523, 24syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  { ( 0g `  W ) } )  =  U )
2620, 25eqtrd 2463 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  U )
27 simplrr 769 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  U  =/=  V )
2826, 27eqnetrd 2713 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =/=  V )
2928ex 435 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V ) )  ->  ( v  =  ( 0g `  W
)  ->  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =/= 
V ) )
3029necon2d 2646 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V ) )  ->  ( ( U 
.(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V  ->  v  =/=  ( 0g `  W
) ) )
3130pm4.71rd 639 . . . . . . . . . . . 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 645 . . . . . . . . . . 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 645 . . . . . . . . . 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 4125 . . . . . . . . . . . 12  |-  ( v  e.  ( V  \  { ( 0g `  W ) } )  <-> 
( v  e.  V  /\  v  =/=  ( 0g `  W ) ) )
3534anbi1i 699 . . . . . . . . . . 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 653 . . . . . . . . . . . 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 804 . . . . . . . . . . . . 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 698 . . . . . . . . . . . 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 252 . . . . . . . . . . 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 253 . . . . . . . . . 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 264 . . . . . . . . 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 1762 . . . . . . . 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 260 . . . . . . 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 5891 . . . . . . . . . 10  |-  ( (
LSpan `  W ) `  { v } )  e.  _V
4544rexcom4b 3103 . . . . . . . . 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 2777 . . . . . . . . 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 253 . . . . . . . 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 451 . . . . . . . . . 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 2924 . . . . . . . . 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 1712 . . . . . . . 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 252 . . . . . . 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 264 . . . . . 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 2977 . . . . . . . 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 6313 . . . . . . . . . . . 12  |-  ( q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  .(+)  q )  =  ( U  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
5554eqeq1d 2424 . . . . . . . . . . 11  |-  ( q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( U  .(+)  q )  =  V  <->  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) )
5655anbi2d 708 . . . . . . . . . 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 641 . . . . . . . . 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 2924 . . . . . . . 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 254 . . . . . . 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 1712 . . . . . 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 266 . . . . 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 32526 . . . . . . . 8  |-  ( W  e.  LMod  ->  ( q  e.  A  <->  E. v  e.  ( V  \  {
( 0g `  W
) } ) q  =  ( ( LSpan `  W ) `  {
v } ) ) )
646, 63syl 17 . . . . . . 7  |-  ( ph  ->  ( q  e.  A  <->  E. v  e.  ( V 
\  { ( 0g
`  W ) } ) q  =  ( ( LSpan `  W ) `  { v } ) ) )
6564anbi1d 709 . . . . . 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 1762 . . . . 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 259 . . . 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 260 . . 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 984 . . . 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 2980 . . . . 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 2777 . . . . 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 254 . . . 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 253 . . 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 264 . 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 256 1  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. q  e.  A  ( U  .(+)  q )  =  V ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2614   E.wrex 2772    \ cdif 3433   {csn 3998   ` cfv 5601  (class class class)co 6305   Basecbs 15120   0gc0g 15337  SubGrpcsubg 16810   LSSumclsm 17285   LModclmod 18090   LSubSpclss 18154   LSpanclspn 18193  LSAtomsclsa 32509  LSHypclsh 32510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-ress 15127  df-plusg 15202  df-0g 15339  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-submnd 16582  df-grp 16672  df-minusg 16673  df-sbg 16674  df-subg 16813  df-cntz 16970  df-lsm 17287  df-cmn 17431  df-abl 17432  df-mgp 17723  df-ur 17735  df-ring 17781  df-lmod 18092  df-lss 18155  df-lsp 18194  df-lsatoms 32511  df-lshyp 32512
This theorem is referenced by:  islshpcv  32588
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