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Theorem islshpat 29207
Description: Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 29170. (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 2283 . . 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 29170 . 2  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V ) ) )
8 df-3an 936 . . . . 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 2694 . . . . 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 243 . . . 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 2549 . . . . . . . 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 447 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
v  =  ( 0g
`  W ) )
1312sneqd 3653 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  { v }  =  { ( 0g `  W ) } )
1413fveq2d 5529 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( ( LSpan `  W
) `  { v } )  =  ( ( LSpan `  W ) `  { ( 0g `  W ) } ) )
156ad3antrrr 710 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  W  e.  LMod )
16 eqid 2283 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0g
`  W )  =  ( 0g `  W
)
1716, 2lspsn0 15765 . . . . . . . . . . . . . . . . . . . 20  |-  ( W  e.  LMod  ->  ( (
LSpan `  W ) `  { ( 0g `  W ) } )  =  { ( 0g
`  W ) } )
1815, 17syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( ( LSpan `  W
) `  { ( 0g `  W ) } )  =  { ( 0g `  W ) } )
1914, 18eqtrd 2315 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( ( LSpan `  W
) `  { v } )  =  {
( 0g `  W
) } )
2019oveq2d 5874 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  ( U 
.(+)  { ( 0g `  W ) } ) )
21 simplrl 736 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  U  e.  S )
223lsssubg 15714 . . . . . . . . . . . . . . . . . . 19  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  e.  (SubGrp `  W )
)
2315, 21, 22syl2anc 642 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  U  e.  (SubGrp `  W
) )
2416, 4lsm01 14980 . . . . . . . . . . . . . . . . . 18  |-  ( U  e.  (SubGrp `  W
)  ->  ( U  .(+)  { ( 0g `  W ) } )  =  U )
2523, 24syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  { ( 0g `  W ) } )  =  U )
2620, 25eqtrd 2315 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =  U )
27 simplrr 737 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  ->  U  =/=  V )
2826, 27eqnetrd 2464 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V
) )  /\  v  =  ( 0g `  W ) )  -> 
( U  .(+)  ( (
LSpan `  W ) `  { v } ) )  =/=  V )
2928ex 423 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V ) )  ->  ( v  =  ( 0g `  W
)  ->  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =/= 
V ) )
3029necon2d 2496 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  v  e.  V )  /\  ( U  e.  S  /\  U  =/=  V ) )  ->  ( ( U 
.(+)  ( ( LSpan `  W ) `  {
v } ) )  =  V  ->  v  =/=  ( 0g `  W
) ) )
3130pm4.71rd 616 . . . . . . . . . . . 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 622 . . . . . . . . . . 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 622 . . . . . . . . . 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 3749 . . . . . . . . . . . 12  |-  ( v  e.  ( V  \  { ( 0g `  W ) } )  <-> 
( v  e.  V  /\  v  =/=  ( 0g `  W ) ) )
3534anbi1i 676 . . . . . . . . . . 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 630 . . . . . . . . . . . 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 772 . . . . . . . . . . . . 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 675 . . . . . . . . . . . 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 240 . . . . . . . . . . 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 241 . . . . . . . . . 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 252 . . . . . . . . 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 1612 . . . . . . . 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 248 . . . . . . 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 5539 . . . . . . . . . 10  |-  ( (
LSpan `  W ) `  { v } )  e.  _V
4544rexcom4b 2809 . . . . . . . . 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 2549 . . . . . . . . 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 241 . . . . . . . 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 437 . . . . . . . . . 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 2568 . . . . . . . . 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 1569 . . . . . . . 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 240 . . . . . . 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 252 . . . . . 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 2693 . . . . . . . 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 5866 . . . . . . . . . . . 12  |-  ( q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( U  .(+)  q )  =  ( U  .(+)  ( ( LSpan `  W ) `  { v } ) ) )
5554eqeq1d 2291 . . . . . . . . . . 11  |-  ( q  =  ( ( LSpan `  W ) `  {
v } )  -> 
( ( U  .(+)  q )  =  V  <->  ( U  .(+) 
( ( LSpan `  W
) `  { v } ) )  =  V ) )
5655anbi2d 684 . . . . . . . . . 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 618 . . . . . . . . 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 2568 . . . . . . . 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 242 . . . . . . 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 1569 . . . . . 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 254 . . . . 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 29181 . . . . . . . 8  |-  ( W  e.  LMod  ->  ( q  e.  A  <->  E. v  e.  ( V  \  {
( 0g `  W
) } ) q  =  ( ( LSpan `  W ) `  {
v } ) ) )
646, 63syl 15 . . . . . . 7  |-  ( ph  ->  ( q  e.  A  <->  E. v  e.  ( V 
\  { ( 0g
`  W ) } ) q  =  ( ( LSpan `  W ) `  { v } ) ) )
6564anbi1d 685 . . . . . 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 1612 . . . . 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 247 . . . 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 248 . . 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 936 . . . 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 2694 . . . . 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 2549 . . . . 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 242 . . . 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 241 . . 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 252 . 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 244 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 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148   0gc0g 13400  SubGrpcsubg 14615   LSSumclsm 14945   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728  LSAtomsclsa 29164  LSHypclsh 29165
This theorem is referenced by:  islshpcv  29243
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-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-uni 3828  df-int 3863  df-iun 3907  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-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lsatoms 29166  df-lshyp 29167
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