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Theorem hgmapadd 32384
Description: Part 15 of [Baer] p. 50 line 13. (Contributed by NM, 6-Jun-2015.)
Hypotheses
Ref Expression
hgmapadd.h  |-  H  =  ( LHyp `  K
)
hgmapadd.u  |-  U  =  ( ( DVecH `  K
) `  W )
hgmapadd.r  |-  R  =  (Scalar `  U )
hgmapadd.b  |-  B  =  ( Base `  R
)
hgmapadd.p  |-  .+  =  ( +g  `  R )
hgmapadd.g  |-  G  =  ( (HGMap `  K
) `  W )
hgmapadd.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hgmapadd.x  |-  ( ph  ->  X  e.  B )
hgmapadd.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
hgmapadd  |-  ( ph  ->  ( G `  ( X  .+  Y ) )  =  ( ( G `
 X )  .+  ( G `  Y ) ) )

Proof of Theorem hgmapadd
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 hgmapadd.h . . . 4  |-  H  =  ( LHyp `  K
)
2 hgmapadd.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
3 eqid 2408 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
4 eqid 2408 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
5 hgmapadd.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 2, 3, 4, 5dvh1dim 31929 . . 3  |-  ( ph  ->  E. t  e.  (
Base `  U )
t  =/=  ( 0g
`  U ) )
7 eqid 2408 . . . . . . . . 9  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
81, 7, 5lcdlmod 32079 . . . . . . . 8  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LMod )
983ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( (LCDual `  K ) `  W
)  e.  LMod )
10 hgmapadd.r . . . . . . . 8  |-  R  =  (Scalar `  U )
11 hgmapadd.b . . . . . . . 8  |-  B  =  ( Base `  R
)
12 eqid 2408 . . . . . . . 8  |-  (Scalar `  ( (LCDual `  K ) `  W ) )  =  (Scalar `  ( (LCDual `  K ) `  W
) )
13 eqid 2408 . . . . . . . 8  |-  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) )
14 hgmapadd.g . . . . . . . 8  |-  G  =  ( (HGMap `  K
) `  W )
1553ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
16 hgmapadd.x . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
17163ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  X  e.  B
)
181, 2, 10, 11, 7, 12, 13, 14, 15, 17hgmapdcl 32380 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( G `  X )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
19 hgmapadd.y . . . . . . . . 9  |-  ( ph  ->  Y  e.  B )
201, 2, 10, 11, 7, 12, 13, 14, 5, 19hgmapdcl 32380 . . . . . . . 8  |-  ( ph  ->  ( G `  Y
)  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
21203ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( G `  Y )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
22 eqid 2408 . . . . . . . 8  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
23 eqid 2408 . . . . . . . 8  |-  ( (HDMap `  K ) `  W
)  =  ( (HDMap `  K ) `  W
)
24 simp2 958 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  t  e.  (
Base `  U )
)
251, 2, 3, 7, 22, 23, 15, 24hdmapcl 32320 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  t )  e.  ( Base `  (
(LCDual `  K ) `  W ) ) )
26 eqid 2408 . . . . . . . 8  |-  ( +g  `  ( (LCDual `  K
) `  W )
)  =  ( +g  `  ( (LCDual `  K
) `  W )
)
27 eqid 2408 . . . . . . . 8  |-  ( .s
`  ( (LCDual `  K ) `  W
) )  =  ( .s `  ( (LCDual `  K ) `  W
) )
28 eqid 2408 . . . . . . . 8  |-  ( +g  `  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) )
2922, 26, 12, 27, 13, 28lmodvsdir 15933 . . . . . . 7  |-  ( ( ( (LCDual `  K
) `  W )  e.  LMod  /\  ( ( G `  X )  e.  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) )  /\  ( G `  Y )  e.  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) )  /\  (
( (HDMap `  K
) `  W ) `  t )  e.  (
Base `  ( (LCDual `  K ) `  W
) ) ) )  ->  ( ( ( G `  X ) ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) ) ( G `
 Y ) ) ( .s `  (
(LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) )  =  ( ( ( G `
 X ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  t
) ) ( +g  `  ( (LCDual `  K
) `  W )
) ( ( G `
 Y ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  t
) ) ) )
309, 18, 21, 25, 29syl13anc 1186 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( G `  X ) ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) ) ( G `
 Y ) ) ( .s `  (
(LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) )  =  ( ( ( G `
 X ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  t
) ) ( +g  `  ( (LCDual `  K
) `  W )
) ( ( G `
 Y ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  t
) ) ) )
311, 2, 5dvhlmod 31597 . . . . . . . . . 10  |-  ( ph  ->  U  e.  LMod )
32313ad2ant1 978 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  U  e.  LMod )
33193ad2ant1 978 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  Y  e.  B
)
34 eqid 2408 . . . . . . . . . 10  |-  ( +g  `  U )  =  ( +g  `  U )
35 eqid 2408 . . . . . . . . . 10  |-  ( .s
`  U )  =  ( .s `  U
)
36 hgmapadd.p . . . . . . . . . 10  |-  .+  =  ( +g  `  R )
373, 34, 10, 35, 11, 36lmodvsdir 15933 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  ( X  e.  B  /\  Y  e.  B  /\  t  e.  ( Base `  U ) ) )  ->  ( ( X 
.+  Y ) ( .s `  U ) t )  =  ( ( X ( .s
`  U ) t ) ( +g  `  U
) ( Y ( .s `  U ) t ) ) )
3832, 17, 33, 24, 37syl13anc 1186 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( X 
.+  Y ) ( .s `  U ) t )  =  ( ( X ( .s
`  U ) t ) ( +g  `  U
) ( Y ( .s `  U ) t ) ) )
3938fveq2d 5695 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  ( ( X  .+  Y ) ( .s `  U ) t ) )  =  ( ( (HDMap `  K ) `  W
) `  ( ( X ( .s `  U ) t ) ( +g  `  U
) ( Y ( .s `  U ) t ) ) ) )
403, 10, 35, 11lmodvscl 15926 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  X  e.  B  /\  t  e.  ( Base `  U
) )  ->  ( X ( .s `  U ) t )  e.  ( Base `  U
) )
4132, 17, 24, 40syl3anc 1184 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( X ( .s `  U ) t )  e.  (
Base `  U )
)
423, 10, 35, 11lmodvscl 15926 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  Y  e.  B  /\  t  e.  ( Base `  U
) )  ->  ( Y ( .s `  U ) t )  e.  ( Base `  U
) )
4332, 33, 24, 42syl3anc 1184 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( Y ( .s `  U ) t )  e.  (
Base `  U )
)
441, 2, 3, 34, 7, 26, 23, 15, 41, 43hdmapadd 32333 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  ( ( X ( .s `  U ) t ) ( +g  `  U
) ( Y ( .s `  U ) t ) ) )  =  ( ( ( (HDMap `  K ) `  W ) `  ( X ( .s `  U ) t ) ) ( +g  `  (
(LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  ( Y ( .s
`  U ) t ) ) ) )
451, 2, 3, 35, 10, 11, 7, 27, 23, 14, 15, 24, 17hgmapvs 32381 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  ( X
( .s `  U
) t ) )  =  ( ( G `
 X ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  t
) ) )
461, 2, 3, 35, 10, 11, 7, 27, 23, 14, 15, 24, 33hgmapvs 32381 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  ( Y
( .s `  U
) t ) )  =  ( ( G `
 Y ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  t
) ) )
4745, 46oveq12d 6062 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  ( X ( .s `  U ) t ) ) ( +g  `  (
(LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  ( Y ( .s
`  U ) t ) ) )  =  ( ( ( G `
 X ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  t
) ) ( +g  `  ( (LCDual `  K
) `  W )
) ( ( G `
 Y ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  t
) ) ) )
4839, 44, 473eqtrrd 2445 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( G `  X ) ( .s `  (
(LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) ) ( +g  `  ( (LCDual `  K ) `  W
) ) ( ( G `  Y ) ( .s `  (
(LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) ) )  =  ( ( (HDMap `  K ) `  W
) `  ( ( X  .+  Y ) ( .s `  U ) t ) ) )
4910, 11, 36lmodacl 15920 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
5031, 16, 19, 49syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( X  .+  Y
)  e.  B )
51503ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( X  .+  Y )  e.  B
)
521, 2, 3, 35, 10, 11, 7, 27, 23, 14, 15, 24, 51hgmapvs 32381 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  ( ( X  .+  Y ) ( .s `  U ) t ) )  =  ( ( G `  ( X  .+  Y ) ) ( .s `  ( (LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) ) )
5330, 48, 523eqtrrd 2445 . . . . 5  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( G `
 ( X  .+  Y ) ) ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  t
) )  =  ( ( ( G `  X ) ( +g  `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( G `  Y ) ) ( .s `  ( (LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) ) )
54 eqid 2408 . . . . . 6  |-  ( 0g
`  ( (LCDual `  K ) `  W
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) )
551, 7, 5lcdlvec 32078 . . . . . . 7  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LVec )
56553ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( (LCDual `  K ) `  W
)  e.  LVec )
571, 2, 10, 11, 7, 12, 13, 14, 5, 50hgmapdcl 32380 . . . . . . 7  |-  ( ph  ->  ( G `  ( X  .+  Y ) )  e.  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) ) )
58573ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( G `  ( X  .+  Y ) )  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
591, 2, 10, 11, 7, 12, 13, 14, 5, 16hgmapdcl 32380 . . . . . . . 8  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
6012, 13, 28lmodacl 15920 . . . . . . . 8  |-  ( ( ( (LCDual `  K
) `  W )  e.  LMod  /\  ( G `  X )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) )  /\  ( G `  Y )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )  ->  ( ( G `  X )
( +g  `  (Scalar `  ( (LCDual `  K ) `  W ) ) ) ( G `  Y
) )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
618, 59, 20, 60syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( ( G `  X ) ( +g  `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( G `  Y ) )  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
62613ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( G `
 X ) ( +g  `  (Scalar `  ( (LCDual `  K ) `  W ) ) ) ( G `  Y
) )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
63 simp3 959 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  t  =/=  ( 0g `  U ) )
641, 2, 3, 4, 7, 54, 23, 15, 24hdmapeq0 32334 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  t
)  =  ( 0g
`  ( (LCDual `  K ) `  W
) )  <->  t  =  ( 0g `  U ) ) )
6564necon3bid 2606 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  t
)  =/=  ( 0g
`  ( (LCDual `  K ) `  W
) )  <->  t  =/=  ( 0g `  U ) ) )
6663, 65mpbird 224 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  t )  =/=  ( 0g `  (
(LCDual `  K ) `  W ) ) )
6722, 27, 12, 13, 54, 56, 58, 62, 25, 66lvecvscan2 16143 . . . . 5  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( G `  ( X 
.+  Y ) ) ( .s `  (
(LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  t ) )  =  ( ( ( G `
 X ) ( +g  `  (Scalar `  ( (LCDual `  K ) `  W ) ) ) ( G `  Y
) ) ( .s
`  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  t
) )  <->  ( G `  ( X  .+  Y
) )  =  ( ( G `  X
) ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) ) ( G `
 Y ) ) ) )
6853, 67mpbid 202 . . . 4  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( G `  ( X  .+  Y ) )  =  ( ( G `  X ) ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) ) ( G `
 Y ) ) )
6968rexlimdv3a 2796 . . 3  |-  ( ph  ->  ( E. t  e.  ( Base `  U
) t  =/=  ( 0g `  U )  -> 
( G `  ( X  .+  Y ) )  =  ( ( G `
 X ) ( +g  `  (Scalar `  ( (LCDual `  K ) `  W ) ) ) ( G `  Y
) ) ) )
706, 69mpd 15 . 2  |-  ( ph  ->  ( G `  ( X  .+  Y ) )  =  ( ( G `
 X ) ( +g  `  (Scalar `  ( (LCDual `  K ) `  W ) ) ) ( G `  Y
) ) )
711, 2, 10, 36, 7, 12, 28, 5lcdsadd 32088 . . 3  |-  ( ph  ->  ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) )  =  .+  )
7271oveqd 6061 . 2  |-  ( ph  ->  ( ( G `  X ) ( +g  `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( G `  Y ) )  =  ( ( G `  X ) 
.+  ( G `  Y ) ) )
7370, 72eqtrd 2440 1  |-  ( ph  ->  ( G `  ( X  .+  Y ) )  =  ( ( G `
 X )  .+  ( G `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   E.wrex 2671   ` cfv 5417  (class class class)co 6044   Basecbs 13428   +g cplusg 13488  Scalarcsca 13491   .scvsca 13492   0gc0g 13682   LModclmod 15909   LVecclvec 16133   HLchlt 29837   LHypclh 30470   DVecHcdvh 31565  LCDualclcd 32073  HDMapchdma 32280  HGMapchg 32373
This theorem is referenced by:  hdmapglem7  32419  hlhilsrnglem  32443
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-ot 3788  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-tpos 6442  df-undef 6506  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-n0 10182  df-z 10243  df-uz 10449  df-fz 11004  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-sca 13504  df-vsca 13505  df-0g 13686  df-mre 13770  df-mrc 13771  df-acs 13773  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-mnd 14649  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-subg 14900  df-cntz 15075  df-oppg 15101  df-lsm 15229  df-cmn 15373  df-abl 15374  df-mgp 15608  df-rng 15622  df-ur 15624  df-oppr 15687  df-dvdsr 15705  df-unit 15706  df-invr 15736  df-dvr 15747  df-drng 15796  df-lmod 15911  df-lss 15968  df-lsp 16007  df-lvec 16134  df-lsatoms 29463  df-lshyp 29464  df-lcv 29506  df-lfl 29545  df-lkr 29573  df-ldual 29611  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986  df-lines 29987  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591  df-trl 30645  df-tgrp 31229  df-tendo 31241  df-edring 31243  df-dveca 31489  df-disoa 31516  df-dvech 31566  df-dib 31626  df-dic 31660  df-dih 31716  df-doch 31835  df-djh 31882  df-lcdual 32074  df-mapd 32112  df-hvmap 32244  df-hdmap1 32281  df-hdmap 32282  df-hgmap 32374
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