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Theorem hgmapadd 32380
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 2404 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
4 eqid 2404 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
5 hgmapadd.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 2, 3, 4, 5dvh1dim 31925 . . 3  |-  ( ph  ->  E. t  e.  (
Base `  U )
t  =/=  ( 0g
`  U ) )
7 eqid 2404 . . . . . . . . 9  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
81, 7, 5lcdlmod 32075 . . . . . . . 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 2404 . . . . . . . 8  |-  (Scalar `  ( (LCDual `  K ) `  W ) )  =  (Scalar `  ( (LCDual `  K ) `  W
) )
13 eqid 2404 . . . . . . . 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 32376 . . . . . . 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 32376 . . . . . . . 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 2404 . . . . . . . 8  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
23 eqid 2404 . . . . . . . 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 32316 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  t )  e.  ( Base `  (
(LCDual `  K ) `  W ) ) )
26 eqid 2404 . . . . . . . 8  |-  ( +g  `  ( (LCDual `  K
) `  W )
)  =  ( +g  `  ( (LCDual `  K
) `  W )
)
27 eqid 2404 . . . . . . . 8  |-  ( .s
`  ( (LCDual `  K ) `  W
) )  =  ( .s `  ( (LCDual `  K ) `  W
) )
28 eqid 2404 . . . . . . . 8  |-  ( +g  `  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) )
2922, 26, 12, 27, 13, 28lmodvsdir 15929 . . . . . . 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 31593 . . . . . . . . . 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 2404 . . . . . . . . . 10  |-  ( +g  `  U )  =  ( +g  `  U )
35 eqid 2404 . . . . . . . . . 10  |-  ( .s
`  U )  =  ( .s `  U
)
36 hgmapadd.p . . . . . . . . . 10  |-  .+  =  ( +g  `  R )
373, 34, 10, 35, 11, 36lmodvsdir 15929 . . . . . . . . 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 5691 . . . . . . 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 15922 . . . . . . . . 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 15922 . . . . . . . . 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 32329 . . . . . . 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 32377 . . . . . . . 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 32377 . . . . . . . 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 6058 . . . . . . 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 2441 . . . . . 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 15916 . . . . . . . . 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 32377 . . . . . 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 2441 . . . . 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 2404 . . . . . 6  |-  ( 0g
`  ( (LCDual `  K ) `  W
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) )
551, 7, 5lcdlvec 32074 . . . . . . 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 32376 . . . . . . 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 32376 . . . . . . . 8  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
6012, 13, 28lmodacl 15916 . . . . . . . 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 32330 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  t
)  =  ( 0g
`  ( (LCDual `  K ) `  W
) )  <->  t  =  ( 0g `  U ) ) )
6564necon3bid 2602 . . . . . . 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 16139 . . . . 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 2792 . . 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 32084 . . 3  |-  ( ph  ->  ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) )  =  .+  )
7271oveqd 6057 . 2  |-  ( ph  ->  ( ( G `  X ) ( +g  `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( G `  Y ) )  =  ( ( G `  X ) 
.+  ( G `  Y ) ) )
7370, 72eqtrd 2436 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 2567   E.wrex 2667   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484  Scalarcsca 13487   .scvsca 13488   0gc0g 13678   LModclmod 15905   LVecclvec 16129   HLchlt 29833   LHypclh 30466   DVecHcdvh 31561  LCDualclcd 32069  HDMapchdma 32276  HGMapchg 32369
This theorem is referenced by:  hdmapglem7  32415  hlhilsrnglem  32439
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-fal 1326  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-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  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-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-undef 6502  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  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-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-0g 13682  df-mre 13766  df-mrc 13767  df-acs 13769  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-oppg 15097  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lvec 16130  df-lsatoms 29459  df-lshyp 29460  df-lcv 29502  df-lfl 29541  df-lkr 29569  df-ldual 29607  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tgrp 31225  df-tendo 31237  df-edring 31239  df-dveca 31485  df-disoa 31512  df-dvech 31562  df-dib 31622  df-dic 31656  df-dih 31712  df-doch 31831  df-djh 31878  df-lcdual 32070  df-mapd 32108  df-hvmap 32240  df-hdmap1 32277  df-hdmap 32278  df-hgmap 32370
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