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Theorem hgmapadd 32632
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 2435 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
4 eqid 2435 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
5 hgmapadd.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 2, 3, 4, 5dvh1dim 32177 . . 3  |-  ( ph  ->  E. t  e.  (
Base `  U )
t  =/=  ( 0g
`  U ) )
7 eqid 2435 . . . . . . . . 9  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
81, 7, 5lcdlmod 32327 . . . . . . . 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 2435 . . . . . . . 8  |-  (Scalar `  ( (LCDual `  K ) `  W ) )  =  (Scalar `  ( (LCDual `  K ) `  W
) )
13 eqid 2435 . . . . . . . 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 32628 . . . . . . 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 32628 . . . . . . . 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 2435 . . . . . . . 8  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
23 eqid 2435 . . . . . . . 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 32568 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  t )  e.  ( Base `  (
(LCDual `  K ) `  W ) ) )
26 eqid 2435 . . . . . . . 8  |-  ( +g  `  ( (LCDual `  K
) `  W )
)  =  ( +g  `  ( (LCDual `  K
) `  W )
)
27 eqid 2435 . . . . . . . 8  |-  ( .s
`  ( (LCDual `  K ) `  W
) )  =  ( .s `  ( (LCDual `  K ) `  W
) )
28 eqid 2435 . . . . . . . 8  |-  ( +g  `  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) )
2922, 26, 12, 27, 13, 28lmodvsdir 15966 . . . . . . 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 31845 . . . . . . . . . 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 2435 . . . . . . . . . 10  |-  ( +g  `  U )  =  ( +g  `  U )
35 eqid 2435 . . . . . . . . . 10  |-  ( .s
`  U )  =  ( .s `  U
)
36 hgmapadd.p . . . . . . . . . 10  |-  .+  =  ( +g  `  R )
373, 34, 10, 35, 11, 36lmodvsdir 15966 . . . . . . . . 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 5724 . . . . . . 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 15959 . . . . . . . . 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 15959 . . . . . . . . 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 32581 . . . . . . 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 32629 . . . . . . . 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 32629 . . . . . . . 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 6091 . . . . . . 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 2472 . . . . . 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 15953 . . . . . . . . 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 32629 . . . . . 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 2472 . . . . 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 2435 . . . . . 6  |-  ( 0g
`  ( (LCDual `  K ) `  W
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) )
551, 7, 5lcdlvec 32326 . . . . . . 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 32628 . . . . . . 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 32628 . . . . . . . 8  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
6012, 13, 28lmodacl 15953 . . . . . . . 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 32582 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  t
)  =  ( 0g
`  ( (LCDual `  K ) `  W
) )  <->  t  =  ( 0g `  U ) ) )
6564necon3bid 2633 . . . . . . 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 16176 . . . . 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 2824 . . 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 32336 . . 3  |-  ( ph  ->  ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) )  =  .+  )
7271oveqd 6090 . 2  |-  ( ph  ->  ( ( G `  X ) ( +g  `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( G `  Y ) )  =  ( ( G `  X ) 
.+  ( G `  Y ) ) )
7370, 72eqtrd 2467 1  |-  ( ph  ->  ( G `  ( X  .+  Y ) )  =  ( ( G `
 X )  .+  ( G `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521  Scalarcsca 13524   .scvsca 13525   0gc0g 13715   LModclmod 15942   LVecclvec 16166   HLchlt 30085   LHypclh 30718   DVecHcdvh 31813  LCDualclcd 32321  HDMapchdma 32528  HGMapchg 32621
This theorem is referenced by:  hdmapglem7  32667  hlhilsrnglem  32691
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-mre 13803  df-mrc 13804  df-acs 13806  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-oppg 15134  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lsatoms 29711  df-lshyp 29712  df-lcv 29754  df-lfl 29793  df-lkr 29821  df-ldual 29859  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-tgrp 31477  df-tendo 31489  df-edring 31491  df-dveca 31737  df-disoa 31764  df-dvech 31814  df-dib 31874  df-dic 31908  df-dih 31964  df-doch 32083  df-djh 32130  df-lcdual 32322  df-mapd 32360  df-hvmap 32492  df-hdmap1 32529  df-hdmap 32530  df-hgmap 32622
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