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Theorem hgmapadd 35174
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 2429 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
4 eqid 2429 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
5 hgmapadd.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 2, 3, 4, 5dvh1dim 34719 . . 3  |-  ( ph  ->  E. t  e.  (
Base `  U )
t  =/=  ( 0g
`  U ) )
7 eqid 2429 . . . . . . . . 9  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
81, 7, 5lcdlmod 34869 . . . . . . . 8  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LMod )
983ad2ant1 1026 . . . . . . 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 2429 . . . . . . . 8  |-  (Scalar `  ( (LCDual `  K ) `  W ) )  =  (Scalar `  ( (LCDual `  K ) `  W
) )
13 eqid 2429 . . . . . . . 8  |-  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) )
14 hgmapadd.g . . . . . . . 8  |-  G  =  ( (HGMap `  K
) `  W )
1553ad2ant1 1026 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
16 hgmapadd.x . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
17163ad2ant1 1026 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  X  e.  B
)
181, 2, 10, 11, 7, 12, 13, 14, 15, 17hgmapdcl 35170 . . . . . . 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 35170 . . . . . . . 8  |-  ( ph  ->  ( G `  Y
)  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
21203ad2ant1 1026 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( G `  Y )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
22 eqid 2429 . . . . . . . 8  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
23 eqid 2429 . . . . . . . 8  |-  ( (HDMap `  K ) `  W
)  =  ( (HDMap `  K ) `  W
)
24 simp2 1006 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  t  e.  (
Base `  U )
)
251, 2, 3, 7, 22, 23, 15, 24hdmapcl 35110 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  t )  e.  ( Base `  (
(LCDual `  K ) `  W ) ) )
26 eqid 2429 . . . . . . . 8  |-  ( +g  `  ( (LCDual `  K
) `  W )
)  =  ( +g  `  ( (LCDual `  K
) `  W )
)
27 eqid 2429 . . . . . . . 8  |-  ( .s
`  ( (LCDual `  K ) `  W
) )  =  ( .s `  ( (LCDual `  K ) `  W
) )
28 eqid 2429 . . . . . . . 8  |-  ( +g  `  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) )
2922, 26, 12, 27, 13, 28lmodvsdir 18050 . . . . . . 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 1266 . . . . . 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 34387 . . . . . . . . . 10  |-  ( ph  ->  U  e.  LMod )
32313ad2ant1 1026 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  U  e.  LMod )
33193ad2ant1 1026 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  Y  e.  B
)
34 eqid 2429 . . . . . . . . . 10  |-  ( +g  `  U )  =  ( +g  `  U )
35 eqid 2429 . . . . . . . . . 10  |-  ( .s
`  U )  =  ( .s `  U
)
36 hgmapadd.p . . . . . . . . . 10  |-  .+  =  ( +g  `  R )
373, 34, 10, 35, 11, 36lmodvsdir 18050 . . . . . . . . 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 1266 . . . . . . . 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 5885 . . . . . . 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 18043 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  X  e.  B  /\  t  e.  ( Base `  U
) )  ->  ( X ( .s `  U ) t )  e.  ( Base `  U
) )
4132, 17, 24, 40syl3anc 1264 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( X ( .s `  U ) t )  e.  (
Base `  U )
)
423, 10, 35, 11lmodvscl 18043 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  Y  e.  B  /\  t  e.  ( Base `  U
) )  ->  ( Y ( .s `  U ) t )  e.  ( Base `  U
) )
4332, 33, 24, 42syl3anc 1264 . . . . . . . 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 35123 . . . . . . 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 35171 . . . . . . . 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 35171 . . . . . . . 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 6323 . . . . . . 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 2475 . . . . . 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 18037 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
5031, 16, 19, 49syl3anc 1264 . . . . . . . 8  |-  ( ph  ->  ( X  .+  Y
)  e.  B )
51503ad2ant1 1026 . . . . . . 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 35171 . . . . . 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 2475 . . . . 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 2429 . . . . . 6  |-  ( 0g
`  ( (LCDual `  K ) `  W
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) )
551, 7, 5lcdlvec 34868 . . . . . . 7  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LVec )
56553ad2ant1 1026 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( (LCDual `  K ) `  W
)  e.  LVec )
571, 2, 10, 11, 7, 12, 13, 14, 5, 50hgmapdcl 35170 . . . . . . 7  |-  ( ph  ->  ( G `  ( X  .+  Y ) )  e.  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) ) )
58573ad2ant1 1026 . . . . . 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 35170 . . . . . . . 8  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
6012, 13, 28lmodacl 18037 . . . . . . . 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 1264 . . . . . . 7  |-  ( ph  ->  ( ( G `  X ) ( +g  `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( G `  Y ) )  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
62613ad2ant1 1026 . . . . . 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 1007 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  t  =/=  ( 0g `  U ) )
641, 2, 3, 4, 7, 54, 23, 15, 24hdmapeq0 35124 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  t
)  =  ( 0g
`  ( (LCDual `  K ) `  W
) )  <->  t  =  ( 0g `  U ) ) )
6564necon3bid 2689 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  t
)  =/=  ( 0g
`  ( (LCDual `  K ) `  W
) )  <->  t  =/=  ( 0g `  U ) ) )
6663, 65mpbird 235 . . . . . 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 18270 . . . . 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 213 . . . 4  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( G `  ( X  .+  Y ) )  =  ( ( G `  X ) ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) ) ( G `
 Y ) ) )
6968rexlimdv3a 2926 . . 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 34878 . . 3  |-  ( ph  ->  ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) )  =  .+  )
7271oveqd 6322 . 2  |-  ( ph  ->  ( ( G `  X ) ( +g  `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( G `  Y ) )  =  ( ( G `  X ) 
.+  ( G `  Y ) ) )
7370, 72eqtrd 2470 1  |-  ( ph  ->  ( G `  ( X  .+  Y ) )  =  ( ( G `
 X )  .+  ( G `  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152  Scalarcsca 15155   .scvsca 15156   0gc0g 15297   LModclmod 18026   LVecclvec 18260   HLchlt 32625   LHypclh 33258   DVecHcdvh 34355  LCDualclcd 34863  HDMapchdma 35070  HGMapchg 35163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-riotaBAD 32234
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-ot 4011  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  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-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-undef 7028  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-0g 15299  df-mre 15443  df-mrc 15444  df-acs 15446  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-subg 16765  df-cntz 16922  df-oppg 16948  df-lsm 17223  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-drng 17912  df-lmod 18028  df-lss 18091  df-lsp 18130  df-lvec 18261  df-lsatoms 32251  df-lshyp 32252  df-lcv 32294  df-lfl 32333  df-lkr 32361  df-ldual 32399  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-llines 32772  df-lplanes 32773  df-lvols 32774  df-lines 32775  df-psubsp 32777  df-pmap 32778  df-padd 33070  df-lhyp 33262  df-laut 33263  df-ldil 33378  df-ltrn 33379  df-trl 33434  df-tgrp 34019  df-tendo 34031  df-edring 34033  df-dveca 34279  df-disoa 34306  df-dvech 34356  df-dib 34416  df-dic 34450  df-dih 34506  df-doch 34625  df-djh 34672  df-lcdual 34864  df-mapd 34902  df-hvmap 35034  df-hdmap1 35071  df-hdmap 35072  df-hgmap 35164
This theorem is referenced by:  hdmapglem7  35209  hlhilsrnglem  35233
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