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Theorem hgmapadd 36712
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 2467 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
4 eqid 2467 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
5 hgmapadd.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
61, 2, 3, 4, 5dvh1dim 36257 . . 3  |-  ( ph  ->  E. t  e.  (
Base `  U )
t  =/=  ( 0g
`  U ) )
7 eqid 2467 . . . . . . . . 9  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
81, 7, 5lcdlmod 36407 . . . . . . . 8  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LMod )
983ad2ant1 1017 . . . . . . 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 2467 . . . . . . . 8  |-  (Scalar `  ( (LCDual `  K ) `  W ) )  =  (Scalar `  ( (LCDual `  K ) `  W
) )
13 eqid 2467 . . . . . . . 8  |-  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) )
14 hgmapadd.g . . . . . . . 8  |-  G  =  ( (HGMap `  K
) `  W )
1553ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
16 hgmapadd.x . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
17163ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  X  e.  B
)
181, 2, 10, 11, 7, 12, 13, 14, 15, 17hgmapdcl 36708 . . . . . . 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 36708 . . . . . . . 8  |-  ( ph  ->  ( G `  Y
)  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
21203ad2ant1 1017 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( G `  Y )  e.  (
Base `  (Scalar `  (
(LCDual `  K ) `  W ) ) ) )
22 eqid 2467 . . . . . . . 8  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
23 eqid 2467 . . . . . . . 8  |-  ( (HDMap `  K ) `  W
)  =  ( (HDMap `  K ) `  W
)
24 simp2 997 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  t  e.  (
Base `  U )
)
251, 2, 3, 7, 22, 23, 15, 24hdmapcl 36648 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( (HDMap `  K ) `  W
) `  t )  e.  ( Base `  (
(LCDual `  K ) `  W ) ) )
26 eqid 2467 . . . . . . . 8  |-  ( +g  `  ( (LCDual `  K
) `  W )
)  =  ( +g  `  ( (LCDual `  K
) `  W )
)
27 eqid 2467 . . . . . . . 8  |-  ( .s
`  ( (LCDual `  K ) `  W
) )  =  ( .s `  ( (LCDual `  K ) `  W
) )
28 eqid 2467 . . . . . . . 8  |-  ( +g  `  (Scalar `  ( (LCDual `  K ) `  W
) ) )  =  ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) )
2922, 26, 12, 27, 13, 28lmodvsdir 17336 . . . . . . 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 1230 . . . . . 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 35925 . . . . . . . . . 10  |-  ( ph  ->  U  e.  LMod )
32313ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  U  e.  LMod )
33193ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  Y  e.  B
)
34 eqid 2467 . . . . . . . . . 10  |-  ( +g  `  U )  =  ( +g  `  U )
35 eqid 2467 . . . . . . . . . 10  |-  ( .s
`  U )  =  ( .s `  U
)
36 hgmapadd.p . . . . . . . . . 10  |-  .+  =  ( +g  `  R )
373, 34, 10, 35, 11, 36lmodvsdir 17336 . . . . . . . . 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 1230 . . . . . . . 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 5870 . . . . . . 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 17329 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  X  e.  B  /\  t  e.  ( Base `  U
) )  ->  ( X ( .s `  U ) t )  e.  ( Base `  U
) )
4132, 17, 24, 40syl3anc 1228 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( X ( .s `  U ) t )  e.  (
Base `  U )
)
423, 10, 35, 11lmodvscl 17329 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  Y  e.  B  /\  t  e.  ( Base `  U
) )  ->  ( Y ( .s `  U ) t )  e.  ( Base `  U
) )
4332, 33, 24, 42syl3anc 1228 . . . . . . . 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 36661 . . . . . . 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 36709 . . . . . . . 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 36709 . . . . . . . 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 6302 . . . . . . 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 2513 . . . . . 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 17323 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
5031, 16, 19, 49syl3anc 1228 . . . . . . . 8  |-  ( ph  ->  ( X  .+  Y
)  e.  B )
51503ad2ant1 1017 . . . . . . 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 36709 . . . . . 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 2513 . . . . 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 2467 . . . . . 6  |-  ( 0g
`  ( (LCDual `  K ) `  W
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) )
551, 7, 5lcdlvec 36406 . . . . . . 7  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LVec )
56553ad2ant1 1017 . . . . . 6  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( (LCDual `  K ) `  W
)  e.  LVec )
571, 2, 10, 11, 7, 12, 13, 14, 5, 50hgmapdcl 36708 . . . . . . 7  |-  ( ph  ->  ( G `  ( X  .+  Y ) )  e.  ( Base `  (Scalar `  ( (LCDual `  K
) `  W )
) ) )
58573ad2ant1 1017 . . . . . 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 36708 . . . . . . . 8  |-  ( ph  ->  ( G `  X
)  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
6012, 13, 28lmodacl 17323 . . . . . . . 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 1228 . . . . . . 7  |-  ( ph  ->  ( ( G `  X ) ( +g  `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( G `  Y ) )  e.  ( Base `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) )
62613ad2ant1 1017 . . . . . 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 998 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  t  =/=  ( 0g `  U ) )
641, 2, 3, 4, 7, 54, 23, 15, 24hdmapeq0 36662 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  t
)  =  ( 0g
`  ( (LCDual `  K ) `  W
) )  <->  t  =  ( 0g `  U ) ) )
6564necon3bid 2725 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( ( ( (HDMap `  K ) `  W ) `  t
)  =/=  ( 0g
`  ( (LCDual `  K ) `  W
) )  <->  t  =/=  ( 0g `  U ) ) )
6663, 65mpbird 232 . . . . . 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 17558 . . . . 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 210 . . . 4  |-  ( (
ph  /\  t  e.  ( Base `  U )  /\  t  =/=  ( 0g `  U ) )  ->  ( G `  ( X  .+  Y ) )  =  ( ( G `  X ) ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) ) ( G `
 Y ) ) )
6968rexlimdv3a 2957 . . 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 36416 . . 3  |-  ( ph  ->  ( +g  `  (Scalar `  ( (LCDual `  K
) `  W )
) )  =  .+  )
7271oveqd 6301 . 2  |-  ( ph  ->  ( ( G `  X ) ( +g  `  (Scalar `  ( (LCDual `  K ) `  W
) ) ) ( G `  Y ) )  =  ( ( G `  X ) 
.+  ( G `  Y ) ) )
7370, 72eqtrd 2508 1  |-  ( ph  ->  ( G `  ( X  .+  Y ) )  =  ( ( G `
 X )  .+  ( G `  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   ` cfv 5588  (class class class)co 6284   Basecbs 14490   +g cplusg 14555  Scalarcsca 14558   .scvsca 14559   0gc0g 14695   LModclmod 17312   LVecclvec 17548   HLchlt 34165   LHypclh 34798   DVecHcdvh 35893  LCDualclcd 36401  HDMapchdma 36608  HGMapchg 36701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-riotaBAD 33774
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-tpos 6955  df-undef 7002  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-sca 14571  df-vsca 14572  df-0g 14697  df-mre 14841  df-mrc 14842  df-acs 14844  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-p1 15527  df-lat 15533  df-clat 15595  df-mnd 15732  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-subg 16003  df-cntz 16160  df-oppg 16186  df-lsm 16462  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-dvr 17133  df-drng 17198  df-lmod 17314  df-lss 17379  df-lsp 17418  df-lvec 17549  df-lsatoms 33791  df-lshyp 33792  df-lcv 33834  df-lfl 33873  df-lkr 33901  df-ldual 33939  df-oposet 33991  df-ol 33993  df-oml 33994  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166  df-llines 34312  df-lplanes 34313  df-lvols 34314  df-lines 34315  df-psubsp 34317  df-pmap 34318  df-padd 34610  df-lhyp 34802  df-laut 34803  df-ldil 34918  df-ltrn 34919  df-trl 34973  df-tgrp 35557  df-tendo 35569  df-edring 35571  df-dveca 35817  df-disoa 35844  df-dvech 35894  df-dib 35954  df-dic 35988  df-dih 36044  df-doch 36163  df-djh 36210  df-lcdual 36402  df-mapd 36440  df-hvmap 36572  df-hdmap1 36609  df-hdmap 36610  df-hgmap 36702
This theorem is referenced by:  hdmapglem7  36747  hlhilsrnglem  36771
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