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Theorem dvhvaddass 31896
Description: Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
Hypotheses
Ref Expression
dvhvaddcl.h  |-  H  =  ( LHyp `  K
)
dvhvaddcl.t  |-  T  =  ( ( LTrn `  K
) `  W )
dvhvaddcl.e  |-  E  =  ( ( TEndo `  K
) `  W )
dvhvaddcl.u  |-  U  =  ( ( DVecH `  K
) `  W )
dvhvaddcl.d  |-  D  =  (Scalar `  U )
dvhvaddcl.p  |-  .+^  =  ( +g  `  D )
dvhvaddcl.a  |-  .+  =  ( +g  `  U )
Assertion
Ref Expression
dvhvaddass  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( ( F  .+  G )  .+  I
)  =  ( F 
.+  ( G  .+  I ) ) )

Proof of Theorem dvhvaddass
StepHypRef Expression
1 coass 5389 . . . 4  |-  ( ( ( 1st `  F
)  o.  ( 1st `  G ) )  o.  ( 1st `  I
) )  =  ( ( 1st `  F
)  o.  ( ( 1st `  G )  o.  ( 1st `  I
) ) )
2 dvhvaddcl.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
3 dvhvaddcl.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
4 dvhvaddcl.e . . . . . . . . 9  |-  E  =  ( ( TEndo `  K
) `  W )
5 dvhvaddcl.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
6 dvhvaddcl.d . . . . . . . . 9  |-  D  =  (Scalar `  U )
7 dvhvaddcl.a . . . . . . . . 9  |-  .+  =  ( +g  `  U )
8 dvhvaddcl.p . . . . . . . . 9  |-  .+^  =  ( +g  `  D )
92, 3, 4, 5, 6, 7, 8dvhvadd 31891 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( F  .+  G
)  =  <. (
( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >. )
1093adantr3 1119 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( F  .+  G
)  =  <. (
( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >. )
1110fveq2d 5733 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( 1st `  ( F  .+  G ) )  =  ( 1st `  <. ( ( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >. )
)
12 fvex 5743 . . . . . . . 8  |-  ( 1st `  F )  e.  _V
13 fvex 5743 . . . . . . . 8  |-  ( 1st `  G )  e.  _V
1412, 13coex 5414 . . . . . . 7  |-  ( ( 1st `  F )  o.  ( 1st `  G
) )  e.  _V
15 ovex 6107 . . . . . . 7  |-  ( ( 2nd `  F ) 
.+^  ( 2nd `  G
) )  e.  _V
1614, 15op1st 6356 . . . . . 6  |-  ( 1st `  <. ( ( 1st `  F )  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F )  .+^  ( 2nd `  G ) ) >.
)  =  ( ( 1st `  F )  o.  ( 1st `  G
) )
1711, 16syl6eq 2485 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( 1st `  ( F  .+  G ) )  =  ( ( 1st `  F )  o.  ( 1st `  G ) ) )
1817coeq1d 5035 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( ( 1st `  ( F  .+  G ) )  o.  ( 1st `  I
) )  =  ( ( ( 1st `  F
)  o.  ( 1st `  G ) )  o.  ( 1st `  I
) ) )
192, 3, 4, 5, 6, 7, 8dvhvadd 31891 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( G  .+  I
)  =  <. (
( 1st `  G
)  o.  ( 1st `  I ) ) ,  ( ( 2nd `  G
)  .+^  ( 2nd `  I
) ) >. )
20193adantr1 1117 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( G  .+  I
)  =  <. (
( 1st `  G
)  o.  ( 1st `  I ) ) ,  ( ( 2nd `  G
)  .+^  ( 2nd `  I
) ) >. )
2120fveq2d 5733 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( 1st `  ( G  .+  I ) )  =  ( 1st `  <. ( ( 1st `  G
)  o.  ( 1st `  I ) ) ,  ( ( 2nd `  G
)  .+^  ( 2nd `  I
) ) >. )
)
22 fvex 5743 . . . . . . . 8  |-  ( 1st `  I )  e.  _V
2313, 22coex 5414 . . . . . . 7  |-  ( ( 1st `  G )  o.  ( 1st `  I
) )  e.  _V
24 ovex 6107 . . . . . . 7  |-  ( ( 2nd `  G ) 
.+^  ( 2nd `  I
) )  e.  _V
2523, 24op1st 6356 . . . . . 6  |-  ( 1st `  <. ( ( 1st `  G )  o.  ( 1st `  I ) ) ,  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) >.
)  =  ( ( 1st `  G )  o.  ( 1st `  I
) )
2621, 25syl6eq 2485 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( 1st `  ( G  .+  I ) )  =  ( ( 1st `  G )  o.  ( 1st `  I ) ) )
2726coeq2d 5036 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( ( 1st `  F
)  o.  ( 1st `  ( G  .+  I
) ) )  =  ( ( 1st `  F
)  o.  ( ( 1st `  G )  o.  ( 1st `  I
) ) ) )
281, 18, 273eqtr4a 2495 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( ( 1st `  ( F  .+  G ) )  o.  ( 1st `  I
) )  =  ( ( 1st `  F
)  o.  ( 1st `  ( G  .+  I
) ) ) )
29 xp2nd 6378 . . . . . 6  |-  ( F  e.  ( T  X.  E )  ->  ( 2nd `  F )  e.  E )
30 xp2nd 6378 . . . . . 6  |-  ( G  e.  ( T  X.  E )  ->  ( 2nd `  G )  e.  E )
31 xp2nd 6378 . . . . . 6  |-  ( I  e.  ( T  X.  E )  ->  ( 2nd `  I )  e.  E )
3229, 30, 313anim123i 1140 . . . . 5  |-  ( ( F  e.  ( T  X.  E )  /\  G  e.  ( T  X.  E )  /\  I  e.  ( T  X.  E
) )  ->  (
( 2nd `  F
)  e.  E  /\  ( 2nd `  G )  e.  E  /\  ( 2nd `  I )  e.  E ) )
33 eqid 2437 . . . . . . . . . 10  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
342, 33, 5, 6dvhsca 31881 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
352, 33erngdv 31791 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
3634, 35eqeltrd 2511 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
37 drnggrp 15844 . . . . . . . 8  |-  ( D  e.  DivRing  ->  D  e.  Grp )
3836, 37syl 16 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
3938adantr 453 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  D  e.  Grp )
40 simpr1 964 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  ( 2nd `  F
)  e.  E )
41 eqid 2437 . . . . . . . . 9  |-  ( Base `  D )  =  (
Base `  D )
422, 4, 5, 6, 41dvhbase 31882 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
4342adantr 453 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  ( Base `  D
)  =  E )
4440, 43eleqtrrd 2514 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  ( 2nd `  F
)  e.  ( Base `  D ) )
45 simpr2 965 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  ( 2nd `  G
)  e.  E )
4645, 43eleqtrrd 2514 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  ( 2nd `  G
)  e.  ( Base `  D ) )
47 simpr3 966 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  ( 2nd `  I
)  e.  E )
4847, 43eleqtrrd 2514 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  ( 2nd `  I
)  e.  ( Base `  D ) )
4941, 8grpass 14820 . . . . . 6  |-  ( ( D  e.  Grp  /\  ( ( 2nd `  F
)  e.  ( Base `  D )  /\  ( 2nd `  G )  e.  ( Base `  D
)  /\  ( 2nd `  I )  e.  (
Base `  D )
) )  ->  (
( ( 2nd `  F
)  .+^  ( 2nd `  G
) )  .+^  ( 2nd `  I ) )  =  ( ( 2nd `  F
)  .+^  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) ) )
5039, 44, 46, 48, 49syl13anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  ( ( ( 2nd `  F ) 
.+^  ( 2nd `  G
) )  .+^  ( 2nd `  I ) )  =  ( ( 2nd `  F
)  .+^  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) ) )
5132, 50sylan2 462 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( ( ( 2nd `  F )  .+^  ( 2nd `  G ) )  .+^  ( 2nd `  I ) )  =  ( ( 2nd `  F ) 
.+^  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) ) )
5210fveq2d 5733 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( 2nd `  ( F  .+  G ) )  =  ( 2nd `  <. ( ( 1st `  F
)  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  G
) ) >. )
)
5314, 15op2nd 6357 . . . . . 6  |-  ( 2nd `  <. ( ( 1st `  F )  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F )  .+^  ( 2nd `  G ) ) >.
)  =  ( ( 2nd `  F ) 
.+^  ( 2nd `  G
) )
5452, 53syl6eq 2485 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( 2nd `  ( F  .+  G ) )  =  ( ( 2nd `  F )  .+^  ( 2nd `  G ) ) )
5554oveq1d 6097 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( ( 2nd `  ( F  .+  G ) ) 
.+^  ( 2nd `  I
) )  =  ( ( ( 2nd `  F
)  .+^  ( 2nd `  G
) )  .+^  ( 2nd `  I ) ) )
5620fveq2d 5733 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( 2nd `  ( G  .+  I ) )  =  ( 2nd `  <. ( ( 1st `  G
)  o.  ( 1st `  I ) ) ,  ( ( 2nd `  G
)  .+^  ( 2nd `  I
) ) >. )
)
5723, 24op2nd 6357 . . . . . 6  |-  ( 2nd `  <. ( ( 1st `  G )  o.  ( 1st `  I ) ) ,  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) >.
)  =  ( ( 2nd `  G ) 
.+^  ( 2nd `  I
) )
5856, 57syl6eq 2485 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( 2nd `  ( G  .+  I ) )  =  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) )
5958oveq2d 6098 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( ( 2nd `  F
)  .+^  ( 2nd `  ( G  .+  I ) ) )  =  ( ( 2nd `  F ) 
.+^  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) ) )
6051, 55, 593eqtr4d 2479 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( ( 2nd `  ( F  .+  G ) ) 
.+^  ( 2nd `  I
) )  =  ( ( 2nd `  F
)  .+^  ( 2nd `  ( G  .+  I ) ) ) )
6128, 60opeq12d 3993 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  ->  <. ( ( 1st `  ( F  .+  G ) )  o.  ( 1st `  I
) ) ,  ( ( 2nd `  ( F  .+  G ) ) 
.+^  ( 2nd `  I
) ) >.  =  <. ( ( 1st `  F
)  o.  ( 1st `  ( G  .+  I
) ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  ( G  .+  I ) ) ) >. )
62 simpl 445 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
632, 3, 4, 5, 6, 8, 7dvhvaddcl 31894 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( F  .+  G
)  e.  ( T  X.  E ) )
64633adantr3 1119 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( F  .+  G
)  e.  ( T  X.  E ) )
65 simpr3 966 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  ->  I  e.  ( T  X.  E ) )
662, 3, 4, 5, 6, 7, 8dvhvadd 31891 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F 
.+  G )  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( ( F  .+  G )  .+  I
)  =  <. (
( 1st `  ( F  .+  G ) )  o.  ( 1st `  I
) ) ,  ( ( 2nd `  ( F  .+  G ) ) 
.+^  ( 2nd `  I
) ) >. )
6762, 64, 65, 66syl12anc 1183 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( ( F  .+  G )  .+  I
)  =  <. (
( 1st `  ( F  .+  G ) )  o.  ( 1st `  I
) ) ,  ( ( 2nd `  ( F  .+  G ) ) 
.+^  ( 2nd `  I
) ) >. )
68 simpr1 964 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  ->  F  e.  ( T  X.  E ) )
692, 3, 4, 5, 6, 8, 7dvhvaddcl 31894 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( G  .+  I
)  e.  ( T  X.  E ) )
70693adantr1 1117 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( G  .+  I
)  e.  ( T  X.  E ) )
712, 3, 4, 5, 6, 7, 8dvhvadd 31891 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  ( G  .+  I )  e.  ( T  X.  E ) ) )  ->  ( F  .+  ( G  .+  I ) )  = 
<. ( ( 1st `  F
)  o.  ( 1st `  ( G  .+  I
) ) ) ,  ( ( 2nd `  F
)  .+^  ( 2nd `  ( G  .+  I ) ) ) >. )
7262, 68, 70, 71syl12anc 1183 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( F  .+  ( G  .+  I ) )  =  <. ( ( 1st `  F )  o.  ( 1st `  ( G  .+  I ) ) ) ,  ( ( 2nd `  F )  .+^  ( 2nd `  ( G  .+  I
) ) ) >.
)
7361, 67, 723eqtr4d 2479 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( ( F  .+  G )  .+  I
)  =  ( F 
.+  ( G  .+  I ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   <.cop 3818    X. cxp 4877    o. ccom 4883   ` cfv 5455  (class class class)co 6082   1stc1st 6348   2ndc2nd 6349   Basecbs 13470   +g cplusg 13530  Scalarcsca 13533   Grpcgrp 14686   DivRingcdr 15836   HLchlt 30149   LHypclh 30782   LTrncltrn 30899   TEndoctendo 31550   EDRingcedring 31551   DVecHcdvh 31877
This theorem is referenced by:  dvhgrp  31906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-tpos 6480  df-undef 6544  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-n0 10223  df-z 10284  df-uz 10490  df-fz 11045  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-sca 13546  df-vsca 13547  df-0g 13728  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-p1 14470  df-lat 14476  df-clat 14538  df-mnd 14691  df-grp 14813  df-minusg 14814  df-mgp 15650  df-rng 15664  df-ur 15666  df-oppr 15729  df-dvdsr 15747  df-unit 15748  df-invr 15778  df-dvr 15789  df-drng 15838  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-llines 30296  df-lplanes 30297  df-lvols 30298  df-lines 30299  df-psubsp 30301  df-pmap 30302  df-padd 30594  df-lhyp 30786  df-laut 30787  df-ldil 30902  df-ltrn 30903  df-trl 30957  df-tendo 31553  df-edring 31555  df-dvech 31878
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