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Theorem dvhvaddass 34736
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 5361 . . . 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 34731 . . . . . . . 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 1191 . . . . . . 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 5883 . . . . . 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 5889 . . . . . . . 8  |-  ( 1st `  F )  e.  _V
13 fvex 5889 . . . . . . . 8  |-  ( 1st `  G )  e.  _V
1412, 13coex 6764 . . . . . . 7  |-  ( ( 1st `  F )  o.  ( 1st `  G
) )  e.  _V
15 ovex 6336 . . . . . . 7  |-  ( ( 2nd `  F ) 
.+^  ( 2nd `  G
) )  e.  _V
1614, 15op1st 6820 . . . . . 6  |-  ( 1st `  <. ( ( 1st `  F )  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F )  .+^  ( 2nd `  G ) ) >.
)  =  ( ( 1st `  F )  o.  ( 1st `  G
) )
1711, 16syl6eq 2521 . . . . 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 5001 . . . 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 34731 . . . . . . . 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 1189 . . . . . . 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 5883 . . . . . 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 5889 . . . . . . . 8  |-  ( 1st `  I )  e.  _V
2313, 22coex 6764 . . . . . . 7  |-  ( ( 1st `  G )  o.  ( 1st `  I
) )  e.  _V
24 ovex 6336 . . . . . . 7  |-  ( ( 2nd `  G ) 
.+^  ( 2nd `  I
) )  e.  _V
2523, 24op1st 6820 . . . . . 6  |-  ( 1st `  <. ( ( 1st `  G )  o.  ( 1st `  I ) ) ,  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) >.
)  =  ( ( 1st `  G )  o.  ( 1st `  I
) )
2621, 25syl6eq 2521 . . . . 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 5002 . . . 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 2531 . . 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 6843 . . . . . 6  |-  ( F  e.  ( T  X.  E )  ->  ( 2nd `  F )  e.  E )
30 xp2nd 6843 . . . . . 6  |-  ( G  e.  ( T  X.  E )  ->  ( 2nd `  G )  e.  E )
31 xp2nd 6843 . . . . . 6  |-  ( I  e.  ( T  X.  E )  ->  ( 2nd `  I )  e.  E )
3229, 30, 313anim123i 1215 . . . . 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 2471 . . . . . . . . . 10  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
342, 33, 5, 6dvhsca 34721 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
352, 33erngdv 34631 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
3634, 35eqeltrd 2549 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
37 drnggrp 18061 . . . . . . . 8  |-  ( D  e.  DivRing  ->  D  e.  Grp )
3836, 37syl 17 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
3938adantr 472 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  D  e.  Grp )
40 simpr1 1036 . . . . . . 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 2471 . . . . . . . . 9  |-  ( Base `  D )  =  (
Base `  D )
422, 4, 5, 6, 41dvhbase 34722 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
4342adantr 472 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  ( Base `  D
)  =  E )
4440, 43eleqtrrd 2552 . . . . . 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 1037 . . . . . . 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 2552 . . . . . 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 1038 . . . . . . 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 2552 . . . . . 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 16758 . . . . . 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 1294 . . . . 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 482 . . . 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 5883 . . . . . 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 6821 . . . . . 6  |-  ( 2nd `  <. ( ( 1st `  F )  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F )  .+^  ( 2nd `  G ) ) >.
)  =  ( ( 2nd `  F ) 
.+^  ( 2nd `  G
) )
5452, 53syl6eq 2521 . . . . 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 6323 . . . 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 5883 . . . . . 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 6821 . . . . . 6  |-  ( 2nd `  <. ( ( 1st `  G )  o.  ( 1st `  I ) ) ,  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) >.
)  =  ( ( 2nd `  G ) 
.+^  ( 2nd `  I
) )
5856, 57syl6eq 2521 . . . . 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 6324 . . . 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 2515 . . 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 4166 . 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 464 . . 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 34734 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( F  .+  G
)  e.  ( T  X.  E ) )
64633adantr3 1191 . . 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 1038 . . 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 34731 . . 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 1290 . 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 1036 . . 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 34734 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( G  .+  I
)  e.  ( T  X.  E ) )
70693adantr1 1189 . . 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 34731 . . 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 1290 . 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 2515 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 setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   <.cop 3965    X. cxp 4837    o. ccom 4843   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811   Basecbs 15199   +g cplusg 15268  Scalarcsca 15271   Grpcgrp 16747   DivRingcdr 18053   HLchlt 32987   LHypclh 33620   LTrncltrn 33737   TEndoctendo 34390   EDRingcedring 34391   DVecHcdvh 34717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-riotaBAD 32589
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-tpos 6991  df-undef 7038  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-0g 15418  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-minusg 16752  df-mgp 17802  df-ur 17814  df-ring 17860  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-drng 18055  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-llines 33134  df-lplanes 33135  df-lvols 33136  df-lines 33137  df-psubsp 33139  df-pmap 33140  df-padd 33432  df-lhyp 33624  df-laut 33625  df-ldil 33740  df-ltrn 33741  df-trl 33796  df-tendo 34393  df-edring 34395  df-dvech 34718
This theorem is referenced by:  dvhgrp  34746
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