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Theorem dvhvaddass 34640
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 5379 . . . 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 34635 . . . . . . . 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 1167 . . . . . . 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 5891 . . . . . 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 5897 . . . . . . . 8  |-  ( 1st `  F )  e.  _V
13 fvex 5897 . . . . . . . 8  |-  ( 1st `  G )  e.  _V
1412, 13coex 6765 . . . . . . 7  |-  ( ( 1st `  F )  o.  ( 1st `  G
) )  e.  _V
15 ovex 6339 . . . . . . 7  |-  ( ( 2nd `  F ) 
.+^  ( 2nd `  G
) )  e.  _V
1614, 15op1st 6821 . . . . . 6  |-  ( 1st `  <. ( ( 1st `  F )  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F )  .+^  ( 2nd `  G ) ) >.
)  =  ( ( 1st `  F )  o.  ( 1st `  G
) )
1711, 16syl6eq 2480 . . . . 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 5021 . . . 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 34635 . . . . . . . 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 1165 . . . . . . 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 5891 . . . . . 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 5897 . . . . . . . 8  |-  ( 1st `  I )  e.  _V
2313, 22coex 6765 . . . . . . 7  |-  ( ( 1st `  G )  o.  ( 1st `  I
) )  e.  _V
24 ovex 6339 . . . . . . 7  |-  ( ( 2nd `  G ) 
.+^  ( 2nd `  I
) )  e.  _V
2523, 24op1st 6821 . . . . . 6  |-  ( 1st `  <. ( ( 1st `  G )  o.  ( 1st `  I ) ) ,  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) >.
)  =  ( ( 1st `  G )  o.  ( 1st `  I
) )
2621, 25syl6eq 2480 . . . . 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 5022 . . . 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 2490 . . 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 6844 . . . . . 6  |-  ( F  e.  ( T  X.  E )  ->  ( 2nd `  F )  e.  E )
30 xp2nd 6844 . . . . . 6  |-  ( G  e.  ( T  X.  E )  ->  ( 2nd `  G )  e.  E )
31 xp2nd 6844 . . . . . 6  |-  ( I  e.  ( T  X.  E )  ->  ( 2nd `  I )  e.  E )
3229, 30, 313anim123i 1191 . . . . 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 2423 . . . . . . . . . 10  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
342, 33, 5, 6dvhsca 34625 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
352, 33erngdv 34535 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
3634, 35eqeltrd 2512 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
37 drnggrp 17988 . . . . . . . 8  |-  ( D  e.  DivRing  ->  D  e.  Grp )
3836, 37syl 17 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
3938adantr 467 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  D  e.  Grp )
40 simpr1 1012 . . . . . . 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 2423 . . . . . . . . 9  |-  ( Base `  D )  =  (
Base `  D )
422, 4, 5, 6, 41dvhbase 34626 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
4342adantr 467 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  ( Base `  D
)  =  E )
4440, 43eleqtrrd 2515 . . . . . 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 1013 . . . . . . 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 2515 . . . . . 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 1014 . . . . . . 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 2515 . . . . . 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 16685 . . . . . 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 1267 . . . . 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 477 . . . 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 5891 . . . . . 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 6822 . . . . . 6  |-  ( 2nd `  <. ( ( 1st `  F )  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F )  .+^  ( 2nd `  G ) ) >.
)  =  ( ( 2nd `  F ) 
.+^  ( 2nd `  G
) )
5452, 53syl6eq 2480 . . . . 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 6326 . . . 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 5891 . . . . . 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 6822 . . . . . 6  |-  ( 2nd `  <. ( ( 1st `  G )  o.  ( 1st `  I ) ) ,  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) >.
)  =  ( ( 2nd `  G ) 
.+^  ( 2nd `  I
) )
5856, 57syl6eq 2480 . . . . 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 6327 . . . 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 2474 . . 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 4201 . 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 459 . . 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 34638 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( F  .+  G
)  e.  ( T  X.  E ) )
64633adantr3 1167 . . 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 1014 . . 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 34635 . . 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 1263 . 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 1012 . . 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 34638 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( G  .+  I
)  e.  ( T  X.  E ) )
70693adantr1 1165 . . 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 34635 . . 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 1263 . 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 2474 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 371    /\ w3a 983    = wceq 1438    e. wcel 1873   <.cop 4010    X. cxp 4857    o. ccom 4863   ` cfv 5607  (class class class)co 6311   1stc1st 6811   2ndc2nd 6812   Basecbs 15126   +g cplusg 15195  Scalarcsca 15198   Grpcgrp 16674   DivRingcdr 17980   HLchlt 32891   LHypclh 33524   LTrncltrn 33641   TEndoctendo 34294   EDRingcedring 34295   DVecHcdvh 34621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-8 1875  ax-9 1877  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402  ax-rep 4542  ax-sep 4552  ax-nul 4561  ax-pow 4608  ax-pr 4666  ax-un 6603  ax-cnex 9608  ax-resscn 9609  ax-1cn 9610  ax-icn 9611  ax-addcl 9612  ax-addrcl 9613  ax-mulcl 9614  ax-mulrcl 9615  ax-mulcom 9616  ax-addass 9617  ax-mulass 9618  ax-distr 9619  ax-i2m1 9620  ax-1ne0 9621  ax-1rid 9622  ax-rnegex 9623  ax-rrecex 9624  ax-cnre 9625  ax-pre-lttri 9626  ax-pre-lttrn 9627  ax-pre-ltadd 9628  ax-pre-mulgt0 9629  ax-riotaBAD 32500
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1659  df-nf 1663  df-sb 1792  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3087  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 3918  df-pw 3989  df-sn 4005  df-pr 4007  df-tp 4009  df-op 4011  df-uni 4226  df-int 4262  df-iun 4307  df-iin 4308  df-br 4430  df-opab 4489  df-mpt 4490  df-tr 4525  df-eprel 4770  df-id 4774  df-po 4780  df-so 4781  df-fr 4818  df-we 4820  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-pred 5405  df-ord 5451  df-on 5452  df-lim 5453  df-suc 5454  df-iota 5571  df-fun 5609  df-fn 5610  df-f 5611  df-f1 5612  df-fo 5613  df-f1o 5614  df-fv 5615  df-riota 6273  df-ov 6314  df-oprab 6315  df-mpt2 6316  df-om 6713  df-1st 6813  df-2nd 6814  df-tpos 6990  df-undef 7037  df-wrecs 7045  df-recs 7107  df-rdg 7145  df-1o 7199  df-oadd 7203  df-er 7380  df-map 7491  df-en 7587  df-dom 7588  df-sdom 7589  df-fin 7590  df-pnf 9690  df-mnf 9691  df-xr 9692  df-ltxr 9693  df-le 9694  df-sub 9875  df-neg 9876  df-nn 10623  df-2 10681  df-3 10682  df-4 10683  df-5 10684  df-6 10685  df-n0 10883  df-z 10951  df-uz 11173  df-fz 11798  df-struct 15128  df-ndx 15129  df-slot 15130  df-base 15131  df-sets 15132  df-ress 15133  df-plusg 15208  df-mulr 15209  df-sca 15211  df-vsca 15212  df-0g 15345  df-preset 16178  df-poset 16196  df-plt 16209  df-lub 16225  df-glb 16226  df-join 16227  df-meet 16228  df-p0 16290  df-p1 16291  df-lat 16297  df-clat 16359  df-mgm 16493  df-sgrp 16532  df-mnd 16542  df-grp 16678  df-minusg 16679  df-mgp 17729  df-ur 17741  df-ring 17787  df-oppr 17856  df-dvdsr 17874  df-unit 17875  df-invr 17905  df-dvr 17916  df-drng 17982  df-oposet 32717  df-ol 32719  df-oml 32720  df-covers 32807  df-ats 32808  df-atl 32839  df-cvlat 32863  df-hlat 32892  df-llines 33038  df-lplanes 33039  df-lvols 33040  df-lines 33041  df-psubsp 33043  df-pmap 33044  df-padd 33336  df-lhyp 33528  df-laut 33529  df-ldil 33644  df-ltrn 33645  df-trl 33700  df-tendo 34297  df-edring 34299  df-dvech 34622
This theorem is referenced by:  dvhgrp  34650
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