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Theorem dvhvaddass 31580
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 5347 . . . 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 31575 . . . . . . . 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 1118 . . . . . . 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 5691 . . . . . 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 5701 . . . . . . . 8  |-  ( 1st `  F )  e.  _V
13 fvex 5701 . . . . . . . 8  |-  ( 1st `  G )  e.  _V
1412, 13coex 5372 . . . . . . 7  |-  ( ( 1st `  F )  o.  ( 1st `  G
) )  e.  _V
15 ovex 6065 . . . . . . 7  |-  ( ( 2nd `  F ) 
.+^  ( 2nd `  G
) )  e.  _V
1614, 15op1st 6314 . . . . . 6  |-  ( 1st `  <. ( ( 1st `  F )  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F )  .+^  ( 2nd `  G ) ) >.
)  =  ( ( 1st `  F )  o.  ( 1st `  G
) )
1711, 16syl6eq 2452 . . . . 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 4993 . . . 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 31575 . . . . . . . 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 1116 . . . . . . 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 5691 . . . . . 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 5701 . . . . . . . 8  |-  ( 1st `  I )  e.  _V
2313, 22coex 5372 . . . . . . 7  |-  ( ( 1st `  G )  o.  ( 1st `  I
) )  e.  _V
24 ovex 6065 . . . . . . 7  |-  ( ( 2nd `  G ) 
.+^  ( 2nd `  I
) )  e.  _V
2523, 24op1st 6314 . . . . . 6  |-  ( 1st `  <. ( ( 1st `  G )  o.  ( 1st `  I ) ) ,  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) >.
)  =  ( ( 1st `  G )  o.  ( 1st `  I
) )
2621, 25syl6eq 2452 . . . . 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 4994 . . . 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 2462 . . 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 6336 . . . . . 6  |-  ( F  e.  ( T  X.  E )  ->  ( 2nd `  F )  e.  E )
30 xp2nd 6336 . . . . . 6  |-  ( G  e.  ( T  X.  E )  ->  ( 2nd `  G )  e.  E )
31 xp2nd 6336 . . . . . 6  |-  ( I  e.  ( T  X.  E )  ->  ( 2nd `  I )  e.  E )
3229, 30, 313anim123i 1139 . . . . 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 2404 . . . . . . . . . 10  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
342, 33, 5, 6dvhsca 31565 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
352, 33erngdv 31475 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
3634, 35eqeltrd 2478 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
37 drnggrp 15798 . . . . . . . 8  |-  ( D  e.  DivRing  ->  D  e.  Grp )
3836, 37syl 16 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
3938adantr 452 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  D  e.  Grp )
40 simpr1 963 . . . . . . 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 2404 . . . . . . . . 9  |-  ( Base `  D )  =  (
Base `  D )
422, 4, 5, 6, 41dvhbase 31566 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
4342adantr 452 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  ( Base `  D
)  =  E )
4440, 43eleqtrrd 2481 . . . . . 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 964 . . . . . . 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 2481 . . . . . 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 965 . . . . . . 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 2481 . . . . . 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 14774 . . . . . 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 1186 . . . . 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 461 . . . 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 5691 . . . . . 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 6315 . . . . . 6  |-  ( 2nd `  <. ( ( 1st `  F )  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F )  .+^  ( 2nd `  G ) ) >.
)  =  ( ( 2nd `  F ) 
.+^  ( 2nd `  G
) )
5452, 53syl6eq 2452 . . . . 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 6055 . . . 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 5691 . . . . . 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 6315 . . . . . 6  |-  ( 2nd `  <. ( ( 1st `  G )  o.  ( 1st `  I ) ) ,  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) >.
)  =  ( ( 2nd `  G ) 
.+^  ( 2nd `  I
) )
5856, 57syl6eq 2452 . . . . 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 6056 . . . 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 2446 . . 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 3952 . 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 444 . . 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 31578 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( F  .+  G
)  e.  ( T  X.  E ) )
64633adantr3 1118 . . 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 965 . . 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 31575 . . 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 1182 . 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 963 . . 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 31578 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( G  .+  I
)  e.  ( T  X.  E ) )
70693adantr1 1116 . . 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 31575 . . 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 1182 . 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 2446 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   <.cop 3777    X. cxp 4835    o. ccom 4841   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   Basecbs 13424   +g cplusg 13484  Scalarcsca 13487   Grpcgrp 14640   DivRingcdr 15790   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   TEndoctendo 31234   EDRingcedring 31235   DVecHcdvh 31561
This theorem is referenced by:  dvhgrp  31590
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-undef 6502  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-0g 13682  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-mnd 14645  df-grp 14767  df-minusg 14768  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tendo 31237  df-edring 31239  df-dvech 31562
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