Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvhvaddass Unicode version

Theorem dvhvaddass 31105
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 5228 . . . 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 31100 . . . . . . . 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 1116 . . . . . . 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 5567 . . . . . 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 5577 . . . . . . . 8  |-  ( 1st `  F )  e.  _V
13 fvex 5577 . . . . . . . 8  |-  ( 1st `  G )  e.  _V
1412, 13coex 5253 . . . . . . 7  |-  ( ( 1st `  F )  o.  ( 1st `  G
) )  e.  _V
15 ovex 5925 . . . . . . 7  |-  ( ( 2nd `  F ) 
.+^  ( 2nd `  G
) )  e.  _V
1614, 15op1st 6170 . . . . . 6  |-  ( 1st `  <. ( ( 1st `  F )  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F )  .+^  ( 2nd `  G ) ) >.
)  =  ( ( 1st `  F )  o.  ( 1st `  G
) )
1711, 16syl6eq 2364 . . . . 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 4882 . . . 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 31100 . . . . . . . 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 1114 . . . . . . 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 5567 . . . . . 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 5577 . . . . . . . 8  |-  ( 1st `  I )  e.  _V
2313, 22coex 5253 . . . . . . 7  |-  ( ( 1st `  G )  o.  ( 1st `  I
) )  e.  _V
24 ovex 5925 . . . . . . 7  |-  ( ( 2nd `  G ) 
.+^  ( 2nd `  I
) )  e.  _V
2523, 24op1st 6170 . . . . . 6  |-  ( 1st `  <. ( ( 1st `  G )  o.  ( 1st `  I ) ) ,  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) >.
)  =  ( ( 1st `  G )  o.  ( 1st `  I
) )
2621, 25syl6eq 2364 . . . . 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 4883 . . . 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 2374 . . 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 6192 . . . . . 6  |-  ( F  e.  ( T  X.  E )  ->  ( 2nd `  F )  e.  E )
30 xp2nd 6192 . . . . . 6  |-  ( G  e.  ( T  X.  E )  ->  ( 2nd `  G )  e.  E )
31 xp2nd 6192 . . . . . 6  |-  ( I  e.  ( T  X.  E )  ->  ( 2nd `  I )  e.  E )
3229, 30, 313anim123i 1137 . . . . 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 2316 . . . . . . . . . 10  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
342, 33, 5, 6dvhsca 31090 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
352, 33erngdv 31000 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
3634, 35eqeltrd 2390 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
37 drnggrp 15569 . . . . . . . 8  |-  ( D  e.  DivRing  ->  D  e.  Grp )
3836, 37syl 15 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Grp )
3938adantr 451 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  D  e.  Grp )
40 simpr1 961 . . . . . . 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 2316 . . . . . . . . 9  |-  ( Base `  D )  =  (
Base `  D )
422, 4, 5, 6, 41dvhbase 31091 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
4342adantr 451 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( 2nd `  F )  e.  E  /\  ( 2nd `  G
)  e.  E  /\  ( 2nd `  I )  e.  E ) )  ->  ( Base `  D
)  =  E )
4440, 43eleqtrrd 2393 . . . . . 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 962 . . . . . . 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 2393 . . . . . 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 963 . . . . . . 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 2393 . . . . . 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 14545 . . . . . 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 1184 . . . . 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 460 . . . 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 5567 . . . . . 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 6171 . . . . . 6  |-  ( 2nd `  <. ( ( 1st `  F )  o.  ( 1st `  G ) ) ,  ( ( 2nd `  F )  .+^  ( 2nd `  G ) ) >.
)  =  ( ( 2nd `  F ) 
.+^  ( 2nd `  G
) )
5452, 53syl6eq 2364 . . . . 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 5915 . . . 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 5567 . . . . . 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 6171 . . . . . 6  |-  ( 2nd `  <. ( ( 1st `  G )  o.  ( 1st `  I ) ) ,  ( ( 2nd `  G )  .+^  ( 2nd `  I ) ) >.
)  =  ( ( 2nd `  G ) 
.+^  ( 2nd `  I
) )
5856, 57syl6eq 2364 . . . . 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 5916 . . . 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 2358 . . 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 3841 . 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 443 . . 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 31103 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  ( T  X.  E
)  /\  G  e.  ( T  X.  E
) ) )  -> 
( F  .+  G
)  e.  ( T  X.  E ) )
64633adantr3 1116 . . 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 963 . . 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 31100 . . 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 1180 . 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 961 . . 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 31103 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  ( T  X.  E
)  /\  I  e.  ( T  X.  E
) ) )  -> 
( G  .+  I
)  e.  ( T  X.  E ) )
70693adantr1 1114 . . 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 31100 . . 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 1180 . 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 2358 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 358    /\ w3a 934    = wceq 1633    e. wcel 1701   <.cop 3677    X. cxp 4724    o. ccom 4730   ` cfv 5292  (class class class)co 5900   1stc1st 6162   2ndc2nd 6163   Basecbs 13195   +g cplusg 13255  Scalarcsca 13258   Grpcgrp 14411   DivRingcdr 15561   HLchlt 29358   LHypclh 29991   LTrncltrn 30108   TEndoctendo 30759   EDRingcedring 30760   DVecHcdvh 31086
This theorem is referenced by:  dvhgrp  31115
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-tpos 6276  df-undef 6340  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-sca 13271  df-vsca 13272  df-0g 13453  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-mnd 14416  df-grp 14538  df-minusg 14539  df-mgp 15375  df-rng 15389  df-ur 15391  df-oppr 15454  df-dvdsr 15472  df-unit 15473  df-invr 15503  df-dvr 15514  df-drng 15563  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-llines 29505  df-lplanes 29506  df-lvols 29507  df-lines 29508  df-psubsp 29510  df-pmap 29511  df-padd 29803  df-lhyp 29995  df-laut 29996  df-ldil 30111  df-ltrn 30112  df-trl 30166  df-tendo 30762  df-edring 30764  df-dvech 31087
  Copyright terms: Public domain W3C validator