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

Theorem dvhlveclem 34475
Description: Lemma for dvhlvec 34476. TODO: proof substituting inner part first shorter/longer than substituting outer part first? TODO: break up into smaller lemmas? TODO: does  ph  -> method shorten proof? (Contributed by NM, 22-Oct-2013.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
dvhgrp.b  |-  B  =  ( Base `  K
)
dvhgrp.h  |-  H  =  ( LHyp `  K
)
dvhgrp.t  |-  T  =  ( ( LTrn `  K
) `  W )
dvhgrp.e  |-  E  =  ( ( TEndo `  K
) `  W )
dvhgrp.u  |-  U  =  ( ( DVecH `  K
) `  W )
dvhgrp.d  |-  D  =  (Scalar `  U )
dvhgrp.p  |-  .+^  =  ( +g  `  D )
dvhgrp.a  |-  .+  =  ( +g  `  U )
dvhgrp.o  |-  .0.  =  ( 0g `  D )
dvhgrp.i  |-  I  =  ( invg `  D )
dvhlvec.m  |-  .X.  =  ( .r `  D )
dvhlvec.s  |-  .x.  =  ( .s `  U )
Assertion
Ref Expression
dvhlveclem  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )

Proof of Theorem dvhlveclem
Dummy variables  t 
f  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvhgrp.h . . . . 5  |-  H  =  ( LHyp `  K
)
2 dvhgrp.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
3 dvhgrp.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
4 dvhgrp.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
5 eqid 2441 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
61, 2, 3, 4, 5dvhvbase 34454 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( T  X.  E ) )
76eqcomd 2446 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( T  X.  E
)  =  ( Base `  U ) )
8 dvhgrp.a . . . 4  |-  .+  =  ( +g  `  U )
98a1i 11 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .+  =  ( +g  `  U ) )
10 dvhgrp.d . . . 4  |-  D  =  (Scalar `  U )
1110a1i 11 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  (Scalar `  U ) )
12 dvhlvec.s . . . 4  |-  .x.  =  ( .s `  U )
1312a1i 11 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .x.  =  ( .s
`  U ) )
14 eqid 2441 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
151, 3, 4, 10, 14dvhbase 34450 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
1615eqcomd 2446 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E  =  ( Base `  D ) )
17 dvhgrp.p . . . 4  |-  .+^  =  ( +g  `  D )
1817a1i 11 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
.+^  =  ( +g  `  D ) )
19 dvhlvec.m . . . 4  |-  .X.  =  ( .r `  D )
2019a1i 11 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .X.  =  ( .r
`  D ) )
21 eqid 2441 . . . . . 6  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
221, 21, 4, 10dvhsca 34449 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
2322fveq2d 5692 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 1r `  D
)  =  ( 1r
`  ( ( EDRing `  K ) `  W
) ) )
24 eqid 2441 . . . . 5  |-  ( 1r
`  ( ( EDRing `  K ) `  W
) )  =  ( 1r `  ( (
EDRing `  K ) `  W ) )
251, 2, 21, 24erng1r 34361 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 1r `  (
( EDRing `  K ) `  W ) )  =  (  _I  |`  T ) )
2623, 25eqtr2d 2474 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  =  ( 1r `  D ) )
271, 21erngdv 34359 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
2822, 27eqeltrd 2515 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
29 drngrng 16819 . . . 4  |-  ( D  e.  DivRing  ->  D  e.  Ring )
3028, 29syl 16 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  Ring )
31 dvhgrp.b . . . 4  |-  B  =  ( Base `  K
)
32 dvhgrp.o . . . 4  |-  .0.  =  ( 0g `  D )
33 dvhgrp.i . . . 4  |-  I  =  ( invg `  D )
3431, 1, 2, 3, 4, 10, 17, 8, 32, 33dvhgrp 34474 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Grp )
351, 2, 3, 4, 12dvhvscacl 34470 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
) ) )  -> 
( s  .x.  t
)  e.  ( T  X.  E ) )
36353impb 1178 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  t  e.  ( T  X.  E ) )  ->  ( s  .x.  t )  e.  ( T  X.  E ) )
37 simpl 454 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
38 simpr1 989 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
s  e.  E )
39 simpr2 990 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
t  e.  ( T  X.  E ) )
40 xp1st 6605 . . . . . . . 8  |-  ( t  e.  ( T  X.  E )  ->  ( 1st `  t )  e.  T )
4139, 40syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  t
)  e.  T )
42 simpr3 991 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
f  e.  ( T  X.  E ) )
43 xp1st 6605 . . . . . . . 8  |-  ( f  e.  ( T  X.  E )  ->  ( 1st `  f )  e.  T )
4442, 43syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  f
)  e.  T )
451, 2, 3tendospdi1 34387 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  ( 1st `  t )  e.  T  /\  ( 1st `  f
)  e.  T ) )  ->  ( s `  ( ( 1st `  t
)  o.  ( 1st `  f ) ) )  =  ( ( s `
 ( 1st `  t
) )  o.  (
s `  ( 1st `  f ) ) ) )
4637, 38, 41, 44, 45syl13anc 1215 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s `  (
( 1st `  t
)  o.  ( 1st `  f ) ) )  =  ( ( s `
 ( 1st `  t
) )  o.  (
s `  ( 1st `  f ) ) ) )
471, 2, 3, 4, 10, 8, 17dvhvadd 34459 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .+  f
)  =  <. (
( 1st `  t
)  o.  ( 1st `  f ) ) ,  ( ( 2nd `  t
)  .+^  ( 2nd `  f
) ) >. )
48473adantr1 1142 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .+  f
)  =  <. (
( 1st `  t
)  o.  ( 1st `  f ) ) ,  ( ( 2nd `  t
)  .+^  ( 2nd `  f
) ) >. )
4948fveq2d 5692 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  (
t  .+  f )
)  =  ( 1st `  <. ( ( 1st `  t )  o.  ( 1st `  f ) ) ,  ( ( 2nd `  t )  .+^  ( 2nd `  f ) ) >.
) )
50 fvex 5698 . . . . . . . . . 10  |-  ( 1st `  t )  e.  _V
51 fvex 5698 . . . . . . . . . 10  |-  ( 1st `  f )  e.  _V
5250, 51coex 6528 . . . . . . . . 9  |-  ( ( 1st `  t )  o.  ( 1st `  f
) )  e.  _V
53 ovex 6115 . . . . . . . . 9  |-  ( ( 2nd `  t ) 
.+^  ( 2nd `  f
) )  e.  _V
5452, 53op1st 6584 . . . . . . . 8  |-  ( 1st `  <. ( ( 1st `  t )  o.  ( 1st `  f ) ) ,  ( ( 2nd `  t )  .+^  ( 2nd `  f ) ) >.
)  =  ( ( 1st `  t )  o.  ( 1st `  f
) )
5549, 54syl6eq 2489 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  (
t  .+  f )
)  =  ( ( 1st `  t )  o.  ( 1st `  f
) ) )
5655fveq2d 5692 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s `  ( 1st `  ( t  .+  f ) ) )  =  ( s `  ( ( 1st `  t
)  o.  ( 1st `  f ) ) ) )
571, 2, 3, 4, 12dvhvsca 34468 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
) ) )  -> 
( s  .x.  t
)  =  <. (
s `  ( 1st `  t ) ) ,  ( s  o.  ( 2nd `  t ) )
>. )
58573adantr3 1144 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  t
)  =  <. (
s `  ( 1st `  t ) ) ,  ( s  o.  ( 2nd `  t ) )
>. )
5958fveq2d 5692 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  (
s  .x.  t )
)  =  ( 1st `  <. ( s `  ( 1st `  t ) ) ,  ( s  o.  ( 2nd `  t
) ) >. )
)
60 fvex 5698 . . . . . . . . 9  |-  ( s `
 ( 1st `  t
) )  e.  _V
61 vex 2973 . . . . . . . . . 10  |-  s  e. 
_V
62 fvex 5698 . . . . . . . . . 10  |-  ( 2nd `  t )  e.  _V
6361, 62coex 6528 . . . . . . . . 9  |-  ( s  o.  ( 2nd `  t
) )  e.  _V
6460, 63op1st 6584 . . . . . . . 8  |-  ( 1st `  <. ( s `  ( 1st `  t ) ) ,  ( s  o.  ( 2nd `  t
) ) >. )  =  ( s `  ( 1st `  t ) )
6559, 64syl6eq 2489 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  (
s  .x.  t )
)  =  ( s `
 ( 1st `  t
) ) )
661, 2, 3, 4, 12dvhvsca 34468 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  f
)  =  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>. )
67663adantr2 1143 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  f
)  =  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>. )
6867fveq2d 5692 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  (
s  .x.  f )
)  =  ( 1st `  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )
)
69 fvex 5698 . . . . . . . . 9  |-  ( s `
 ( 1st `  f
) )  e.  _V
70 fvex 5698 . . . . . . . . . 10  |-  ( 2nd `  f )  e.  _V
7161, 70coex 6528 . . . . . . . . 9  |-  ( s  o.  ( 2nd `  f
) )  e.  _V
7269, 71op1st 6584 . . . . . . . 8  |-  ( 1st `  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )  =  ( s `  ( 1st `  f ) )
7368, 72syl6eq 2489 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  (
s  .x.  f )
)  =  ( s `
 ( 1st `  f
) ) )
7465, 73coeq12d 5000 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( 1st `  (
s  .x.  t )
)  o.  ( 1st `  ( s  .x.  f
) ) )  =  ( ( s `  ( 1st `  t ) )  o.  ( s `
 ( 1st `  f
) ) ) )
7546, 56, 743eqtr4d 2483 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s `  ( 1st `  ( t  .+  f ) ) )  =  ( ( 1st `  ( s  .x.  t
) )  o.  ( 1st `  ( s  .x.  f ) ) ) )
7630adantr 462 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  ->  D  e.  Ring )
7716adantr 462 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  ->  E  =  ( Base `  D ) )
7838, 77eleqtrd 2517 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
s  e.  ( Base `  D ) )
79 xp2nd 6606 . . . . . . . . . 10  |-  ( t  e.  ( T  X.  E )  ->  ( 2nd `  t )  e.  E )
8039, 79syl 16 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  t
)  e.  E )
8180, 77eleqtrd 2517 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  t
)  e.  ( Base `  D ) )
82 xp2nd 6606 . . . . . . . . . 10  |-  ( f  e.  ( T  X.  E )  ->  ( 2nd `  f )  e.  E )
8342, 82syl 16 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  f
)  e.  E )
8483, 77eleqtrd 2517 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  f
)  e.  ( Base `  D ) )
8514, 17, 19rngdi 16653 . . . . . . . 8  |-  ( ( D  e.  Ring  /\  (
s  e.  ( Base `  D )  /\  ( 2nd `  t )  e.  ( Base `  D
)  /\  ( 2nd `  f )  e.  (
Base `  D )
) )  ->  (
s  .X.  ( ( 2nd `  t )  .+^  ( 2nd `  f ) ) )  =  ( ( s  .X.  ( 2nd `  t ) ) 
.+^  ( s  .X.  ( 2nd `  f ) ) ) )
8676, 78, 81, 84, 85syl13anc 1215 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .X.  (
( 2nd `  t
)  .+^  ( 2nd `  f
) ) )  =  ( ( s  .X.  ( 2nd `  t ) )  .+^  ( s  .X.  ( 2nd `  f
) ) ) )
8714, 17rngacl 16662 . . . . . . . . . 10  |-  ( ( D  e.  Ring  /\  ( 2nd `  t )  e.  ( Base `  D
)  /\  ( 2nd `  f )  e.  (
Base `  D )
)  ->  ( ( 2nd `  t )  .+^  ( 2nd `  f ) )  e.  ( Base `  D ) )
8876, 81, 84, 87syl3anc 1213 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( 2nd `  t
)  .+^  ( 2nd `  f
) )  e.  (
Base `  D )
)
8988, 77eleqtrrd 2518 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( 2nd `  t
)  .+^  ( 2nd `  f
) )  e.  E
)
901, 2, 3, 4, 10, 19dvhmulr 34453 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  ( ( 2nd `  t ) 
.+^  ( 2nd `  f
) )  e.  E
) )  ->  (
s  .X.  ( ( 2nd `  t )  .+^  ( 2nd `  f ) ) )  =  ( s  o.  ( ( 2nd `  t ) 
.+^  ( 2nd `  f
) ) ) )
9137, 38, 89, 90syl12anc 1211 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .X.  (
( 2nd `  t
)  .+^  ( 2nd `  f
) ) )  =  ( s  o.  (
( 2nd `  t
)  .+^  ( 2nd `  f
) ) ) )
921, 2, 3, 4, 10, 19dvhmulr 34453 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  ( 2nd `  t )  e.  E
) )  ->  (
s  .X.  ( 2nd `  t ) )  =  ( s  o.  ( 2nd `  t ) ) )
9337, 38, 80, 92syl12anc 1211 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .X.  ( 2nd `  t ) )  =  ( s  o.  ( 2nd `  t
) ) )
941, 2, 3, 4, 10, 19dvhmulr 34453 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  ( 2nd `  f )  e.  E
) )  ->  (
s  .X.  ( 2nd `  f ) )  =  ( s  o.  ( 2nd `  f ) ) )
9537, 38, 83, 94syl12anc 1211 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .X.  ( 2nd `  f ) )  =  ( s  o.  ( 2nd `  f
) ) )
9693, 95oveq12d 6108 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .X.  ( 2nd `  t ) )  .+^  ( s  .X.  ( 2nd `  f
) ) )  =  ( ( s  o.  ( 2nd `  t
) )  .+^  ( s  o.  ( 2nd `  f
) ) ) )
9786, 91, 963eqtr3d 2481 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  o.  (
( 2nd `  t
)  .+^  ( 2nd `  f
) ) )  =  ( ( s  o.  ( 2nd `  t
) )  .+^  ( s  o.  ( 2nd `  f
) ) ) )
9848fveq2d 5692 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  (
t  .+  f )
)  =  ( 2nd `  <. ( ( 1st `  t )  o.  ( 1st `  f ) ) ,  ( ( 2nd `  t )  .+^  ( 2nd `  f ) ) >.
) )
9952, 53op2nd 6585 . . . . . . . 8  |-  ( 2nd `  <. ( ( 1st `  t )  o.  ( 1st `  f ) ) ,  ( ( 2nd `  t )  .+^  ( 2nd `  f ) ) >.
)  =  ( ( 2nd `  t ) 
.+^  ( 2nd `  f
) )
10098, 99syl6eq 2489 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  (
t  .+  f )
)  =  ( ( 2nd `  t ) 
.+^  ( 2nd `  f
) ) )
101100coeq2d 4998 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  o.  ( 2nd `  ( t  .+  f ) ) )  =  ( s  o.  ( ( 2nd `  t
)  .+^  ( 2nd `  f
) ) ) )
10258fveq2d 5692 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  (
s  .x.  t )
)  =  ( 2nd `  <. ( s `  ( 1st `  t ) ) ,  ( s  o.  ( 2nd `  t
) ) >. )
)
10360, 63op2nd 6585 . . . . . . . 8  |-  ( 2nd `  <. ( s `  ( 1st `  t ) ) ,  ( s  o.  ( 2nd `  t
) ) >. )  =  ( s  o.  ( 2nd `  t
) )
104102, 103syl6eq 2489 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  (
s  .x.  t )
)  =  ( s  o.  ( 2nd `  t
) ) )
10567fveq2d 5692 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  (
s  .x.  f )
)  =  ( 2nd `  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )
)
10669, 71op2nd 6585 . . . . . . . 8  |-  ( 2nd `  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )  =  ( s  o.  ( 2nd `  f
) )
107105, 106syl6eq 2489 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  (
s  .x.  f )
)  =  ( s  o.  ( 2nd `  f
) ) )
108104, 107oveq12d 6108 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( 2nd `  (
s  .x.  t )
)  .+^  ( 2nd `  (
s  .x.  f )
) )  =  ( ( s  o.  ( 2nd `  t ) ) 
.+^  ( s  o.  ( 2nd `  f
) ) ) )
10997, 101, 1083eqtr4d 2483 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  o.  ( 2nd `  ( t  .+  f ) ) )  =  ( ( 2nd `  ( s  .x.  t
) )  .+^  ( 2nd `  ( s  .x.  f
) ) ) )
11075, 109opeq12d 4064 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  ->  <. ( s `  ( 1st `  ( t  .+  f ) ) ) ,  ( s  o.  ( 2nd `  (
t  .+  f )
) ) >.  =  <. ( ( 1st `  (
s  .x.  t )
)  o.  ( 1st `  ( s  .x.  f
) ) ) ,  ( ( 2nd `  (
s  .x.  t )
)  .+^  ( 2nd `  (
s  .x.  f )
) ) >. )
1111, 2, 3, 4, 10, 17, 8dvhvaddcl 34462 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .+  f
)  e.  ( T  X.  E ) )
1121113adantr1 1142 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .+  f
)  e.  ( T  X.  E ) )
1131, 2, 3, 4, 12dvhvsca 34468 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  ( t 
.+  f )  e.  ( T  X.  E
) ) )  -> 
( s  .x.  (
t  .+  f )
)  =  <. (
s `  ( 1st `  ( t  .+  f
) ) ) ,  ( s  o.  ( 2nd `  ( t  .+  f ) ) )
>. )
11437, 38, 112, 113syl12anc 1211 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  (
t  .+  f )
)  =  <. (
s `  ( 1st `  ( t  .+  f
) ) ) ,  ( s  o.  ( 2nd `  ( t  .+  f ) ) )
>. )
115353adantr3 1144 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  t
)  e.  ( T  X.  E ) )
1161, 2, 3, 4, 12dvhvscacl 34470 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  f
)  e.  ( T  X.  E ) )
1171163adantr2 1143 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  f
)  e.  ( T  X.  E ) )
1181, 2, 3, 4, 10, 8, 17dvhvadd 34459 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( s 
.x.  t )  e.  ( T  X.  E
)  /\  ( s  .x.  f )  e.  ( T  X.  E ) ) )  ->  (
( s  .x.  t
)  .+  ( s  .x.  f ) )  = 
<. ( ( 1st `  (
s  .x.  t )
)  o.  ( 1st `  ( s  .x.  f
) ) ) ,  ( ( 2nd `  (
s  .x.  t )
)  .+^  ( 2nd `  (
s  .x.  f )
) ) >. )
11937, 115, 117, 118syl12anc 1211 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .x.  t )  .+  (
s  .x.  f )
)  =  <. (
( 1st `  (
s  .x.  t )
)  o.  ( 1st `  ( s  .x.  f
) ) ) ,  ( ( 2nd `  (
s  .x.  t )
)  .+^  ( 2nd `  (
s  .x.  f )
) ) >. )
120110, 114, 1193eqtr4d 2483 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  (
t  .+  f )
)  =  ( ( s  .x.  t ) 
.+  ( s  .x.  f ) ) )
121 simpl 454 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
122 simpr1 989 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
s  e.  E )
123 simpr2 990 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
t  e.  E )
124 simpr3 991 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
f  e.  ( T  X.  E ) )
125124, 43syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  f
)  e.  T )
126 eqid 2441 . . . . . . . 8  |-  ( +g  `  ( ( EDRing `  K
) `  W )
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
)
1271, 2, 3, 21, 126erngplus2 34170 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  ( 1st `  f )  e.  T
) )  ->  (
( s ( +g  `  ( ( EDRing `  K
) `  W )
) t ) `  ( 1st `  f ) )  =  ( ( s `  ( 1st `  f ) )  o.  ( t `  ( 1st `  f ) ) ) )
128121, 122, 123, 125, 127syl13anc 1215 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s ( +g  `  ( (
EDRing `  K ) `  W ) ) t ) `  ( 1st `  f ) )  =  ( ( s `  ( 1st `  f ) )  o.  ( t `
 ( 1st `  f
) ) ) )
12922fveq2d 5692 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( +g  `  D
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
13017, 129syl5eq 2485 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
.+^  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
131130oveqd 6107 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( s  .+^  t )  =  ( s ( +g  `  ( (
EDRing `  K ) `  W ) ) t ) )
132131fveq1d 5690 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( s  .+^  t ) `  ( 1st `  f ) )  =  ( ( s ( +g  `  (
( EDRing `  K ) `  W ) ) t ) `  ( 1st `  f ) ) )
133132adantr 462 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .+^  t ) `  ( 1st `  f ) )  =  ( ( s ( +g  `  (
( EDRing `  K ) `  W ) ) t ) `  ( 1st `  f ) ) )
134663adantr2 1143 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  f
)  =  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>. )
135134fveq2d 5692 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  (
s  .x.  f )
)  =  ( 1st `  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )
)
136135, 72syl6eq 2489 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  (
s  .x.  f )
)  =  ( s `
 ( 1st `  f
) ) )
1371, 2, 3, 4, 12dvhvsca 34468 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .x.  f
)  =  <. (
t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f ) )
>. )
1381373adantr1 1142 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .x.  f
)  =  <. (
t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f ) )
>. )
139138fveq2d 5692 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  (
t  .x.  f )
)  =  ( 1st `  <. ( t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f
) ) >. )
)
140 fvex 5698 . . . . . . . . 9  |-  ( t `
 ( 1st `  f
) )  e.  _V
141 vex 2973 . . . . . . . . . 10  |-  t  e. 
_V
142141, 70coex 6528 . . . . . . . . 9  |-  ( t  o.  ( 2nd `  f
) )  e.  _V
143140, 142op1st 6584 . . . . . . . 8  |-  ( 1st `  <. ( t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f
) ) >. )  =  ( t `  ( 1st `  f ) )
144139, 143syl6eq 2489 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  (
t  .x.  f )
)  =  ( t `
 ( 1st `  f
) ) )
145136, 144coeq12d 5000 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( 1st `  (
s  .x.  f )
)  o.  ( 1st `  ( t  .x.  f
) ) )  =  ( ( s `  ( 1st `  f ) )  o.  ( t `
 ( 1st `  f
) ) ) )
146128, 133, 1453eqtr4d 2483 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .+^  t ) `  ( 1st `  f ) )  =  ( ( 1st `  ( s  .x.  f
) )  o.  ( 1st `  ( t  .x.  f ) ) ) )
14730adantr 462 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  ->  D  e.  Ring )
14816adantr 462 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  ->  E  =  ( Base `  D ) )
149122, 148eleqtrd 2517 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
s  e.  ( Base `  D ) )
150123, 148eleqtrd 2517 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
t  e.  ( Base `  D ) )
151124, 82syl 16 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  f
)  e.  E )
152151, 148eleqtrd 2517 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  f
)  e.  ( Base `  D ) )
15314, 17, 19rngdir 16654 . . . . . . . 8  |-  ( ( D  e.  Ring  /\  (
s  e.  ( Base `  D )  /\  t  e.  ( Base `  D
)  /\  ( 2nd `  f )  e.  (
Base `  D )
) )  ->  (
( s  .+^  t ) 
.X.  ( 2nd `  f
) )  =  ( ( s  .X.  ( 2nd `  f ) ) 
.+^  ( t  .X.  ( 2nd `  f ) ) ) )
154147, 149, 150, 152, 153syl13anc 1215 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .+^  t )  .X.  ( 2nd `  f ) )  =  ( ( s 
.X.  ( 2nd `  f
) )  .+^  ( t 
.X.  ( 2nd `  f
) ) ) )
15514, 17rngacl 16662 . . . . . . . . . 10  |-  ( ( D  e.  Ring  /\  s  e.  ( Base `  D
)  /\  t  e.  ( Base `  D )
)  ->  ( s  .+^  t )  e.  (
Base `  D )
)
156147, 149, 150, 155syl3anc 1213 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .+^  t )  e.  ( Base `  D
) )
157156, 148eleqtrrd 2518 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .+^  t )  e.  E )
1581, 2, 3, 4, 10, 19dvhmulr 34453 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( s 
.+^  t )  e.  E  /\  ( 2nd `  f )  e.  E
) )  ->  (
( s  .+^  t ) 
.X.  ( 2nd `  f
) )  =  ( ( s  .+^  t )  o.  ( 2nd `  f
) ) )
159121, 157, 151, 158syl12anc 1211 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .+^  t )  .X.  ( 2nd `  f ) )  =  ( ( s 
.+^  t )  o.  ( 2nd `  f
) ) )
160121, 122, 151, 94syl12anc 1211 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .X.  ( 2nd `  f ) )  =  ( s  o.  ( 2nd `  f
) ) )
1611, 2, 3, 4, 10, 19dvhmulr 34453 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( t  e.  E  /\  ( 2nd `  f )  e.  E
) )  ->  (
t  .X.  ( 2nd `  f ) )  =  ( t  o.  ( 2nd `  f ) ) )
162121, 123, 151, 161syl12anc 1211 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .X.  ( 2nd `  f ) )  =  ( t  o.  ( 2nd `  f
) ) )
163160, 162oveq12d 6108 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .X.  ( 2nd `  f ) )  .+^  ( t  .X.  ( 2nd `  f
) ) )  =  ( ( s  o.  ( 2nd `  f
) )  .+^  ( t  o.  ( 2nd `  f
) ) ) )
164154, 159, 1633eqtr3d 2481 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .+^  t )  o.  ( 2nd `  f ) )  =  ( ( s  o.  ( 2nd `  f
) )  .+^  ( t  o.  ( 2nd `  f
) ) ) )
165134fveq2d 5692 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  (
s  .x.  f )
)  =  ( 2nd `  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )
)
166165, 106syl6eq 2489 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  (
s  .x.  f )
)  =  ( s  o.  ( 2nd `  f
) ) )
167138fveq2d 5692 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  (
t  .x.  f )
)  =  ( 2nd `  <. ( t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f
) ) >. )
)
168140, 142op2nd 6585 . . . . . . . 8  |-  ( 2nd `  <. ( t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f
) ) >. )  =  ( t  o.  ( 2nd `  f
) )
169167, 168syl6eq 2489 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  (
t  .x.  f )
)  =  ( t  o.  ( 2nd `  f
) ) )
170166, 169oveq12d 6108 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( 2nd `  (
s  .x.  f )
)  .+^  ( 2nd `  (
t  .x.  f )
) )  =  ( ( s  o.  ( 2nd `  f ) ) 
.+^  ( t  o.  ( 2nd `  f
) ) ) )
171164, 170eqtr4d 2476 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .+^  t )  o.  ( 2nd `  f ) )  =  ( ( 2nd `  ( s  .x.  f
) )  .+^  ( 2nd `  ( t  .x.  f
) ) ) )
172146, 171opeq12d 4064 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  ->  <. ( ( s  .+^  t ) `  ( 1st `  f ) ) ,  ( ( s 
.+^  t )  o.  ( 2nd `  f
) ) >.  =  <. ( ( 1st `  (
s  .x.  f )
)  o.  ( 1st `  ( t  .x.  f
) ) ) ,  ( ( 2nd `  (
s  .x.  f )
)  .+^  ( 2nd `  (
t  .x.  f )
) ) >. )
1731, 2, 3, 4, 12dvhvsca 34468 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( s 
.+^  t )  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .+^  t )  .x.  f
)  =  <. (
( s  .+^  t ) `
 ( 1st `  f
) ) ,  ( ( s  .+^  t )  o.  ( 2nd `  f
) ) >. )
174121, 157, 124, 173syl12anc 1211 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .+^  t )  .x.  f
)  =  <. (
( s  .+^  t ) `
 ( 1st `  f
) ) ,  ( ( s  .+^  t )  o.  ( 2nd `  f
) ) >. )
1751163adantr2 1143 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  f
)  e.  ( T  X.  E ) )
1761, 2, 3, 4, 12dvhvscacl 34470 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .x.  f
)  e.  ( T  X.  E ) )
1771763adantr1 1142 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .x.  f
)  e.  ( T  X.  E ) )
1781, 2, 3, 4, 10, 8, 17dvhvadd 34459 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( s 
.x.  f )  e.  ( T  X.  E
)  /\  ( t  .x.  f )  e.  ( T  X.  E ) ) )  ->  (
( s  .x.  f
)  .+  ( t  .x.  f ) )  = 
<. ( ( 1st `  (
s  .x.  f )
)  o.  ( 1st `  ( t  .x.  f
) ) ) ,  ( ( 2nd `  (
s  .x.  f )
)  .+^  ( 2nd `  (
t  .x.  f )
) ) >. )
179121, 175, 177, 178syl12anc 1211 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .x.  f )  .+  (
t  .x.  f )
)  =  <. (
( 1st `  (
s  .x.  f )
)  o.  ( 1st `  ( t  .x.  f
) ) ) ,  ( ( 2nd `  (
s  .x.  f )
)  .+^  ( 2nd `  (
t  .x.  f )
) ) >. )
180172, 174, 1793eqtr4d 2483 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .+^  t )  .x.  f
)  =  ( ( s  .x.  f ) 
.+  ( t  .x.  f ) ) )
1811, 2, 3tendocoval 34132 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E )  /\  ( 1st `  f )  e.  T )  ->  (
( s  o.  t
) `  ( 1st `  f ) )  =  ( s `  (
t `  ( 1st `  f ) ) ) )
182121, 122, 123, 125, 181syl121anc 1218 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  o.  t ) `  ( 1st `  f ) )  =  ( s `  ( t `  ( 1st `  f ) ) ) )
183 coass 5353 . . . . . . 7  |-  ( ( s  o.  t )  o.  ( 2nd `  f
) )  =  ( s  o.  ( t  o.  ( 2nd `  f
) ) )
184183a1i 11 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  o.  t )  o.  ( 2nd `  f ) )  =  ( s  o.  ( t  o.  ( 2nd `  f ) ) ) )
185182, 184opeq12d 4064 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  ->  <. ( ( s  o.  t ) `  ( 1st `  f ) ) ,  ( ( s  o.  t )  o.  ( 2nd `  f
) ) >.  =  <. ( s `  ( t `
 ( 1st `  f
) ) ) ,  ( s  o.  (
t  o.  ( 2nd `  f ) ) )
>. )
1861, 3tendococl 34138 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  t  e.  E
)  ->  ( s  o.  t )  e.  E
)
187121, 122, 123, 186syl3anc 1213 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  o.  t
)  e.  E )
1881, 2, 3, 4, 12dvhvsca 34468 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( s  o.  t )  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  o.  t )  .x.  f
)  =  <. (
( s  o.  t
) `  ( 1st `  f ) ) ,  ( ( s  o.  t )  o.  ( 2nd `  f ) )
>. )
189121, 187, 124, 188syl12anc 1211 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  o.  t )  .x.  f
)  =  <. (
( s  o.  t
) `  ( 1st `  f ) ) ,  ( ( s  o.  t )  o.  ( 2nd `  f ) )
>. )
1901, 2, 3tendocl 34133 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  t  e.  E  /\  ( 1st `  f
)  e.  T )  ->  ( t `  ( 1st `  f ) )  e.  T )
191121, 123, 125, 190syl3anc 1213 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( t `  ( 1st `  f ) )  e.  T )
1921, 3tendococl 34138 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  t  e.  E  /\  ( 2nd `  f
)  e.  E )  ->  ( t  o.  ( 2nd `  f
) )  e.  E
)
193121, 123, 151, 192syl3anc 1213 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  o.  ( 2nd `  f ) )  e.  E )
1941, 2, 3, 4, 12dvhopvsca 34469 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  ( t `
 ( 1st `  f
) )  e.  T  /\  ( t  o.  ( 2nd `  f ) )  e.  E ) )  ->  ( s  .x.  <.
( t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f
) ) >. )  =  <. ( s `  ( t `  ( 1st `  f ) ) ) ,  ( s  o.  ( t  o.  ( 2nd `  f
) ) ) >.
)
195121, 122, 191, 193, 194syl13anc 1215 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  <. (
t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f ) )
>. )  =  <. ( s `  ( t `
 ( 1st `  f
) ) ) ,  ( s  o.  (
t  o.  ( 2nd `  f ) ) )
>. )
196185, 189, 1953eqtr4d 2483 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  o.  t )  .x.  f
)  =  ( s 
.x.  <. ( t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f
) ) >. )
)
1971, 2, 3, 4, 10, 19dvhmulr 34453 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E ) )  -> 
( s  .X.  t
)  =  ( s  o.  t ) )
1981973adantr3 1144 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .X.  t
)  =  ( s  o.  t ) )
199198oveq1d 6105 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .X.  t )  .x.  f
)  =  ( ( s  o.  t ) 
.x.  f ) )
200138oveq2d 6106 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  (
t  .x.  f )
)  =  ( s 
.x.  <. ( t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f
) ) >. )
)
201196, 199, 2003eqtr4d 2483 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( ( s  .X.  t )  .x.  f
)  =  ( s 
.x.  ( t  .x.  f ) ) )
202 xp1st 6605 . . . . . . 7  |-  ( s  e.  ( T  X.  E )  ->  ( 1st `  s )  e.  T )
203202adantl 463 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( 1st `  s )  e.  T
)
204 tendospid 34384 . . . . . 6  |-  ( ( 1st `  s )  e.  T  ->  (
(  _I  |`  T ) `
 ( 1st `  s
) )  =  ( 1st `  s ) )
205203, 204syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( (  _I  |`  T ) `  ( 1st `  s ) )  =  ( 1st `  s ) )
206 xp2nd 6606 . . . . . . 7  |-  ( s  e.  ( T  X.  E )  ->  ( 2nd `  s )  e.  E )
2071, 2, 3tendof 34129 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 2nd `  s
)  e.  E )  ->  ( 2nd `  s
) : T --> T )
208206, 207sylan2 471 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( 2nd `  s ) : T --> T )
209 fcoi2 5583 . . . . . 6  |-  ( ( 2nd `  s ) : T --> T  -> 
( (  _I  |`  T )  o.  ( 2nd `  s
) )  =  ( 2nd `  s ) )
210208, 209syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( (  _I  |`  T )  o.  ( 2nd `  s
) )  =  ( 2nd `  s ) )
211205, 210opeq12d 4064 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  <. ( (  _I  |`  T ) `  ( 1st `  s
) ) ,  ( (  _I  |`  T )  o.  ( 2nd `  s
) ) >.  =  <. ( 1st `  s ) ,  ( 2nd `  s
) >. )
2121, 2, 3tendoidcl 34135 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
213212anim1i 565 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( (  _I  |`  T )  e.  E  /\  s  e.  ( T  X.  E
) ) )
2141, 2, 3, 4, 12dvhvsca 34468 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( (  _I  |`  T )  e.  E  /\  s  e.  ( T  X.  E ) ) )  ->  ( (  _I  |`  T )  .x.  s )  =  <. ( (  _I  |`  T ) `
 ( 1st `  s
) ) ,  ( (  _I  |`  T )  o.  ( 2nd `  s
) ) >. )
215213, 214syldan 467 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( (  _I  |`  T )  .x.  s )  =  <. ( (  _I  |`  T ) `
 ( 1st `  s
) ) ,  ( (  _I  |`  T )  o.  ( 2nd `  s
) ) >. )
216 1st2nd2 6612 . . . . 5  |-  ( s  e.  ( T  X.  E )  ->  s  =  <. ( 1st `  s
) ,  ( 2nd `  s ) >. )
217216adantl 463 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  s  =  <. ( 1st `  s
) ,  ( 2nd `  s ) >. )
218211, 215, 2173eqtr4d 2483 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( (  _I  |`  T )  .x.  s )  =  s )
2197, 9, 11, 13, 16, 18, 20, 26, 30, 34, 36, 120, 180, 201, 218islmodd 16934 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LMod )
22010islvec 17163 . 2  |-  ( U  e.  LVec  <->  ( U  e. 
LMod  /\  D  e.  DivRing ) )
221219, 28, 220sylanbrc 659 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   <.cop 3880    _I cid 4627    X. cxp 4834    |` cres 4838    o. ccom 4840   -->wf 5411   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   Basecbs 14170   +g cplusg 14234   .rcmulr 14235  Scalarcsca 14237   .scvsca 14238   0gc0g 14374   invgcminusg 15407   1rcur 16593   Ringcrg 16635   DivRingcdr 16812   LModclmod 16928   LVecclvec 17161   HLchlt 32717   LHypclh 33350   LTrncltrn 33467   TEndoctendo 34118   EDRingcedring 34119   DVecHcdvh 34445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-riotaBAD 32326
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-undef 6788  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-0g 14376  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-p1 15206  df-lat 15212  df-clat 15274  df-mnd 15411  df-grp 15538  df-minusg 15539  df-mgp 16582  df-ur 16594  df-rng 16637  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-drng 16814  df-lmod 16930  df-lvec 17162  df-oposet 32543  df-ol 32545  df-oml 32546  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718  df-llines 32864  df-lplanes 32865  df-lvols 32866  df-lines 32867  df-psubsp 32869  df-pmap 32870  df-padd 33162  df-lhyp 33354  df-laut 33355  df-ldil 33470  df-ltrn 33471  df-trl 33525  df-tendo 34121  df-edring 34123  df-dvech 34446
This theorem is referenced by:  dvhlvec  34476
  Copyright terms: Public domain W3C validator