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Theorem dvhlveclem 34388
Description: Lemma for dvhlvec 34389. 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 2429 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
61, 2, 3, 4, 5dvhvbase 34367 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( T  X.  E ) )
76eqcomd 2437 . . 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 2429 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
151, 3, 4, 10, 14dvhbase 34363 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
1615eqcomd 2437 . . 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 2429 . . . . . 6  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
221, 21, 4, 10dvhsca 34362 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
2322fveq2d 5885 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 1r `  D
)  =  ( 1r
`  ( ( EDRing `  K ) `  W
) ) )
24 eqid 2429 . . . . 5  |-  ( 1r
`  ( ( EDRing `  K ) `  W
) )  =  ( 1r `  ( (
EDRing `  K ) `  W ) )
251, 2, 21, 24erng1r 34274 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 1r `  (
( EDRing `  K ) `  W ) )  =  (  _I  |`  T ) )
2623, 25eqtr2d 2471 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  =  ( 1r `  D ) )
271, 21erngdv 34272 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
2822, 27eqeltrd 2517 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
29 drngring 17917 . . . 4  |-  ( D  e.  DivRing  ->  D  e.  Ring )
3028, 29syl 17 . . 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 34387 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Grp )
351, 2, 3, 4, 12dvhvscacl 34383 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
) ) )  -> 
( s  .x.  t
)  e.  ( T  X.  E ) )
36353impb 1201 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  t  e.  ( T  X.  E ) )  ->  ( s  .x.  t )  e.  ( T  X.  E ) )
37 simpl 458 . . . . . . 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 1011 . . . . . . 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 1012 . . . . . . . 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 6837 . . . . . . . 8  |-  ( t  e.  ( T  X.  E )  ->  ( 1st `  t )  e.  T )
4139, 40syl 17 . . . . . . 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 1013 . . . . . . . 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 6837 . . . . . . . 8  |-  ( f  e.  ( T  X.  E )  ->  ( 1st `  f )  e.  T )
4442, 43syl 17 . . . . . . 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 34300 . . . . . . 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 1266 . . . . . 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 34372 . . . . . . . . . 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 1164 . . . . . . . . 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 5885 . . . . . . . 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 5891 . . . . . . . . . 10  |-  ( 1st `  t )  e.  _V
51 fvex 5891 . . . . . . . . . 10  |-  ( 1st `  f )  e.  _V
5250, 51coex 6759 . . . . . . . . 9  |-  ( ( 1st `  t )  o.  ( 1st `  f
) )  e.  _V
53 ovex 6333 . . . . . . . . 9  |-  ( ( 2nd `  t ) 
.+^  ( 2nd `  f
) )  e.  _V
5452, 53op1st 6815 . . . . . . . 8  |-  ( 1st `  <. ( ( 1st `  t )  o.  ( 1st `  f ) ) ,  ( ( 2nd `  t )  .+^  ( 2nd `  f ) ) >.
)  =  ( ( 1st `  t )  o.  ( 1st `  f
) )
5549, 54syl6eq 2486 . . . . . . 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 5885 . . . . . 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 34381 . . . . . . . . . 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 1166 . . . . . . . . 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 5885 . . . . . . . 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 5891 . . . . . . . . 9  |-  ( s `
 ( 1st `  t
) )  e.  _V
61 vex 3090 . . . . . . . . . 10  |-  s  e. 
_V
62 fvex 5891 . . . . . . . . . 10  |-  ( 2nd `  t )  e.  _V
6361, 62coex 6759 . . . . . . . . 9  |-  ( s  o.  ( 2nd `  t
) )  e.  _V
6460, 63op1st 6815 . . . . . . . 8  |-  ( 1st `  <. ( s `  ( 1st `  t ) ) ,  ( s  o.  ( 2nd `  t
) ) >. )  =  ( s `  ( 1st `  t ) )
6559, 64syl6eq 2486 . . . . . . 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 34381 . . . . . . . . . 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 1165 . . . . . . . . 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 5885 . . . . . . . 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 5891 . . . . . . . . 9  |-  ( s `
 ( 1st `  f
) )  e.  _V
70 fvex 5891 . . . . . . . . . 10  |-  ( 2nd `  f )  e.  _V
7161, 70coex 6759 . . . . . . . . 9  |-  ( s  o.  ( 2nd `  f
) )  e.  _V
7269, 71op1st 6815 . . . . . . . 8  |-  ( 1st `  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )  =  ( s `  ( 1st `  f ) )
7368, 72syl6eq 2486 . . . . . . 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 5019 . . . . . 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 2480 . . . . 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 466 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  ->  D  e.  Ring )
7716adantr 466 . . . . . . . . 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 2519 . . . . . . . 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 6838 . . . . . . . . . 10  |-  ( t  e.  ( T  X.  E )  ->  ( 2nd `  t )  e.  E )
8039, 79syl 17 . . . . . . . . 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 2519 . . . . . . . 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 6838 . . . . . . . . . 10  |-  ( f  e.  ( T  X.  E )  ->  ( 2nd `  f )  e.  E )
8342, 82syl 17 . . . . . . . . 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 2519 . . . . . . . 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, 19ringdi 17734 . . . . . . . 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 1266 . . . . . . 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, 17ringacl 17743 . . . . . . . . . 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 1264 . . . . . . . . 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 2520 . . . . . . . 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 34366 . . . . . . . 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 1262 . . . . . . 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 34366 . . . . . . . . 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 1262 . . . . . . . 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 34366 . . . . . . . . 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 1262 . . . . . . . 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 6323 . . . . . . 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 2478 . . . . . 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 5885 . . . . . . . 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 6816 . . . . . . . 8  |-  ( 2nd `  <. ( ( 1st `  t )  o.  ( 1st `  f ) ) ,  ( ( 2nd `  t )  .+^  ( 2nd `  f ) ) >.
)  =  ( ( 2nd `  t ) 
.+^  ( 2nd `  f
) )
10098, 99syl6eq 2486 . . . . . . 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 5017 . . . . . 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 5885 . . . . . . . 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 6816 . . . . . . . 8  |-  ( 2nd `  <. ( s `  ( 1st `  t ) ) ,  ( s  o.  ( 2nd `  t
) ) >. )  =  ( s  o.  ( 2nd `  t
) )
104102, 103syl6eq 2486 . . . . . . 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 5885 . . . . . . . 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 6816 . . . . . . . 8  |-  ( 2nd `  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )  =  ( s  o.  ( 2nd `  f
) )
107105, 106syl6eq 2486 . . . . . . 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 6323 . . . . . 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 2480 . . . . 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 4198 . . . 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 34375 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .+  f
)  e.  ( T  X.  E ) )
1121113adantr1 1164 . . . . 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 34381 . . . . 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 1262 . . . 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 1166 . . . . 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 34383 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  f
)  e.  ( T  X.  E ) )
1171163adantr2 1165 . . . . 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 34372 . . . . 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 1262 . . . 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 2480 . . 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 458 . . . . . . 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 1011 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
s  e.  E )
123 simpr2 1012 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
t  e.  E )
124 simpr3 1013 . . . . . . . 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 17 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  f
)  e.  T )
126 eqid 2429 . . . . . . . 8  |-  ( +g  `  ( ( EDRing `  K
) `  W )
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
)
1271, 2, 3, 21, 126erngplus2 34083 . . . . . . 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 1266 . . . . . 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 5885 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( +g  `  D
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
13017, 129syl5eq 2482 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
.+^  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
131130oveqd 6322 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( s  .+^  t )  =  ( s ( +g  `  ( (
EDRing `  K ) `  W ) ) t ) )
132131fveq1d 5883 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( s  .+^  t ) `  ( 1st `  f ) )  =  ( ( s ( +g  `  (
( EDRing `  K ) `  W ) ) t ) `  ( 1st `  f ) ) )
133132adantr 466 . . . . . 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 1165 . . . . . . . . 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 5885 . . . . . . . 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 2486 . . . . . . 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 34381 . . . . . . . . . 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 1164 . . . . . . . . 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 5885 . . . . . . . 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 5891 . . . . . . . . 9  |-  ( t `
 ( 1st `  f
) )  e.  _V
141 vex 3090 . . . . . . . . . 10  |-  t  e. 
_V
142141, 70coex 6759 . . . . . . . . 9  |-  ( t  o.  ( 2nd `  f
) )  e.  _V
143140, 142op1st 6815 . . . . . . . 8  |-  ( 1st `  <. ( t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f
) ) >. )  =  ( t `  ( 1st `  f ) )
144139, 143syl6eq 2486 . . . . . . 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 5019 . . . . . 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 2480 . . . . 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 466 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  ->  D  e.  Ring )
14816adantr 466 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  ->  E  =  ( Base `  D ) )
149122, 148eleqtrd 2519 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
s  e.  ( Base `  D ) )
150123, 148eleqtrd 2519 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
t  e.  ( Base `  D ) )
151124, 82syl 17 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  f
)  e.  E )
152151, 148eleqtrd 2519 . . . . . . . 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, 19ringdir 17735 . . . . . . . 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 1266 . . . . . . 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, 17ringacl 17743 . . . . . . . . . 10  |-  ( ( D  e.  Ring  /\  s  e.  ( Base `  D
)  /\  t  e.  ( Base `  D )
)  ->  ( s  .+^  t )  e.  (
Base `  D )
)
156147, 149, 150, 155syl3anc 1264 . . . . . . . . 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 2520 . . . . . . . 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 34366 . . . . . . . 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 1262 . . . . . . 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 1262 . . . . . . . 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 34366 . . . . . . . . 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 1262 . . . . . . . 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 6323 . . . . . . 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 2478 . . . . . 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 5885 . . . . . . . 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 2486 . . . . . . 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 5885 . . . . . . . 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 6816 . . . . . . . 8  |-  ( 2nd `  <. ( t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f
) ) >. )  =  ( t  o.  ( 2nd `  f
) )
169167, 168syl6eq 2486 . . . . . . 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 6323 . . . . . 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 2473 . . . . 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 4198 . . . 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 34381 . . . . 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 1262 . . . 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 1165 . . . . 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 34383 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .x.  f
)  e.  ( T  X.  E ) )
1771763adantr1 1164 . . . . 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 34372 . . . . 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 1262 . . . 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 2480 . . 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 34045 . . . . . . 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 1269 . . . . . 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 5374 . . . . . . 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 4198 . . . . 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 34051 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  t  e.  E
)  ->  ( s  o.  t )  e.  E
)
187121, 122, 123, 186syl3anc 1264 . . . . . 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 34381 . . . . . 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 1262 . . . . 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 34046 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  t  e.  E  /\  ( 1st `  f
)  e.  T )  ->  ( t `  ( 1st `  f ) )  e.  T )
191121, 123, 125, 190syl3anc 1264 . . . . . 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 34051 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  t  e.  E  /\  ( 2nd `  f
)  e.  E )  ->  ( t  o.  ( 2nd `  f
) )  e.  E
)
193121, 123, 151, 192syl3anc 1264 . . . . . 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 34382 . . . . . 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 1266 . . . . 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 2480 . . . 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 34366 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E ) )  -> 
( s  .X.  t
)  =  ( s  o.  t ) )
1981973adantr3 1166 . . . . 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 6320 . . . 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 6321 . . . 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 2480 . . 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 6837 . . . . . . 7  |-  ( s  e.  ( T  X.  E )  ->  ( 1st `  s )  e.  T )
203202adantl 467 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( 1st `  s )  e.  T
)
204 tendospid 34297 . . . . . 6  |-  ( ( 1st `  s )  e.  T  ->  (
(  _I  |`  T ) `
 ( 1st `  s
) )  =  ( 1st `  s ) )
205203, 204syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( (  _I  |`  T ) `  ( 1st `  s ) )  =  ( 1st `  s ) )
206 xp2nd 6838 . . . . . . 7  |-  ( s  e.  ( T  X.  E )  ->  ( 2nd `  s )  e.  E )
2071, 2, 3tendof 34042 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 2nd `  s
)  e.  E )  ->  ( 2nd `  s
) : T --> T )
208206, 207sylan2 476 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( 2nd `  s ) : T --> T )
209 fcoi2 5775 . . . . . 6  |-  ( ( 2nd `  s ) : T --> T  -> 
( (  _I  |`  T )  o.  ( 2nd `  s
) )  =  ( 2nd `  s ) )
210208, 209syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( (  _I  |`  T )  o.  ( 2nd `  s
) )  =  ( 2nd `  s ) )
211205, 210opeq12d 4198 . . . 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 34048 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
213212anim1i 570 . . . . 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 34381 . . . . 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 472 . . . 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 6844 . . . . 5  |-  ( s  e.  ( T  X.  E )  ->  s  =  <. ( 1st `  s
) ,  ( 2nd `  s ) >. )
217216adantl 467 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  s  =  <. ( 1st `  s
) ,  ( 2nd `  s ) >. )
218211, 215, 2173eqtr4d 2480 . . 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 18032 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LMod )
22010islvec 18262 . 2  |-  ( U  e.  LVec  <->  ( U  e. 
LMod  /\  D  e.  DivRing ) )
221219, 28, 220sylanbrc 668 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   <.cop 4008    _I cid 4764    X. cxp 4852    |` cres 4856    o. ccom 4858   -->wf 5597   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   Basecbs 15084   +g cplusg 15152   .rcmulr 15153  Scalarcsca 15155   .scvsca 15156   0gc0g 15297   invgcminusg 16621   1rcur 17670   Ringcrg 17715   DivRingcdr 17910   LModclmod 18026   LVecclvec 18260   HLchlt 32628   LHypclh 33261   LTrncltrn 33378   TEndoctendo 34031   EDRingcedring 34032   DVecHcdvh 34358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-riotaBAD 32237
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  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 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-undef 7028  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-0g 15299  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-drng 17912  df-lmod 18028  df-lvec 18261  df-oposet 32454  df-ol 32456  df-oml 32457  df-covers 32544  df-ats 32545  df-atl 32576  df-cvlat 32600  df-hlat 32629  df-llines 32775  df-lplanes 32776  df-lvols 32777  df-lines 32778  df-psubsp 32780  df-pmap 32781  df-padd 33073  df-lhyp 33265  df-laut 33266  df-ldil 33381  df-ltrn 33382  df-trl 33437  df-tendo 34034  df-edring 34036  df-dvech 34359
This theorem is referenced by:  dvhlvec  34389
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