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Theorem dvhlveclem 31920
Description: Lemma for dvhlvec 31921. 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  =  ( inv g `  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 2296 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
61, 2, 3, 4, 5dvhvbase 31899 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( T  X.  E ) )
76eqcomd 2301 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( T  X.  E
)  =  ( Base `  U ) )
8 dvhgrp.a . . . 4  |-  .+  =  ( +g  `  U )
98a1i 10 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .+  =  ( +g  `  U ) )
10 dvhgrp.d . . . 4  |-  D  =  (Scalar `  U )
1110a1i 10 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  (Scalar `  U ) )
12 dvhlvec.s . . . 4  |-  .x.  =  ( .s `  U )
1312a1i 10 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .x.  =  ( .s
`  U ) )
14 eqid 2296 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
151, 3, 4, 10, 14dvhbase 31895 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
1615eqcomd 2301 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E  =  ( Base `  D ) )
17 dvhgrp.p . . . 4  |-  .+^  =  ( +g  `  D )
1817a1i 10 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
.+^  =  ( +g  `  D ) )
19 dvhlvec.m . . . 4  |-  .X.  =  ( .r `  D )
2019a1i 10 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .X.  =  ( .r
`  D ) )
21 eqid 2296 . . . . . 6  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
221, 21, 4, 10dvhsca 31894 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
2322fveq2d 5545 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 1r `  D
)  =  ( 1r
`  ( ( EDRing `  K ) `  W
) ) )
24 eqid 2296 . . . . 5  |-  ( 1r
`  ( ( EDRing `  K ) `  W
) )  =  ( 1r `  ( (
EDRing `  K ) `  W ) )
251, 2, 21, 24erng1r 31806 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 1r `  (
( EDRing `  K ) `  W ) )  =  (  _I  |`  T ) )
2623, 25eqtr2d 2329 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  =  ( 1r `  D ) )
271, 21erngdv 31804 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
2822, 27eqeltrd 2370 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
29 drngrng 15535 . . . 4  |-  ( D  e.  DivRing  ->  D  e.  Ring )
3028, 29syl 15 . . 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  =  ( inv g `  D )
3431, 1, 2, 3, 4, 10, 17, 8, 32, 33dvhgrp 31919 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Grp )
351, 2, 3, 4, 12dvhvscacl 31915 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
) ) )  -> 
( s  .x.  t
)  e.  ( T  X.  E ) )
36353impb 1147 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  t  e.  ( T  X.  E ) )  ->  ( s  .x.  t )  e.  ( T  X.  E ) )
37 simpl 443 . . . . . . 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 961 . . . . . . 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 962 . . . . . . . 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 6165 . . . . . . . 8  |-  ( t  e.  ( T  X.  E )  ->  ( 1st `  t )  e.  T )
4139, 40syl 15 . . . . . . 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 963 . . . . . . . 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 6165 . . . . . . . 8  |-  ( f  e.  ( T  X.  E )  ->  ( 1st `  f )  e.  T )
4442, 43syl 15 . . . . . . 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 31832 . . . . . . 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 1184 . . . . . 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 31904 . . . . . . . . . 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 1114 . . . . . . . . 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 5545 . . . . . . . 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 5555 . . . . . . . . . 10  |-  ( 1st `  t )  e.  _V
51 fvex 5555 . . . . . . . . . 10  |-  ( 1st `  f )  e.  _V
5250, 51coex 5232 . . . . . . . . 9  |-  ( ( 1st `  t )  o.  ( 1st `  f
) )  e.  _V
53 ovex 5899 . . . . . . . . 9  |-  ( ( 2nd `  t ) 
.+^  ( 2nd `  f
) )  e.  _V
5452, 53op1st 6144 . . . . . . . 8  |-  ( 1st `  <. ( ( 1st `  t )  o.  ( 1st `  f ) ) ,  ( ( 2nd `  t )  .+^  ( 2nd `  f ) ) >.
)  =  ( ( 1st `  t )  o.  ( 1st `  f
) )
5549, 54syl6eq 2344 . . . . . . 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 5545 . . . . . 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 31913 . . . . . . . . . 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 1116 . . . . . . . . 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 5545 . . . . . . . 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 5555 . . . . . . . . 9  |-  ( s `
 ( 1st `  t
) )  e.  _V
61 vex 2804 . . . . . . . . . 10  |-  s  e. 
_V
62 fvex 5555 . . . . . . . . . 10  |-  ( 2nd `  t )  e.  _V
6361, 62coex 5232 . . . . . . . . 9  |-  ( s  o.  ( 2nd `  t
) )  e.  _V
6460, 63op1st 6144 . . . . . . . 8  |-  ( 1st `  <. ( s `  ( 1st `  t ) ) ,  ( s  o.  ( 2nd `  t
) ) >. )  =  ( s `  ( 1st `  t ) )
6559, 64syl6eq 2344 . . . . . . 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 31913 . . . . . . . . . 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 1115 . . . . . . . . 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 5545 . . . . . . . 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 5555 . . . . . . . . 9  |-  ( s `
 ( 1st `  f
) )  e.  _V
70 fvex 5555 . . . . . . . . . 10  |-  ( 2nd `  f )  e.  _V
7161, 70coex 5232 . . . . . . . . 9  |-  ( s  o.  ( 2nd `  f
) )  e.  _V
7269, 71op1st 6144 . . . . . . . 8  |-  ( 1st `  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )  =  ( s `  ( 1st `  f ) )
7368, 72syl6eq 2344 . . . . . . 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 4864 . . . . . 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 2338 . . . . 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 451 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  ->  D  e.  Ring )
7716adantr 451 . . . . . . . . 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 2372 . . . . . . . 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 6166 . . . . . . . . . 10  |-  ( t  e.  ( T  X.  E )  ->  ( 2nd `  t )  e.  E )
8039, 79syl 15 . . . . . . . . 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 2372 . . . . . . . 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 6166 . . . . . . . . . 10  |-  ( f  e.  ( T  X.  E )  ->  ( 2nd `  f )  e.  E )
8342, 82syl 15 . . . . . . . . 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 2372 . . . . . . . 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 15375 . . . . . . . 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 1184 . . . . . . 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 15384 . . . . . . . . . 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 1182 . . . . . . . . 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 2373 . . . . . . . 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 31898 . . . . . . . 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 1180 . . . . . . 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 31898 . . . . . . . . 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 1180 . . . . . . . 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 31898 . . . . . . . . 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 1180 . . . . . . . 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 5892 . . . . . . 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 2336 . . . . . 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 5545 . . . . . . . 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 6145 . . . . . . . 8  |-  ( 2nd `  <. ( ( 1st `  t )  o.  ( 1st `  f ) ) ,  ( ( 2nd `  t )  .+^  ( 2nd `  f ) ) >.
)  =  ( ( 2nd `  t ) 
.+^  ( 2nd `  f
) )
10098, 99syl6eq 2344 . . . . . . 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 4862 . . . . . 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 5545 . . . . . . . 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 6145 . . . . . . . 8  |-  ( 2nd `  <. ( s `  ( 1st `  t ) ) ,  ( s  o.  ( 2nd `  t
) ) >. )  =  ( s  o.  ( 2nd `  t
) )
104102, 103syl6eq 2344 . . . . . . 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 5545 . . . . . . . 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 6145 . . . . . . . 8  |-  ( 2nd `  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )  =  ( s  o.  ( 2nd `  f
) )
107105, 106syl6eq 2344 . . . . . . 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 5892 . . . . . 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 2338 . . . . 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 3820 . . . 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 31907 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .+  f
)  e.  ( T  X.  E ) )
1121113adantr1 1114 . . . . 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 31913 . . . . 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 1180 . . . 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 1116 . . . . 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 31915 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  f
)  e.  ( T  X.  E ) )
1171163adantr2 1115 . . . . 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 31904 . . . . 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 1180 . . . 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 2338 . . 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 443 . . . . . . 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 961 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
s  e.  E )
123 simpr2 962 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
t  e.  E )
124 simpr3 963 . . . . . . . 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 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 1st `  f
)  e.  T )
126 eqid 2296 . . . . . . . 8  |-  ( +g  `  ( ( EDRing `  K
) `  W )
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
)
1271, 2, 3, 21, 126erngplus2 31615 . . . . . . 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 1184 . . . . . 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 5545 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( +g  `  D
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
13017, 129syl5eq 2340 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
.+^  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
131130oveqd 5891 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( s  .+^  t )  =  ( s ( +g  `  ( (
EDRing `  K ) `  W ) ) t ) )
132131fveq1d 5543 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( s  .+^  t ) `  ( 1st `  f ) )  =  ( ( s ( +g  `  (
( EDRing `  K ) `  W ) ) t ) `  ( 1st `  f ) ) )
133132adantr 451 . . . . . 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 1115 . . . . . . . . 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 5545 . . . . . . . 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 2344 . . . . . . 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 31913 . . . . . . . . . 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 1114 . . . . . . . . 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 5545 . . . . . . . 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 5555 . . . . . . . . 9  |-  ( t `
 ( 1st `  f
) )  e.  _V
141 vex 2804 . . . . . . . . . 10  |-  t  e. 
_V
142141, 70coex 5232 . . . . . . . . 9  |-  ( t  o.  ( 2nd `  f
) )  e.  _V
143140, 142op1st 6144 . . . . . . . 8  |-  ( 1st `  <. ( t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f
) ) >. )  =  ( t `  ( 1st `  f ) )
144139, 143syl6eq 2344 . . . . . . 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 4864 . . . . . 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 2338 . . . . 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 451 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  ->  D  e.  Ring )
14816adantr 451 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  ->  E  =  ( Base `  D ) )
149122, 148eleqtrd 2372 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
s  e.  ( Base `  D ) )
150123, 148eleqtrd 2372 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
t  e.  ( Base `  D ) )
151124, 82syl 15 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( 2nd `  f
)  e.  E )
152151, 148eleqtrd 2372 . . . . . . . 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 15376 . . . . . . . 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 1184 . . . . . . 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 15384 . . . . . . . . . 10  |-  ( ( D  e.  Ring  /\  s  e.  ( Base `  D
)  /\  t  e.  ( Base `  D )
)  ->  ( s  .+^  t )  e.  (
Base `  D )
)
156147, 149, 150, 155syl3anc 1182 . . . . . . . . 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 2373 . . . . . . . 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 31898 . . . . . . . 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 1180 . . . . . . 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 1180 . . . . . . . 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 31898 . . . . . . . . 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 1180 . . . . . . . 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 5892 . . . . . . 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 2336 . . . . . 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 5545 . . . . . . . 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 2344 . . . . . . 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 5545 . . . . . . . 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 6145 . . . . . . . 8  |-  ( 2nd `  <. ( t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f
) ) >. )  =  ( t  o.  ( 2nd `  f
) )
169167, 168syl6eq 2344 . . . . . . 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 5892 . . . . . 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 2331 . . . . 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 3820 . . . 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 31913 . . . . 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 1180 . . . 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 1115 . . . . 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 31915 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .x.  f
)  e.  ( T  X.  E ) )
1771763adantr1 1114 . . . . 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 31904 . . . . 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 1180 . . . 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 2338 . . 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 31577 . . . . . . 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 1187 . . . . . 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 5207 . . . . . . 7  |-  ( ( s  o.  t )  o.  ( 2nd `  f
) )  =  ( s  o.  ( t  o.  ( 2nd `  f
) ) )
184183a1i 10 . . . . . 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 3820 . . . . 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 31583 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  t  e.  E
)  ->  ( s  o.  t )  e.  E
)
187121, 122, 123, 186syl3anc 1182 . . . . . 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 31913 . . . . . 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 1180 . . . . 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 31578 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  t  e.  E  /\  ( 1st `  f
)  e.  T )  ->  ( t `  ( 1st `  f ) )  e.  T )
191121, 123, 125, 190syl3anc 1182 . . . . . 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 31583 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  t  e.  E  /\  ( 2nd `  f
)  e.  E )  ->  ( t  o.  ( 2nd `  f
) )  e.  E
)
193121, 123, 151, 192syl3anc 1182 . . . . . 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 31914 . . . . . 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 1184 . . . . 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 2338 . . . 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 31898 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E ) )  -> 
( s  .X.  t
)  =  ( s  o.  t ) )
1981973adantr3 1116 . . . . 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 5889 . . . 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 5890 . . . 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 2338 . . 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 6165 . . . . . . 7  |-  ( s  e.  ( T  X.  E )  ->  ( 1st `  s )  e.  T )
203202adantl 452 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( 1st `  s )  e.  T
)
204 tendospid 31829 . . . . . 6  |-  ( ( 1st `  s )  e.  T  ->  (
(  _I  |`  T ) `
 ( 1st `  s
) )  =  ( 1st `  s ) )
205203, 204syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( (  _I  |`  T ) `  ( 1st `  s ) )  =  ( 1st `  s ) )
206 xp2nd 6166 . . . . . . 7  |-  ( s  e.  ( T  X.  E )  ->  ( 2nd `  s )  e.  E )
2071, 2, 3tendof 31574 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 2nd `  s
)  e.  E )  ->  ( 2nd `  s
) : T --> T )
208206, 207sylan2 460 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( 2nd `  s ) : T --> T )
209 fcoi2 5432 . . . . . 6  |-  ( ( 2nd `  s ) : T --> T  -> 
( (  _I  |`  T )  o.  ( 2nd `  s
) )  =  ( 2nd `  s ) )
210208, 209syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( (  _I  |`  T )  o.  ( 2nd `  s
) )  =  ( 2nd `  s ) )
211205, 210opeq12d 3820 . . . 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 31580 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
213212anim1i 551 . . . . 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 31913 . . . . 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 456 . . . 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 6175 . . . . 5  |-  ( s  e.  ( T  X.  E )  ->  s  =  <. ( 1st `  s
) ,  ( 2nd `  s ) >. )
217216adantl 452 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  s  =  <. ( 1st `  s
) ,  ( 2nd `  s ) >. )
218211, 215, 2173eqtr4d 2338 . . 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 15649 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LMod )
22010islvec 15873 . 2  |-  ( U  e.  LVec  <->  ( U  e. 
LMod  /\  D  e.  DivRing ) )
221219, 28, 220sylanbrc 645 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   <.cop 3656    _I cid 4320    X. cxp 4703    |` cres 4707    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Basecbs 13164   +g cplusg 13224   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   inv gcminusg 14379   Ringcrg 15353   1rcur 15355   DivRingcdr 15528   LModclmod 15643   LVecclvec 15871   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   TEndoctendo 31563   EDRingcedring 31564   DVecHcdvh 31890
This theorem is referenced by:  dvhlvec  31921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-undef 6314  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-lmod 15645  df-lvec 15872  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566  df-edring 31568  df-dvech 31891
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