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Theorem dvhlveclem 31906
Description: Lemma for dvhlvec 31907. 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 2436 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
61, 2, 3, 4, 5dvhvbase 31885 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( T  X.  E ) )
76eqcomd 2441 . . 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 2436 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
151, 3, 4, 10, 14dvhbase 31881 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  D
)  =  E )
1615eqcomd 2441 . . 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 2436 . . . . . 6  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
221, 21, 4, 10dvhsca 31880 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  =  ( (
EDRing `  K ) `  W ) )
2322fveq2d 5732 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 1r `  D
)  =  ( 1r
`  ( ( EDRing `  K ) `  W
) ) )
24 eqid 2436 . . . . 5  |-  ( 1r
`  ( ( EDRing `  K ) `  W
) )  =  ( 1r `  ( (
EDRing `  K ) `  W ) )
251, 2, 21, 24erng1r 31792 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 1r `  (
( EDRing `  K ) `  W ) )  =  (  _I  |`  T ) )
2623, 25eqtr2d 2469 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  =  ( 1r `  D ) )
271, 21erngdv 31790 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( EDRing `  K
) `  W )  e.  DivRing )
2822, 27eqeltrd 2510 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  D  e.  DivRing )
29 drngrng 15842 . . . 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  =  ( inv g `  D )
3431, 1, 2, 3, 4, 10, 17, 8, 32, 33dvhgrp 31905 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  Grp )
351, 2, 3, 4, 12dvhvscacl 31901 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
) ) )  -> 
( s  .x.  t
)  e.  ( T  X.  E ) )
36353impb 1149 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  t  e.  ( T  X.  E ) )  ->  ( s  .x.  t )  e.  ( T  X.  E ) )
37 simpl 444 . . . . . . 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 963 . . . . . . 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 964 . . . . . . . 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 6376 . . . . . . . 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 965 . . . . . . . 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 6376 . . . . . . . 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 31818 . . . . . . 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 1186 . . . . . 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 31890 . . . . . . . . . 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 1116 . . . . . . . . 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 5732 . . . . . . . 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 5742 . . . . . . . . . 10  |-  ( 1st `  t )  e.  _V
51 fvex 5742 . . . . . . . . . 10  |-  ( 1st `  f )  e.  _V
5250, 51coex 5413 . . . . . . . . 9  |-  ( ( 1st `  t )  o.  ( 1st `  f
) )  e.  _V
53 ovex 6106 . . . . . . . . 9  |-  ( ( 2nd `  t ) 
.+^  ( 2nd `  f
) )  e.  _V
5452, 53op1st 6355 . . . . . . . 8  |-  ( 1st `  <. ( ( 1st `  t )  o.  ( 1st `  f ) ) ,  ( ( 2nd `  t )  .+^  ( 2nd `  f ) ) >.
)  =  ( ( 1st `  t )  o.  ( 1st `  f
) )
5549, 54syl6eq 2484 . . . . . . 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 5732 . . . . . 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 31899 . . . . . . . . . 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 1118 . . . . . . . . 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 5732 . . . . . . . 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 5742 . . . . . . . . 9  |-  ( s `
 ( 1st `  t
) )  e.  _V
61 vex 2959 . . . . . . . . . 10  |-  s  e. 
_V
62 fvex 5742 . . . . . . . . . 10  |-  ( 2nd `  t )  e.  _V
6361, 62coex 5413 . . . . . . . . 9  |-  ( s  o.  ( 2nd `  t
) )  e.  _V
6460, 63op1st 6355 . . . . . . . 8  |-  ( 1st `  <. ( s `  ( 1st `  t ) ) ,  ( s  o.  ( 2nd `  t
) ) >. )  =  ( s `  ( 1st `  t ) )
6559, 64syl6eq 2484 . . . . . . 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 31899 . . . . . . . . . 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 1117 . . . . . . . . 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 5732 . . . . . . . 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 5742 . . . . . . . . 9  |-  ( s `
 ( 1st `  f
) )  e.  _V
70 fvex 5742 . . . . . . . . . 10  |-  ( 2nd `  f )  e.  _V
7161, 70coex 5413 . . . . . . . . 9  |-  ( s  o.  ( 2nd `  f
) )  e.  _V
7269, 71op1st 6355 . . . . . . . 8  |-  ( 1st `  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )  =  ( s `  ( 1st `  f ) )
7368, 72syl6eq 2484 . . . . . . 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 5037 . . . . . 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 2478 . . . . 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 452 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  ->  D  e.  Ring )
7716adantr 452 . . . . . . . . 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 2512 . . . . . . . 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 6377 . . . . . . . . . 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 2512 . . . . . . . 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 6377 . . . . . . . . . 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 2512 . . . . . . . 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 15682 . . . . . . . 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 1186 . . . . . . 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 15691 . . . . . . . . . 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 1184 . . . . . . . . 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 2513 . . . . . . . 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 31884 . . . . . . . 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 1182 . . . . . . 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 31884 . . . . . . . . 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 1182 . . . . . . . 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 31884 . . . . . . . . 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 1182 . . . . . . . 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 6099 . . . . . . 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 2476 . . . . . 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 5732 . . . . . . . 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 6356 . . . . . . . 8  |-  ( 2nd `  <. ( ( 1st `  t )  o.  ( 1st `  f ) ) ,  ( ( 2nd `  t )  .+^  ( 2nd `  f ) ) >.
)  =  ( ( 2nd `  t ) 
.+^  ( 2nd `  f
) )
10098, 99syl6eq 2484 . . . . . . 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 5035 . . . . . 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 5732 . . . . . . . 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 6356 . . . . . . . 8  |-  ( 2nd `  <. ( s `  ( 1st `  t ) ) ,  ( s  o.  ( 2nd `  t
) ) >. )  =  ( s  o.  ( 2nd `  t
) )
104102, 103syl6eq 2484 . . . . . . 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 5732 . . . . . . . 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 6356 . . . . . . . 8  |-  ( 2nd `  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )  =  ( s  o.  ( 2nd `  f
) )
107105, 106syl6eq 2484 . . . . . . 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 6099 . . . . . 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 2478 . . . . 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 3992 . . . 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 31893 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( t  e.  ( T  X.  E
)  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .+  f
)  e.  ( T  X.  E ) )
1121113adantr1 1116 . . . . 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 31899 . . . . 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 1182 . . . 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 1118 . . . . 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 31901 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( s  .x.  f
)  e.  ( T  X.  E ) )
1171163adantr2 1117 . . . . 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 31890 . . . . 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 1182 . . . 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 2478 . . 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 444 . . . . . . 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 963 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
s  e.  E )
123 simpr2 964 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
t  e.  E )
124 simpr3 965 . . . . . . . 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 2436 . . . . . . . 8  |-  ( +g  `  ( ( EDRing `  K
) `  W )
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
)
1271, 2, 3, 21, 126erngplus2 31601 . . . . . . 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 1186 . . . . . 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 5732 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( +g  `  D
)  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
13017, 129syl5eq 2480 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
.+^  =  ( +g  `  ( ( EDRing `  K
) `  W )
) )
131130oveqd 6098 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( s  .+^  t )  =  ( s ( +g  `  ( (
EDRing `  K ) `  W ) ) t ) )
132131fveq1d 5730 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( s  .+^  t ) `  ( 1st `  f ) )  =  ( ( s ( +g  `  (
( EDRing `  K ) `  W ) ) t ) `  ( 1st `  f ) ) )
133132adantr 452 . . . . . 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 1117 . . . . . . . . 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 5732 . . . . . . . 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 2484 . . . . . . 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 31899 . . . . . . . . . 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 1116 . . . . . . . . 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 5732 . . . . . . . 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 5742 . . . . . . . . 9  |-  ( t `
 ( 1st `  f
) )  e.  _V
141 vex 2959 . . . . . . . . . 10  |-  t  e. 
_V
142141, 70coex 5413 . . . . . . . . 9  |-  ( t  o.  ( 2nd `  f
) )  e.  _V
143140, 142op1st 6355 . . . . . . . 8  |-  ( 1st `  <. ( t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f
) ) >. )  =  ( t `  ( 1st `  f ) )
144139, 143syl6eq 2484 . . . . . . 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 5037 . . . . . 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 2478 . . . . 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 452 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  ->  D  e.  Ring )
14816adantr 452 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  ->  E  =  ( Base `  D ) )
149122, 148eleqtrd 2512 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
s  e.  ( Base `  D ) )
150123, 148eleqtrd 2512 . . . . . . . 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 2512 . . . . . . . 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 15683 . . . . . . . 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 1186 . . . . . . 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 15691 . . . . . . . . . 10  |-  ( ( D  e.  Ring  /\  s  e.  ( Base `  D
)  /\  t  e.  ( Base `  D )
)  ->  ( s  .+^  t )  e.  (
Base `  D )
)
156147, 149, 150, 155syl3anc 1184 . . . . . . . . 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 2513 . . . . . . . 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 31884 . . . . . . . 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 1182 . . . . . . 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 1182 . . . . . . . 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 31884 . . . . . . . . 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 1182 . . . . . . . 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 6099 . . . . . . 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 2476 . . . . . 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 5732 . . . . . . . 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 2484 . . . . . . 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 5732 . . . . . . . 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 6356 . . . . . . . 8  |-  ( 2nd `  <. ( t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f
) ) >. )  =  ( t  o.  ( 2nd `  f
) )
169167, 168syl6eq 2484 . . . . . . 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 6099 . . . . . 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 2471 . . . . 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 3992 . . . 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 31899 . . . . 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 1182 . . . 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 1117 . . . . 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 31901 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( t  e.  E  /\  f  e.  ( T  X.  E
) ) )  -> 
( t  .x.  f
)  e.  ( T  X.  E ) )
1771763adantr1 1116 . . . . 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 31890 . . . . 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 1182 . . . 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 2478 . . 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 31563 . . . . . . 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 1189 . . . . . 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 5388 . . . . . . 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 3992 . . . . 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 31569 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  t  e.  E
)  ->  ( s  o.  t )  e.  E
)
187121, 122, 123, 186syl3anc 1184 . . . . . 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 31899 . . . . . 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 1182 . . . . 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 31564 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  t  e.  E  /\  ( 1st `  f
)  e.  T )  ->  ( t `  ( 1st `  f ) )  e.  T )
191121, 123, 125, 190syl3anc 1184 . . . . . 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 31569 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  t  e.  E  /\  ( 2nd `  f
)  e.  E )  ->  ( t  o.  ( 2nd `  f
) )  e.  E
)
193121, 123, 151, 192syl3anc 1184 . . . . . 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 31900 . . . . . 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 1186 . . . . 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 2478 . . . 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 31884 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  E  /\  t  e.  E ) )  -> 
( s  .X.  t
)  =  ( s  o.  t ) )
1981973adantr3 1118 . . . . 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 6096 . . . 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 6097 . . . 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 2478 . . 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 6376 . . . . . . 7  |-  ( s  e.  ( T  X.  E )  ->  ( 1st `  s )  e.  T )
203202adantl 453 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( 1st `  s )  e.  T
)
204 tendospid 31815 . . . . . 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 6377 . . . . . . 7  |-  ( s  e.  ( T  X.  E )  ->  ( 2nd `  s )  e.  E )
2071, 2, 3tendof 31560 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 2nd `  s
)  e.  E )  ->  ( 2nd `  s
) : T --> T )
208206, 207sylan2 461 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  ( 2nd `  s ) : T --> T )
209 fcoi2 5618 . . . . . 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 3992 . . . 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 31566 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
213212anim1i 552 . . . . 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 31899 . . . . 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 457 . . . 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 6386 . . . . 5  |-  ( s  e.  ( T  X.  E )  ->  s  =  <. ( 1st `  s
) ,  ( 2nd `  s ) >. )
217216adantl 453 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  ( T  X.  E ) )  ->  s  =  <. ( 1st `  s
) ,  ( 2nd `  s ) >. )
218211, 215, 2173eqtr4d 2478 . . 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 15956 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LMod )
22010islvec 16176 . 2  |-  ( U  e.  LVec  <->  ( U  e. 
LMod  /\  D  e.  DivRing ) )
221219, 28, 220sylanbrc 646 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LVec )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   <.cop 3817    _I cid 4493    X. cxp 4876    |` cres 4880    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   Basecbs 13469   +g cplusg 13529   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533   0gc0g 13723   inv gcminusg 14686   Ringcrg 15660   1rcur 15662   DivRingcdr 15835   LModclmod 15950   LVecclvec 16174   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   TEndoctendo 31549   EDRingcedring 31550   DVecHcdvh 31876
This theorem is referenced by:  dvhlvec  31907
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-undef 6543  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-0g 13727  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-mnd 14690  df-grp 14812  df-minusg 14813  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-dvr 15788  df-drng 15837  df-lmod 15952  df-lvec 16175  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956  df-tendo 31552  df-edring 31554  df-dvech 31877
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