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Theorem ditgsplit 19426
Description: This theorem is the raison d'être for the directed integral, because unlike itgspliticc 19406, there is no constraint on the ordering of the points  A ,  B ,  C in the domain. (Contributed by Mario Carneiro, 13-Aug-2014.)
Hypotheses
Ref Expression
ditgsplit.x  |-  ( ph  ->  X  e.  RR )
ditgsplit.y  |-  ( ph  ->  Y  e.  RR )
ditgsplit.a  |-  ( ph  ->  A  e.  ( X [,] Y ) )
ditgsplit.b  |-  ( ph  ->  B  e.  ( X [,] Y ) )
ditgsplit.c  |-  ( ph  ->  C  e.  ( X [,] Y ) )
ditgsplit.d  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  D  e.  V )
ditgsplit.i  |-  ( ph  ->  ( x  e.  ( X (,) Y ) 
|->  D )  e.  L ^1 )
Assertion
Ref Expression
ditgsplit  |-  ( ph  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
Distinct variable groups:    x, A    x, B    x, C    ph, x    x, V    x, X    x, Y
Allowed substitution hint:    D( x)

Proof of Theorem ditgsplit
StepHypRef Expression
1 ditgsplit.a . . . 4  |-  ( ph  ->  A  e.  ( X [,] Y ) )
2 ditgsplit.x . . . . 5  |-  ( ph  ->  X  e.  RR )
3 ditgsplit.y . . . . 5  |-  ( ph  ->  Y  e.  RR )
4 elicc2 10868 . . . . 5  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
52, 3, 4syl2anc 642 . . . 4  |-  ( ph  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
61, 5mpbid 201 . . 3  |-  ( ph  ->  ( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) )
76simp1d 968 . 2  |-  ( ph  ->  A  e.  RR )
8 ditgsplit.b . . . 4  |-  ( ph  ->  B  e.  ( X [,] Y ) )
9 elicc2 10868 . . . . 5  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
102, 3, 9syl2anc 642 . . . 4  |-  ( ph  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
118, 10mpbid 201 . . 3  |-  ( ph  ->  ( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) )
1211simp1d 968 . 2  |-  ( ph  ->  B  e.  RR )
137adantr 451 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  A  e.  RR )
14 ditgsplit.c . . . . . 6  |-  ( ph  ->  C  e.  ( X [,] Y ) )
15 elicc2 10868 . . . . . . 7  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( C  e.  ( X [,] Y )  <-> 
( C  e.  RR  /\  X  <_  C  /\  C  <_  Y ) ) )
162, 3, 15syl2anc 642 . . . . . 6  |-  ( ph  ->  ( C  e.  ( X [,] Y )  <-> 
( C  e.  RR  /\  X  <_  C  /\  C  <_  Y ) ) )
1714, 16mpbid 201 . . . . 5  |-  ( ph  ->  ( C  e.  RR  /\  X  <_  C  /\  C  <_  Y ) )
1817simp1d 968 . . . 4  |-  ( ph  ->  C  e.  RR )
1918adantr 451 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  C  e.  RR )
2012ad2antrr 706 . . . 4  |-  ( ( ( ph  /\  A  <_  B )  /\  A  <_  C )  ->  B  e.  RR )
2118ad2antrr 706 . . . 4  |-  ( ( ( ph  /\  A  <_  B )  /\  A  <_  C )  ->  C  e.  RR )
22 ditgsplit.d . . . . . 6  |-  ( (
ph  /\  x  e.  ( X (,) Y ) )  ->  D  e.  V )
23 ditgsplit.i . . . . . 6  |-  ( ph  ->  ( x  e.  ( X (,) Y ) 
|->  D )  e.  L ^1 )
24 biid 227 . . . . . 6  |-  ( ( A  <_  B  /\  B  <_  C )  <->  ( A  <_  B  /\  B  <_  C ) )
252, 3, 1, 8, 14, 22, 23, 24ditgsplitlem 19425 . . . . 5  |-  ( ( ( ph  /\  A  <_  B )  /\  B  <_  C )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
2625adantlr 695 . . . 4  |-  ( ( ( ( ph  /\  A  <_  B )  /\  A  <_  C )  /\  B  <_  C )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
27 biid 227 . . . . . . . 8  |-  ( ( A  <_  C  /\  C  <_  B )  <->  ( A  <_  C  /\  C  <_  B ) )
282, 3, 1, 14, 8, 22, 23, 27ditgsplitlem 19425 . . . . . . 7  |-  ( ( ( ph  /\  A  <_  C )  /\  C  <_  B )  ->  S__ [ A  ->  B ] D  _d x  =  ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  B ] D  _d x ) )
2928oveq1d 5996 . . . . . 6  |-  ( ( ( ph  /\  A  <_  C )  /\  C  <_  B )  ->  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x )  =  ( ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  B ] D  _d x )  +  S__ [ B  ->  C ] D  _d x ) )
302, 3, 1, 14, 22, 23ditgcl 19423 . . . . . . . . 9  |-  ( ph  ->  S__ [ A  ->  C ] D  _d x  e.  CC )
312, 3, 14, 8, 22, 23ditgcl 19423 . . . . . . . . 9  |-  ( ph  ->  S__ [ C  ->  B ] D  _d x  e.  CC )
322, 3, 8, 14, 22, 23ditgcl 19423 . . . . . . . . 9  |-  ( ph  ->  S__ [ B  ->  C ] D  _d x  e.  CC )
3330, 31, 32addassd 9004 . . . . . . . 8  |-  ( ph  ->  ( ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  B ] D  _d x )  +  S__ [ B  ->  C ] D  _d x )  =  ( S__
[ A  ->  C ] D  _d x  +  ( S__ [ C  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) ) )
342, 3, 14, 8, 22, 23ditgswap 19424 . . . . . . . . . . 11  |-  ( ph  ->  S__ [ B  ->  C ] D  _d x  =  -u S__ [ C  ->  B ] D  _d x )
3534oveq2d 5997 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x )  =  ( S__ [ C  ->  B ] D  _d x  +  -u S__ [ C  ->  B ] D  _d x ) )
3631negidd 9294 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  -u S__ [ C  ->  B ] D  _d x )  =  0 )
3735, 36eqtrd 2398 . . . . . . . . 9  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x )  =  0 )
3837oveq2d 5997 . . . . . . . 8  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  ( S__
[ C  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )  =  ( S__ [ A  ->  C ] D  _d x  +  0 ) )
3930addid1d 9159 . . . . . . . 8  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  0 )  =  S__ [ A  ->  C ] D  _d x )
4033, 38, 393eqtrd 2402 . . . . . . 7  |-  ( ph  ->  ( ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  B ] D  _d x )  +  S__ [ B  ->  C ] D  _d x )  =  S__ [ A  ->  C ] D  _d x )
4140ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  A  <_  C )  /\  C  <_  B )  ->  (
( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  B ] D  _d x )  +  S__ [ B  ->  C ] D  _d x )  =  S__ [ A  ->  C ] D  _d x )
4229, 41eqtr2d 2399 . . . . 5  |-  ( ( ( ph  /\  A  <_  C )  /\  C  <_  B )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
4342adantllr 699 . . . 4  |-  ( ( ( ( ph  /\  A  <_  B )  /\  A  <_  C )  /\  C  <_  B )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
4420, 21, 26, 43lecasei 9073 . . 3  |-  ( ( ( ph  /\  A  <_  B )  /\  A  <_  C )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
4540ad2antrr 706 . . . 4  |-  ( ( ( ph  /\  A  <_  B )  /\  C  <_  A )  ->  (
( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  B ] D  _d x )  +  S__ [ B  ->  C ] D  _d x )  =  S__ [ A  ->  C ] D  _d x )
46 ancom 437 . . . . . . . 8  |-  ( ( A  <_  B  /\  C  <_  A )  <->  ( C  <_  A  /\  A  <_  B ) )
472, 3, 14, 1, 8, 22, 23, 46ditgsplitlem 19425 . . . . . . 7  |-  ( ( ( ph  /\  A  <_  B )  /\  C  <_  A )  ->  S__ [ C  ->  B ] D  _d x  =  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x ) )
4847oveq2d 5997 . . . . . 6  |-  ( ( ( ph  /\  A  <_  B )  /\  C  <_  A )  ->  ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  B ] D  _d x )  =  ( S__
[ A  ->  C ] D  _d x  +  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x ) ) )
492, 3, 1, 14, 22, 23ditgswap 19424 . . . . . . . . . . 11  |-  ( ph  ->  S__ [ C  ->  A ] D  _d x  =  -u S__ [ A  ->  C ] D  _d x )
5049oveq2d 5997 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  A ] D  _d x )  =  ( S__ [ A  ->  C ] D  _d x  +  -u S__ [ A  ->  C ] D  _d x ) )
5130negidd 9294 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  -u S__ [ A  ->  C ] D  _d x )  =  0 )
5250, 51eqtrd 2398 . . . . . . . . 9  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  A ] D  _d x )  =  0 )
5352oveq1d 5996 . . . . . . . 8  |-  ( ph  ->  ( ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  A ] D  _d x )  +  S__ [ A  ->  B ] D  _d x )  =  ( 0  +  S__ [ A  ->  B ] D  _d x ) )
542, 3, 14, 1, 22, 23ditgcl 19423 . . . . . . . . 9  |-  ( ph  ->  S__ [ C  ->  A ] D  _d x  e.  CC )
552, 3, 1, 8, 22, 23ditgcl 19423 . . . . . . . . 9  |-  ( ph  ->  S__ [ A  ->  B ] D  _d x  e.  CC )
5630, 54, 55addassd 9004 . . . . . . . 8  |-  ( ph  ->  ( ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  A ] D  _d x )  +  S__ [ A  ->  B ] D  _d x )  =  ( S__
[ A  ->  C ] D  _d x  +  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x ) ) )
5755addid2d 9160 . . . . . . . 8  |-  ( ph  ->  ( 0  +  S__ [ A  ->  B ] D  _d x )  =  S__ [ A  ->  B ] D  _d x )
5853, 56, 573eqtr3d 2406 . . . . . . 7  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  ( S__
[ C  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x ) )  =  S__
[ A  ->  B ] D  _d x
)
5958ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  A  <_  B )  /\  C  <_  A )  ->  ( S__ [ A  ->  C ] D  _d x  +  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x ) )  =  S__ [ A  ->  B ] D  _d x )
6048, 59eqtrd 2398 . . . . 5  |-  ( ( ( ph  /\  A  <_  B )  /\  C  <_  A )  ->  ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  B ] D  _d x )  =  S__ [ A  ->  B ] D  _d x )
6160oveq1d 5996 . . . 4  |-  ( ( ( ph  /\  A  <_  B )  /\  C  <_  A )  ->  (
( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  B ] D  _d x )  +  S__ [ B  ->  C ] D  _d x )  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
6245, 61eqtr3d 2400 . . 3  |-  ( ( ( ph  /\  A  <_  B )  /\  C  <_  A )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
6313, 19, 44, 62lecasei 9073 . 2  |-  ( (
ph  /\  A  <_  B )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__
[ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
647adantr 451 . . 3  |-  ( (
ph  /\  B  <_  A )  ->  A  e.  RR )
6518adantr 451 . . 3  |-  ( (
ph  /\  B  <_  A )  ->  C  e.  RR )
66 biid 227 . . . . . 6  |-  ( ( B  <_  A  /\  A  <_  C )  <->  ( B  <_  A  /\  A  <_  C ) )
672, 3, 8, 1, 14, 22, 23, 66ditgsplitlem 19425 . . . . 5  |-  ( ( ( ph  /\  B  <_  A )  /\  A  <_  C )  ->  S__ [ B  ->  C ] D  _d x  =  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x ) )
6867oveq2d 5997 . . . 4  |-  ( ( ( ph  /\  B  <_  A )  /\  A  <_  C )  ->  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x )  =  ( S__
[ A  ->  B ] D  _d x  +  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x ) ) )
692, 3, 1, 8, 22, 23ditgswap 19424 . . . . . . . . 9  |-  ( ph  ->  S__ [ B  ->  A ] D  _d x  =  -u S__ [ A  ->  B ] D  _d x )
7069oveq2d 5997 . . . . . . . 8  |-  ( ph  ->  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  A ] D  _d x )  =  ( S__ [ A  ->  B ] D  _d x  +  -u S__ [ A  ->  B ] D  _d x ) )
7155negidd 9294 . . . . . . . 8  |-  ( ph  ->  ( S__ [ A  ->  B ] D  _d x  +  -u S__ [ A  ->  B ] D  _d x )  =  0 )
7270, 71eqtrd 2398 . . . . . . 7  |-  ( ph  ->  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  A ] D  _d x )  =  0 )
7372oveq1d 5996 . . . . . 6  |-  ( ph  ->  ( ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  A ] D  _d x )  +  S__ [ A  ->  C ] D  _d x )  =  ( 0  +  S__ [ A  ->  C ] D  _d x ) )
742, 3, 8, 1, 22, 23ditgcl 19423 . . . . . . 7  |-  ( ph  ->  S__ [ B  ->  A ] D  _d x  e.  CC )
7555, 74, 30addassd 9004 . . . . . 6  |-  ( ph  ->  ( ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  A ] D  _d x )  +  S__ [ A  ->  C ] D  _d x )  =  ( S__
[ A  ->  B ] D  _d x  +  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x ) ) )
7630addid2d 9160 . . . . . 6  |-  ( ph  ->  ( 0  +  S__ [ A  ->  C ] D  _d x )  =  S__ [ A  ->  C ] D  _d x )
7773, 75, 763eqtr3d 2406 . . . . 5  |-  ( ph  ->  ( S__ [ A  ->  B ] D  _d x  +  ( S__
[ B  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x ) )  =  S__
[ A  ->  C ] D  _d x
)
7877ad2antrr 706 . . . 4  |-  ( ( ( ph  /\  B  <_  A )  /\  A  <_  C )  ->  ( S__ [ A  ->  B ] D  _d x  +  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x ) )  =  S__ [ A  ->  C ] D  _d x )
7968, 78eqtr2d 2399 . . 3  |-  ( ( ( ph  /\  B  <_  A )  /\  A  <_  C )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
8012ad2antrr 706 . . . 4  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  A )  ->  B  e.  RR )
8118ad2antrr 706 . . . 4  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  A )  ->  C  e.  RR )
82 ancom 437 . . . . . . . . . 10  |-  ( ( C  <_  A  /\  B  <_  C )  <->  ( B  <_  C  /\  C  <_  A ) )
832, 3, 8, 14, 1, 22, 23, 82ditgsplitlem 19425 . . . . . . . . 9  |-  ( ( ( ph  /\  C  <_  A )  /\  B  <_  C )  ->  S__ [ B  ->  A ] D  _d x  =  ( S__ [ B  ->  C ] D  _d x  +  S__ [ C  ->  A ] D  _d x ) )
8483oveq1d 5996 . . . . . . . 8  |-  ( ( ( ph  /\  C  <_  A )  /\  B  <_  C )  ->  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x )  =  ( ( S__ [ B  ->  C ] D  _d x  +  S__ [ C  ->  A ] D  _d x )  +  S__ [ A  ->  C ] D  _d x ) )
8532, 54, 30addassd 9004 . . . . . . . . . 10  |-  ( ph  ->  ( ( S__ [ B  ->  C ] D  _d x  +  S__ [ C  ->  A ] D  _d x )  +  S__ [ A  ->  C ] D  _d x )  =  ( S__
[ B  ->  C ] D  _d x  +  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x ) ) )
862, 3, 14, 1, 22, 23ditgswap 19424 . . . . . . . . . . . . 13  |-  ( ph  ->  S__ [ A  ->  C ] D  _d x  =  -u S__ [ C  ->  A ] D  _d x )
8786oveq2d 5997 . . . . . . . . . . . 12  |-  ( ph  ->  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x )  =  ( S__ [ C  ->  A ] D  _d x  +  -u S__ [ C  ->  A ] D  _d x ) )
8854negidd 9294 . . . . . . . . . . . 12  |-  ( ph  ->  ( S__ [ C  ->  A ] D  _d x  +  -u S__ [ C  ->  A ] D  _d x )  =  0 )
8987, 88eqtrd 2398 . . . . . . . . . . 11  |-  ( ph  ->  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x )  =  0 )
9089oveq2d 5997 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ B  ->  C ] D  _d x  +  ( S__
[ C  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x ) )  =  ( S__ [ B  ->  C ] D  _d x  +  0 ) )
9132addid1d 9159 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ B  ->  C ] D  _d x  +  0 )  =  S__ [ B  ->  C ] D  _d x )
9285, 90, 913eqtrd 2402 . . . . . . . . 9  |-  ( ph  ->  ( ( S__ [ B  ->  C ] D  _d x  +  S__ [ C  ->  A ] D  _d x )  +  S__ [ A  ->  C ] D  _d x )  =  S__ [ B  ->  C ] D  _d x )
9392ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  C  <_  A )  /\  B  <_  C )  ->  (
( S__ [ B  ->  C ] D  _d x  +  S__ [ C  ->  A ] D  _d x )  +  S__ [ A  ->  C ] D  _d x )  =  S__ [ B  ->  C ] D  _d x )
9484, 93eqtr2d 2399 . . . . . . 7  |-  ( ( ( ph  /\  C  <_  A )  /\  B  <_  C )  ->  S__ [ B  ->  C ] D  _d x  =  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x ) )
9594oveq2d 5997 . . . . . 6  |-  ( ( ( ph  /\  C  <_  A )  /\  B  <_  C )  ->  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x )  =  ( S__
[ A  ->  B ] D  _d x  +  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x ) ) )
9677ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  C  <_  A )  /\  B  <_  C )  ->  ( S__ [ A  ->  B ] D  _d x  +  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x ) )  =  S__ [ A  ->  C ] D  _d x )
9795, 96eqtr2d 2399 . . . . 5  |-  ( ( ( ph  /\  C  <_  A )  /\  B  <_  C )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
9897adantllr 699 . . . 4  |-  ( ( ( ( ph  /\  B  <_  A )  /\  C  <_  A )  /\  B  <_  C )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
99 ancom 437 . . . . . . . . . . . 12  |-  ( ( B  <_  A  /\  C  <_  B )  <->  ( C  <_  B  /\  B  <_  A ) )
1002, 3, 14, 8, 1, 22, 23, 99ditgsplitlem 19425 . . . . . . . . . . 11  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  B )  ->  S__ [ C  ->  A ] D  _d x  =  ( S__ [ C  ->  B ] D  _d x  +  S__ [ B  ->  A ] D  _d x ) )
101100oveq1d 5996 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  B )  ->  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x )  =  ( ( S__ [ C  ->  B ] D  _d x  +  S__ [ B  ->  A ] D  _d x )  +  S__ [ A  ->  B ] D  _d x ) )
10231, 74, 55addassd 9004 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( S__ [ C  ->  B ] D  _d x  +  S__ [ B  ->  A ] D  _d x )  +  S__ [ A  ->  B ] D  _d x )  =  ( S__
[ C  ->  B ] D  _d x  +  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x ) ) )
1032, 3, 8, 1, 22, 23ditgswap 19424 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S__ [ A  ->  B ] D  _d x  =  -u S__ [ B  ->  A ] D  _d x )
104103oveq2d 5997 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x )  =  ( S__ [ B  ->  A ] D  _d x  +  -u S__ [ B  ->  A ] D  _d x ) )
10574negidd 9294 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S__ [ B  ->  A ] D  _d x  +  -u S__ [ B  ->  A ] D  _d x )  =  0 )
106104, 105eqtrd 2398 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x )  =  0 )
107106oveq2d 5997 . . . . . . . . . . . 12  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  ( S__
[ B  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x ) )  =  ( S__ [ C  ->  B ] D  _d x  +  0 ) )
10831addid1d 9159 . . . . . . . . . . . 12  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  0 )  =  S__ [ C  ->  B ] D  _d x )
109102, 107, 1083eqtrd 2402 . . . . . . . . . . 11  |-  ( ph  ->  ( ( S__ [ C  ->  B ] D  _d x  +  S__ [ B  ->  A ] D  _d x )  +  S__ [ A  ->  B ] D  _d x )  =  S__ [ C  ->  B ] D  _d x )
110109ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  B )  ->  (
( S__ [ C  ->  B ] D  _d x  +  S__ [ B  ->  A ] D  _d x )  +  S__ [ A  ->  B ] D  _d x )  =  S__ [ C  ->  B ] D  _d x )
111101, 110eqtr2d 2399 . . . . . . . . 9  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  B )  ->  S__ [ C  ->  B ] D  _d x  =  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x ) )
112111oveq2d 5997 . . . . . . . 8  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  B )  ->  ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  B ] D  _d x )  =  ( S__
[ A  ->  C ] D  _d x  +  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x ) ) )
11358ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  B )  ->  ( S__ [ A  ->  C ] D  _d x  +  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x ) )  =  S__ [ A  ->  B ] D  _d x )
114112, 113eqtr2d 2399 . . . . . . 7  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  B )  ->  S__ [ A  ->  B ] D  _d x  =  ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  B ] D  _d x ) )
115114oveq1d 5996 . . . . . 6  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  B )  ->  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x )  =  ( ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  B ] D  _d x )  +  S__ [ B  ->  C ] D  _d x ) )
11640ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  B )  ->  (
( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  B ] D  _d x )  +  S__ [ B  ->  C ] D  _d x )  =  S__ [ A  ->  C ] D  _d x )
117115, 116eqtr2d 2399 . . . . 5  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  B )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
118117adantlr 695 . . . 4  |-  ( ( ( ( ph  /\  B  <_  A )  /\  C  <_  A )  /\  C  <_  B )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
11980, 81, 98, 118lecasei 9073 . . 3  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  A )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
12064, 65, 79, 119lecasei 9073 . 2  |-  ( (
ph  /\  B  <_  A )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__
[ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
1217, 12, 63, 120lecasei 9073 1  |-  ( ph  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   class class class wbr 4125    e. cmpt 4179  (class class class)co 5981   RRcr 8883   0cc0 8884    + caddc 8887    <_ cle 9015   -ucneg 9185   (,)cioo 10809   [,]cicc 10812   L ^1cibl 19187   S__cdit 19189
This theorem is referenced by:  itgsubstlem  19610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-addf 8963
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-disj 4096  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-ofr 6206  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-fi 7312  df-sup 7341  df-oi 7372  df-card 7719  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-n0 10115  df-z 10176  df-uz 10382  df-q 10468  df-rp 10506  df-xneg 10603  df-xadd 10604  df-xmul 10605  df-ioo 10813  df-ico 10815  df-icc 10816  df-fz 10936  df-fzo 11026  df-fl 11089  df-mod 11138  df-seq 11211  df-exp 11270  df-hash 11506  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-clim 12169  df-rlim 12170  df-sum 12367  df-rest 13537  df-topgen 13554  df-xmet 16586  df-met 16587  df-bl 16588  df-mopn 16589  df-top 16853  df-bases 16855  df-topon 16856  df-cmp 17331  df-ovol 19039  df-vol 19040  df-mbf 19190  df-itg1 19191  df-itg2 19192  df-ibl 19193  df-itg 19194  df-ditg 19195  df-0p 19240
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