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Theorem ditgsplit 21178
Description: This theorem is the raison d'être for the directed integral, because unlike itgspliticc 21156, 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 11348 . . . . 5  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
52, 3, 4syl2anc 654 . . . 4  |-  ( ph  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
61, 5mpbid 210 . . 3  |-  ( ph  ->  ( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) )
76simp1d 993 . 2  |-  ( ph  ->  A  e.  RR )
8 ditgsplit.b . . . 4  |-  ( ph  ->  B  e.  ( X [,] Y ) )
9 elicc2 11348 . . . . 5  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
102, 3, 9syl2anc 654 . . . 4  |-  ( ph  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
118, 10mpbid 210 . . 3  |-  ( ph  ->  ( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) )
1211simp1d 993 . 2  |-  ( ph  ->  B  e.  RR )
137adantr 462 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  A  e.  RR )
14 ditgsplit.c . . . . . 6  |-  ( ph  ->  C  e.  ( X [,] Y ) )
15 elicc2 11348 . . . . . . 7  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( C  e.  ( X [,] Y )  <-> 
( C  e.  RR  /\  X  <_  C  /\  C  <_  Y ) ) )
162, 3, 15syl2anc 654 . . . . . 6  |-  ( ph  ->  ( C  e.  ( X [,] Y )  <-> 
( C  e.  RR  /\  X  <_  C  /\  C  <_  Y ) ) )
1714, 16mpbid 210 . . . . 5  |-  ( ph  ->  ( C  e.  RR  /\  X  <_  C  /\  C  <_  Y ) )
1817simp1d 993 . . . 4  |-  ( ph  ->  C  e.  RR )
1918adantr 462 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  C  e.  RR )
2012ad2antrr 718 . . . 4  |-  ( ( ( ph  /\  A  <_  B )  /\  A  <_  C )  ->  B  e.  RR )
2118ad2antrr 718 . . . 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 236 . . . . . 6  |-  ( ( A  <_  B  /\  B  <_  C )  <->  ( A  <_  B  /\  B  <_  C ) )
252, 3, 1, 8, 14, 22, 23, 24ditgsplitlem 21177 . . . . 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 707 . . . 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 236 . . . . . . . 8  |-  ( ( A  <_  C  /\  C  <_  B )  <->  ( A  <_  C  /\  C  <_  B ) )
282, 3, 1, 14, 8, 22, 23, 27ditgsplitlem 21177 . . . . . . 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 6095 . . . . . 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 21175 . . . . . . . . 9  |-  ( ph  ->  S__ [ A  ->  C ] D  _d x  e.  CC )
312, 3, 14, 8, 22, 23ditgcl 21175 . . . . . . . . 9  |-  ( ph  ->  S__ [ C  ->  B ] D  _d x  e.  CC )
322, 3, 8, 14, 22, 23ditgcl 21175 . . . . . . . . 9  |-  ( ph  ->  S__ [ B  ->  C ] D  _d x  e.  CC )
3330, 31, 32addassd 9396 . . . . . . . 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 21176 . . . . . . . . . . 11  |-  ( ph  ->  S__ [ B  ->  C ] D  _d x  =  -u S__ [ C  ->  B ] D  _d x )
3534oveq2d 6096 . . . . . . . . . 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 9697 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  -u S__ [ C  ->  B ] D  _d x )  =  0 )
3735, 36eqtrd 2465 . . . . . . . . 9  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x )  =  0 )
3837oveq2d 6096 . . . . . . . 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 9557 . . . . . . . 8  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  0 )  =  S__ [ A  ->  C ] D  _d x )
4033, 38, 393eqtrd 2469 . . . . . . 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 718 . . . . . 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 2466 . . . . 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 711 . . . 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 9468 . . 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 718 . . . 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 448 . . . . . . . 8  |-  ( ( A  <_  B  /\  C  <_  A )  <->  ( C  <_  A  /\  A  <_  B ) )
472, 3, 14, 1, 8, 22, 23, 46ditgsplitlem 21177 . . . . . . 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 6096 . . . . . 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 21176 . . . . . . . . . . 11  |-  ( ph  ->  S__ [ C  ->  A ] D  _d x  =  -u S__ [ A  ->  C ] D  _d x )
5049oveq2d 6096 . . . . . . . . . 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 9697 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  -u S__ [ A  ->  C ] D  _d x )  =  0 )
5250, 51eqtrd 2465 . . . . . . . . 9  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  A ] D  _d x )  =  0 )
5352oveq1d 6095 . . . . . . . 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 21175 . . . . . . . . 9  |-  ( ph  ->  S__ [ C  ->  A ] D  _d x  e.  CC )
552, 3, 1, 8, 22, 23ditgcl 21175 . . . . . . . . 9  |-  ( ph  ->  S__ [ A  ->  B ] D  _d x  e.  CC )
5630, 54, 55addassd 9396 . . . . . . . 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 9558 . . . . . . . 8  |-  ( ph  ->  ( 0  +  S__ [ A  ->  B ] D  _d x )  =  S__ [ A  ->  B ] D  _d x )
5853, 56, 573eqtr3d 2473 . . . . . . 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 718 . . . . . 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 2465 . . . . 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 6095 . . . 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 2467 . . 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 9468 . 2  |-  ( (
ph  /\  A  <_  B )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__
[ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
647adantr 462 . . 3  |-  ( (
ph  /\  B  <_  A )  ->  A  e.  RR )
6518adantr 462 . . 3  |-  ( (
ph  /\  B  <_  A )  ->  C  e.  RR )
66 biid 236 . . . . . 6  |-  ( ( B  <_  A  /\  A  <_  C )  <->  ( B  <_  A  /\  A  <_  C ) )
672, 3, 8, 1, 14, 22, 23, 66ditgsplitlem 21177 . . . . 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 6096 . . . 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 21176 . . . . . . . . 9  |-  ( ph  ->  S__ [ B  ->  A ] D  _d x  =  -u S__ [ A  ->  B ] D  _d x )
7069oveq2d 6096 . . . . . . . 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 9697 . . . . . . . 8  |-  ( ph  ->  ( S__ [ A  ->  B ] D  _d x  +  -u S__ [ A  ->  B ] D  _d x )  =  0 )
7270, 71eqtrd 2465 . . . . . . 7  |-  ( ph  ->  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  A ] D  _d x )  =  0 )
7372oveq1d 6095 . . . . . 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 21175 . . . . . . 7  |-  ( ph  ->  S__ [ B  ->  A ] D  _d x  e.  CC )
7555, 74, 30addassd 9396 . . . . . 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 9558 . . . . . 6  |-  ( ph  ->  ( 0  +  S__ [ A  ->  C ] D  _d x )  =  S__ [ A  ->  C ] D  _d x )
7773, 75, 763eqtr3d 2473 . . . . 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 718 . . . 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 2466 . . 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 718 . . . 4  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  A )  ->  B  e.  RR )
8118ad2antrr 718 . . . 4  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  A )  ->  C  e.  RR )
82 ancom 448 . . . . . . . . . 10  |-  ( ( C  <_  A  /\  B  <_  C )  <->  ( B  <_  C  /\  C  <_  A ) )
832, 3, 8, 14, 1, 22, 23, 82ditgsplitlem 21177 . . . . . . . . 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 6095 . . . . . . . 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 9396 . . . . . . . . . 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 21176 . . . . . . . . . . . . 13  |-  ( ph  ->  S__ [ A  ->  C ] D  _d x  =  -u S__ [ C  ->  A ] D  _d x )
8786oveq2d 6096 . . . . . . . . . . . 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 9697 . . . . . . . . . . . 12  |-  ( ph  ->  ( S__ [ C  ->  A ] D  _d x  +  -u S__ [ C  ->  A ] D  _d x )  =  0 )
8987, 88eqtrd 2465 . . . . . . . . . . 11  |-  ( ph  ->  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x )  =  0 )
9089oveq2d 6096 . . . . . . . . . 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 9557 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ B  ->  C ] D  _d x  +  0 )  =  S__ [ B  ->  C ] D  _d x )
9285, 90, 913eqtrd 2469 . . . . . . . . 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 718 . . . . . . . 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 2466 . . . . . . 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 6096 . . . . . 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 718 . . . . . 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 2466 . . . . 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 711 . . . 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 448 . . . . . . . . . . . 12  |-  ( ( B  <_  A  /\  C  <_  B )  <->  ( C  <_  B  /\  B  <_  A ) )
1002, 3, 14, 8, 1, 22, 23, 99ditgsplitlem 21177 . . . . . . . . . . 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 6095 . . . . . . . . . 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 9396 . . . . . . . . . . . 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 21176 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S__ [ A  ->  B ] D  _d x  =  -u S__ [ B  ->  A ] D  _d x )
104103oveq2d 6096 . . . . . . . . . . . . . 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 9697 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S__ [ B  ->  A ] D  _d x  +  -u S__ [ B  ->  A ] D  _d x )  =  0 )
106104, 105eqtrd 2465 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x )  =  0 )
107106oveq2d 6096 . . . . . . . . . . . 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 9557 . . . . . . . . . . . 12  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  0 )  =  S__ [ C  ->  B ] D  _d x )
109102, 107, 1083eqtrd 2469 . . . . . . . . . . 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 718 . . . . . . . . . 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 2466 . . . . . . . . 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 6096 . . . . . . . 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 718 . . . . . . . 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 2466 . . . . . . 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 6095 . . . . . 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 718 . . . . . 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 2466 . . . . 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 707 . . . 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 9468 . . 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 9468 . 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 9468 1  |-  ( ph  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755   class class class wbr 4280    e. cmpt 4338  (class class class)co 6080   RRcr 9269   0cc0 9270    + caddc 9273    <_ cle 9407   -ucneg 9584   (,)cioo 11288   [,]cicc 11291   L^1cibl 20939   S__cdit 21163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-disj 4251  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-ofr 6310  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-n0 10568  df-z 10635  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-rlim 12951  df-sum 13148  df-rest 14344  df-topgen 14365  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-top 18345  df-bases 18347  df-topon 18348  df-cmp 18832  df-ovol 20790  df-vol 20791  df-mbf 20941  df-itg1 20942  df-itg2 20943  df-ibl 20944  df-itg 20945  df-0p 20990  df-ditg 21164
This theorem is referenced by:  itgsubstlem  21362
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