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Theorem ditgsplit 19753
Description: This theorem is the raison d'être for the directed integral, because unlike itgspliticc 19731, 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 10980 . . . . 5  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
52, 3, 4syl2anc 644 . . . 4  |-  ( ph  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
61, 5mpbid 203 . . 3  |-  ( ph  ->  ( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) )
76simp1d 970 . 2  |-  ( ph  ->  A  e.  RR )
8 ditgsplit.b . . . 4  |-  ( ph  ->  B  e.  ( X [,] Y ) )
9 elicc2 10980 . . . . 5  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
102, 3, 9syl2anc 644 . . . 4  |-  ( ph  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
118, 10mpbid 203 . . 3  |-  ( ph  ->  ( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) )
1211simp1d 970 . 2  |-  ( ph  ->  B  e.  RR )
137adantr 453 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  A  e.  RR )
14 ditgsplit.c . . . . . 6  |-  ( ph  ->  C  e.  ( X [,] Y ) )
15 elicc2 10980 . . . . . . 7  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( C  e.  ( X [,] Y )  <-> 
( C  e.  RR  /\  X  <_  C  /\  C  <_  Y ) ) )
162, 3, 15syl2anc 644 . . . . . 6  |-  ( ph  ->  ( C  e.  ( X [,] Y )  <-> 
( C  e.  RR  /\  X  <_  C  /\  C  <_  Y ) ) )
1714, 16mpbid 203 . . . . 5  |-  ( ph  ->  ( C  e.  RR  /\  X  <_  C  /\  C  <_  Y ) )
1817simp1d 970 . . . 4  |-  ( ph  ->  C  e.  RR )
1918adantr 453 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  C  e.  RR )
2012ad2antrr 708 . . . 4  |-  ( ( ( ph  /\  A  <_  B )  /\  A  <_  C )  ->  B  e.  RR )
2118ad2antrr 708 . . . 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 229 . . . . . 6  |-  ( ( A  <_  B  /\  B  <_  C )  <->  ( A  <_  B  /\  B  <_  C ) )
252, 3, 1, 8, 14, 22, 23, 24ditgsplitlem 19752 . . . . 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 697 . . . 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 229 . . . . . . . 8  |-  ( ( A  <_  C  /\  C  <_  B )  <->  ( A  <_  C  /\  C  <_  B ) )
282, 3, 1, 14, 8, 22, 23, 27ditgsplitlem 19752 . . . . . . 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 6099 . . . . . 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 19750 . . . . . . . . 9  |-  ( ph  ->  S__ [ A  ->  C ] D  _d x  e.  CC )
312, 3, 14, 8, 22, 23ditgcl 19750 . . . . . . . . 9  |-  ( ph  ->  S__ [ C  ->  B ] D  _d x  e.  CC )
322, 3, 8, 14, 22, 23ditgcl 19750 . . . . . . . . 9  |-  ( ph  ->  S__ [ B  ->  C ] D  _d x  e.  CC )
3330, 31, 32addassd 9115 . . . . . . . 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 19751 . . . . . . . . . . 11  |-  ( ph  ->  S__ [ B  ->  C ] D  _d x  =  -u S__ [ C  ->  B ] D  _d x )
3534oveq2d 6100 . . . . . . . . . 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 9406 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  -u S__ [ C  ->  B ] D  _d x )  =  0 )
3735, 36eqtrd 2470 . . . . . . . . 9  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x )  =  0 )
3837oveq2d 6100 . . . . . . . 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 9271 . . . . . . . 8  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  0 )  =  S__ [ A  ->  C ] D  _d x )
4033, 38, 393eqtrd 2474 . . . . . . 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 708 . . . . . 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 2471 . . . . 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 701 . . . 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 9184 . . 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 708 . . . 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 439 . . . . . . . 8  |-  ( ( A  <_  B  /\  C  <_  A )  <->  ( C  <_  A  /\  A  <_  B ) )
472, 3, 14, 1, 8, 22, 23, 46ditgsplitlem 19752 . . . . . . 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 6100 . . . . . 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 19751 . . . . . . . . . . 11  |-  ( ph  ->  S__ [ C  ->  A ] D  _d x  =  -u S__ [ A  ->  C ] D  _d x )
5049oveq2d 6100 . . . . . . . . . 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 9406 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  -u S__ [ A  ->  C ] D  _d x )  =  0 )
5250, 51eqtrd 2470 . . . . . . . . 9  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  A ] D  _d x )  =  0 )
5352oveq1d 6099 . . . . . . . 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 19750 . . . . . . . . 9  |-  ( ph  ->  S__ [ C  ->  A ] D  _d x  e.  CC )
552, 3, 1, 8, 22, 23ditgcl 19750 . . . . . . . . 9  |-  ( ph  ->  S__ [ A  ->  B ] D  _d x  e.  CC )
5630, 54, 55addassd 9115 . . . . . . . 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 9272 . . . . . . . 8  |-  ( ph  ->  ( 0  +  S__ [ A  ->  B ] D  _d x )  =  S__ [ A  ->  B ] D  _d x )
5853, 56, 573eqtr3d 2478 . . . . . . 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 708 . . . . . 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 2470 . . . . 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 6099 . . . 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 2472 . . 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 9184 . 2  |-  ( (
ph  /\  A  <_  B )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__
[ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
647adantr 453 . . 3  |-  ( (
ph  /\  B  <_  A )  ->  A  e.  RR )
6518adantr 453 . . 3  |-  ( (
ph  /\  B  <_  A )  ->  C  e.  RR )
66 biid 229 . . . . . 6  |-  ( ( B  <_  A  /\  A  <_  C )  <->  ( B  <_  A  /\  A  <_  C ) )
672, 3, 8, 1, 14, 22, 23, 66ditgsplitlem 19752 . . . . 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 6100 . . . 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 19751 . . . . . . . . 9  |-  ( ph  ->  S__ [ B  ->  A ] D  _d x  =  -u S__ [ A  ->  B ] D  _d x )
7069oveq2d 6100 . . . . . . . 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 9406 . . . . . . . 8  |-  ( ph  ->  ( S__ [ A  ->  B ] D  _d x  +  -u S__ [ A  ->  B ] D  _d x )  =  0 )
7270, 71eqtrd 2470 . . . . . . 7  |-  ( ph  ->  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  A ] D  _d x )  =  0 )
7372oveq1d 6099 . . . . . 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 19750 . . . . . . 7  |-  ( ph  ->  S__ [ B  ->  A ] D  _d x  e.  CC )
7555, 74, 30addassd 9115 . . . . . 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 9272 . . . . . 6  |-  ( ph  ->  ( 0  +  S__ [ A  ->  C ] D  _d x )  =  S__ [ A  ->  C ] D  _d x )
7773, 75, 763eqtr3d 2478 . . . . 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 708 . . . 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 2471 . . 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 708 . . . 4  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  A )  ->  B  e.  RR )
8118ad2antrr 708 . . . 4  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  A )  ->  C  e.  RR )
82 ancom 439 . . . . . . . . . 10  |-  ( ( C  <_  A  /\  B  <_  C )  <->  ( B  <_  C  /\  C  <_  A ) )
832, 3, 8, 14, 1, 22, 23, 82ditgsplitlem 19752 . . . . . . . . 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 6099 . . . . . . . 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 9115 . . . . . . . . . 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 19751 . . . . . . . . . . . . 13  |-  ( ph  ->  S__ [ A  ->  C ] D  _d x  =  -u S__ [ C  ->  A ] D  _d x )
8786oveq2d 6100 . . . . . . . . . . . 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 9406 . . . . . . . . . . . 12  |-  ( ph  ->  ( S__ [ C  ->  A ] D  _d x  +  -u S__ [ C  ->  A ] D  _d x )  =  0 )
8987, 88eqtrd 2470 . . . . . . . . . . 11  |-  ( ph  ->  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x )  =  0 )
9089oveq2d 6100 . . . . . . . . . 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 9271 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ B  ->  C ] D  _d x  +  0 )  =  S__ [ B  ->  C ] D  _d x )
9285, 90, 913eqtrd 2474 . . . . . . . . 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 708 . . . . . . . 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 2471 . . . . . . 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 6100 . . . . . 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 708 . . . . . 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 2471 . . . . 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 701 . . . 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 439 . . . . . . . . . . . 12  |-  ( ( B  <_  A  /\  C  <_  B )  <->  ( C  <_  B  /\  B  <_  A ) )
1002, 3, 14, 8, 1, 22, 23, 99ditgsplitlem 19752 . . . . . . . . . . 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 6099 . . . . . . . . . 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 9115 . . . . . . . . . . . 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 19751 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S__ [ A  ->  B ] D  _d x  =  -u S__ [ B  ->  A ] D  _d x )
104103oveq2d 6100 . . . . . . . . . . . . . 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 9406 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S__ [ B  ->  A ] D  _d x  +  -u S__ [ B  ->  A ] D  _d x )  =  0 )
106104, 105eqtrd 2470 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x )  =  0 )
107106oveq2d 6100 . . . . . . . . . . . 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 9271 . . . . . . . . . . . 12  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  0 )  =  S__ [ C  ->  B ] D  _d x )
109102, 107, 1083eqtrd 2474 . . . . . . . . . . 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 708 . . . . . . . . . 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 2471 . . . . . . . . 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 6100 . . . . . . . 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 708 . . . . . . . 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 2471 . . . . . . 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 6099 . . . . . 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 708 . . . . . 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 2471 . . . . 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 697 . . . 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 9184 . . 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 9184 . 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 9184 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4215    e. cmpt 4269  (class class class)co 6084   RRcr 8994   0cc0 8995    + caddc 8998    <_ cle 9126   -ucneg 9297   (,)cioo 10921   [,]cicc 10924   L ^1cibl 19514   S__cdit 19738
This theorem is referenced by:  itgsubstlem  19937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-disj 4186  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-ofr 6309  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-rest 13655  df-topgen 13672  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-top 16968  df-bases 16970  df-topon 16971  df-cmp 17455  df-ovol 19366  df-vol 19367  df-mbf 19516  df-itg1 19517  df-itg2 19518  df-ibl 19519  df-itg 19520  df-0p 19565  df-ditg 19739
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