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Theorem ditgsplit 19709
Description: This theorem is the raison d'être for the directed integral, because unlike itgspliticc 19689, 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 10939 . . . . 5  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
52, 3, 4syl2anc 643 . . . 4  |-  ( ph  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
61, 5mpbid 202 . . 3  |-  ( ph  ->  ( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) )
76simp1d 969 . 2  |-  ( ph  ->  A  e.  RR )
8 ditgsplit.b . . . 4  |-  ( ph  ->  B  e.  ( X [,] Y ) )
9 elicc2 10939 . . . . 5  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
102, 3, 9syl2anc 643 . . . 4  |-  ( ph  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
118, 10mpbid 202 . . 3  |-  ( ph  ->  ( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) )
1211simp1d 969 . 2  |-  ( ph  ->  B  e.  RR )
137adantr 452 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  A  e.  RR )
14 ditgsplit.c . . . . . 6  |-  ( ph  ->  C  e.  ( X [,] Y ) )
15 elicc2 10939 . . . . . . 7  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( C  e.  ( X [,] Y )  <-> 
( C  e.  RR  /\  X  <_  C  /\  C  <_  Y ) ) )
162, 3, 15syl2anc 643 . . . . . 6  |-  ( ph  ->  ( C  e.  ( X [,] Y )  <-> 
( C  e.  RR  /\  X  <_  C  /\  C  <_  Y ) ) )
1714, 16mpbid 202 . . . . 5  |-  ( ph  ->  ( C  e.  RR  /\  X  <_  C  /\  C  <_  Y ) )
1817simp1d 969 . . . 4  |-  ( ph  ->  C  e.  RR )
1918adantr 452 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  C  e.  RR )
2012ad2antrr 707 . . . 4  |-  ( ( ( ph  /\  A  <_  B )  /\  A  <_  C )  ->  B  e.  RR )
2118ad2antrr 707 . . . 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 228 . . . . . 6  |-  ( ( A  <_  B  /\  B  <_  C )  <->  ( A  <_  B  /\  B  <_  C ) )
252, 3, 1, 8, 14, 22, 23, 24ditgsplitlem 19708 . . . . 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 696 . . . 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 228 . . . . . . . 8  |-  ( ( A  <_  C  /\  C  <_  B )  <->  ( A  <_  C  /\  C  <_  B ) )
282, 3, 1, 14, 8, 22, 23, 27ditgsplitlem 19708 . . . . . . 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 6063 . . . . . 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 19706 . . . . . . . . 9  |-  ( ph  ->  S__ [ A  ->  C ] D  _d x  e.  CC )
312, 3, 14, 8, 22, 23ditgcl 19706 . . . . . . . . 9  |-  ( ph  ->  S__ [ C  ->  B ] D  _d x  e.  CC )
322, 3, 8, 14, 22, 23ditgcl 19706 . . . . . . . . 9  |-  ( ph  ->  S__ [ B  ->  C ] D  _d x  e.  CC )
3330, 31, 32addassd 9074 . . . . . . . 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 19707 . . . . . . . . . . 11  |-  ( ph  ->  S__ [ B  ->  C ] D  _d x  =  -u S__ [ C  ->  B ] D  _d x )
3534oveq2d 6064 . . . . . . . . . 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 9365 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  -u S__ [ C  ->  B ] D  _d x )  =  0 )
3735, 36eqtrd 2444 . . . . . . . . 9  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x )  =  0 )
3837oveq2d 6064 . . . . . . . 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 9230 . . . . . . . 8  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  0 )  =  S__ [ A  ->  C ] D  _d x )
4033, 38, 393eqtrd 2448 . . . . . . 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 707 . . . . . 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 2445 . . . . 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 700 . . . 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 9143 . . 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 707 . . . 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 438 . . . . . . . 8  |-  ( ( A  <_  B  /\  C  <_  A )  <->  ( C  <_  A  /\  A  <_  B ) )
472, 3, 14, 1, 8, 22, 23, 46ditgsplitlem 19708 . . . . . . 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 6064 . . . . . 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 19707 . . . . . . . . . . 11  |-  ( ph  ->  S__ [ C  ->  A ] D  _d x  =  -u S__ [ A  ->  C ] D  _d x )
5049oveq2d 6064 . . . . . . . . . 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 9365 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  -u S__ [ A  ->  C ] D  _d x )  =  0 )
5250, 51eqtrd 2444 . . . . . . . . 9  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  A ] D  _d x )  =  0 )
5352oveq1d 6063 . . . . . . . 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 19706 . . . . . . . . 9  |-  ( ph  ->  S__ [ C  ->  A ] D  _d x  e.  CC )
552, 3, 1, 8, 22, 23ditgcl 19706 . . . . . . . . 9  |-  ( ph  ->  S__ [ A  ->  B ] D  _d x  e.  CC )
5630, 54, 55addassd 9074 . . . . . . . 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 9231 . . . . . . . 8  |-  ( ph  ->  ( 0  +  S__ [ A  ->  B ] D  _d x )  =  S__ [ A  ->  B ] D  _d x )
5853, 56, 573eqtr3d 2452 . . . . . . 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 707 . . . . . 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 2444 . . . . 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 6063 . . . 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 2446 . . 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 9143 . 2  |-  ( (
ph  /\  A  <_  B )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__
[ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
647adantr 452 . . 3  |-  ( (
ph  /\  B  <_  A )  ->  A  e.  RR )
6518adantr 452 . . 3  |-  ( (
ph  /\  B  <_  A )  ->  C  e.  RR )
66 biid 228 . . . . . 6  |-  ( ( B  <_  A  /\  A  <_  C )  <->  ( B  <_  A  /\  A  <_  C ) )
672, 3, 8, 1, 14, 22, 23, 66ditgsplitlem 19708 . . . . 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 6064 . . . 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 19707 . . . . . . . . 9  |-  ( ph  ->  S__ [ B  ->  A ] D  _d x  =  -u S__ [ A  ->  B ] D  _d x )
7069oveq2d 6064 . . . . . . . 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 9365 . . . . . . . 8  |-  ( ph  ->  ( S__ [ A  ->  B ] D  _d x  +  -u S__ [ A  ->  B ] D  _d x )  =  0 )
7270, 71eqtrd 2444 . . . . . . 7  |-  ( ph  ->  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  A ] D  _d x )  =  0 )
7372oveq1d 6063 . . . . . 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 19706 . . . . . . 7  |-  ( ph  ->  S__ [ B  ->  A ] D  _d x  e.  CC )
7555, 74, 30addassd 9074 . . . . . 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 9231 . . . . . 6  |-  ( ph  ->  ( 0  +  S__ [ A  ->  C ] D  _d x )  =  S__ [ A  ->  C ] D  _d x )
7773, 75, 763eqtr3d 2452 . . . . 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 707 . . . 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 2445 . . 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 707 . . . 4  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  A )  ->  B  e.  RR )
8118ad2antrr 707 . . . 4  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  A )  ->  C  e.  RR )
82 ancom 438 . . . . . . . . . 10  |-  ( ( C  <_  A  /\  B  <_  C )  <->  ( B  <_  C  /\  C  <_  A ) )
832, 3, 8, 14, 1, 22, 23, 82ditgsplitlem 19708 . . . . . . . . 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 6063 . . . . . . . 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 9074 . . . . . . . . . 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 19707 . . . . . . . . . . . . 13  |-  ( ph  ->  S__ [ A  ->  C ] D  _d x  =  -u S__ [ C  ->  A ] D  _d x )
8786oveq2d 6064 . . . . . . . . . . . 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 9365 . . . . . . . . . . . 12  |-  ( ph  ->  ( S__ [ C  ->  A ] D  _d x  +  -u S__ [ C  ->  A ] D  _d x )  =  0 )
8987, 88eqtrd 2444 . . . . . . . . . . 11  |-  ( ph  ->  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x )  =  0 )
9089oveq2d 6064 . . . . . . . . . 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 9230 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ B  ->  C ] D  _d x  +  0 )  =  S__ [ B  ->  C ] D  _d x )
9285, 90, 913eqtrd 2448 . . . . . . . . 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 707 . . . . . . . 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 2445 . . . . . . 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 6064 . . . . . 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 707 . . . . . 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 2445 . . . . 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 700 . . . 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 438 . . . . . . . . . . . 12  |-  ( ( B  <_  A  /\  C  <_  B )  <->  ( C  <_  B  /\  B  <_  A ) )
1002, 3, 14, 8, 1, 22, 23, 99ditgsplitlem 19708 . . . . . . . . . . 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 6063 . . . . . . . . . 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 9074 . . . . . . . . . . . 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 19707 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S__ [ A  ->  B ] D  _d x  =  -u S__ [ B  ->  A ] D  _d x )
104103oveq2d 6064 . . . . . . . . . . . . . 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 9365 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S__ [ B  ->  A ] D  _d x  +  -u S__ [ B  ->  A ] D  _d x )  =  0 )
106104, 105eqtrd 2444 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x )  =  0 )
107106oveq2d 6064 . . . . . . . . . . . 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 9230 . . . . . . . . . . . 12  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  0 )  =  S__ [ C  ->  B ] D  _d x )
109102, 107, 1083eqtrd 2448 . . . . . . . . . . 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 707 . . . . . . . . . 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 2445 . . . . . . . . 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 6064 . . . . . . . 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 707 . . . . . . . 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 2445 . . . . . . 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 6063 . . . . . 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 707 . . . . . 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 2445 . . . . 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 696 . . . 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 9143 . . 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 9143 . 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 9143 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4180    e. cmpt 4234  (class class class)co 6048   RRcr 8953   0cc0 8954    + caddc 8957    <_ cle 9085   -ucneg 9256   (,)cioo 10880   [,]cicc 10883   L ^1cibl 19470   S__cdit 19472
This theorem is referenced by:  itgsubstlem  19893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-disj 4151  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-ofr 6273  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-rlim 12246  df-sum 12443  df-rest 13613  df-topgen 13630  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-top 16926  df-bases 16928  df-topon 16929  df-cmp 17412  df-ovol 19322  df-vol 19323  df-mbf 19473  df-itg1 19474  df-itg2 19475  df-ibl 19476  df-itg 19477  df-ditg 19478  df-0p 19523
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