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Theorem ditgsplit 22431
Description: This theorem is the raison d'être for the directed integral, because unlike itgspliticc 22409, 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 11592 . . . . 5  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
52, 3, 4syl2anc 659 . . . 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 1006 . 2  |-  ( ph  ->  A  e.  RR )
8 ditgsplit.b . . . 4  |-  ( ph  ->  B  e.  ( X [,] Y ) )
9 elicc2 11592 . . . . 5  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
102, 3, 9syl2anc 659 . . . 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 1006 . 2  |-  ( ph  ->  B  e.  RR )
137adantr 463 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  A  e.  RR )
14 ditgsplit.c . . . . . 6  |-  ( ph  ->  C  e.  ( X [,] Y ) )
15 elicc2 11592 . . . . . . 7  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( C  e.  ( X [,] Y )  <-> 
( C  e.  RR  /\  X  <_  C  /\  C  <_  Y ) ) )
162, 3, 15syl2anc 659 . . . . . 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 1006 . . . 4  |-  ( ph  ->  C  e.  RR )
1918adantr 463 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  C  e.  RR )
2012ad2antrr 723 . . . 4  |-  ( ( ( ph  /\  A  <_  B )  /\  A  <_  C )  ->  B  e.  RR )
2118ad2antrr 723 . . . 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 22430 . . . . 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 712 . . . 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 22430 . . . . . . 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 6285 . . . . . 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 22428 . . . . . . . . 9  |-  ( ph  ->  S__ [ A  ->  C ] D  _d x  e.  CC )
312, 3, 14, 8, 22, 23ditgcl 22428 . . . . . . . . 9  |-  ( ph  ->  S__ [ C  ->  B ] D  _d x  e.  CC )
322, 3, 8, 14, 22, 23ditgcl 22428 . . . . . . . . 9  |-  ( ph  ->  S__ [ B  ->  C ] D  _d x  e.  CC )
3330, 31, 32addassd 9607 . . . . . . . 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 22429 . . . . . . . . . . 11  |-  ( ph  ->  S__ [ B  ->  C ] D  _d x  =  -u S__ [ C  ->  B ] D  _d x )
3534oveq2d 6286 . . . . . . . . . 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 9912 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  -u S__ [ C  ->  B ] D  _d x )  =  0 )
3735, 36eqtrd 2495 . . . . . . . . 9  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x )  =  0 )
3837oveq2d 6286 . . . . . . . 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 9769 . . . . . . . 8  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  0 )  =  S__ [ A  ->  C ] D  _d x )
4033, 38, 393eqtrd 2499 . . . . . . 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 723 . . . . . 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 2496 . . . . 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 716 . . . 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 9679 . . 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 723 . . . 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 22430 . . . . . . 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 6286 . . . . . 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 22429 . . . . . . . . . . 11  |-  ( ph  ->  S__ [ C  ->  A ] D  _d x  =  -u S__ [ A  ->  C ] D  _d x )
5049oveq2d 6286 . . . . . . . . . 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 9912 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  -u S__ [ A  ->  C ] D  _d x )  =  0 )
5250, 51eqtrd 2495 . . . . . . . . 9  |-  ( ph  ->  ( S__ [ A  ->  C ] D  _d x  +  S__ [ C  ->  A ] D  _d x )  =  0 )
5352oveq1d 6285 . . . . . . . 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 22428 . . . . . . . . 9  |-  ( ph  ->  S__ [ C  ->  A ] D  _d x  e.  CC )
552, 3, 1, 8, 22, 23ditgcl 22428 . . . . . . . . 9  |-  ( ph  ->  S__ [ A  ->  B ] D  _d x  e.  CC )
5630, 54, 55addassd 9607 . . . . . . . 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 9770 . . . . . . . 8  |-  ( ph  ->  ( 0  +  S__ [ A  ->  B ] D  _d x )  =  S__ [ A  ->  B ] D  _d x )
5853, 56, 573eqtr3d 2503 . . . . . . 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 723 . . . . . 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 2495 . . . . 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 6285 . . . 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 2497 . . 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 9679 . 2  |-  ( (
ph  /\  A  <_  B )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__
[ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
647adantr 463 . . 3  |-  ( (
ph  /\  B  <_  A )  ->  A  e.  RR )
6518adantr 463 . . 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 22430 . . . . 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 6286 . . . 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 22429 . . . . . . . . 9  |-  ( ph  ->  S__ [ B  ->  A ] D  _d x  =  -u S__ [ A  ->  B ] D  _d x )
7069oveq2d 6286 . . . . . . . 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 9912 . . . . . . . 8  |-  ( ph  ->  ( S__ [ A  ->  B ] D  _d x  +  -u S__ [ A  ->  B ] D  _d x )  =  0 )
7270, 71eqtrd 2495 . . . . . . 7  |-  ( ph  ->  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  A ] D  _d x )  =  0 )
7372oveq1d 6285 . . . . . 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 22428 . . . . . . 7  |-  ( ph  ->  S__ [ B  ->  A ] D  _d x  e.  CC )
7555, 74, 30addassd 9607 . . . . . 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 9770 . . . . . 6  |-  ( ph  ->  ( 0  +  S__ [ A  ->  C ] D  _d x )  =  S__ [ A  ->  C ] D  _d x )
7773, 75, 763eqtr3d 2503 . . . . 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 723 . . . 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 2496 . . 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 723 . . . 4  |-  ( ( ( ph  /\  B  <_  A )  /\  C  <_  A )  ->  B  e.  RR )
8118ad2antrr 723 . . . 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 22430 . . . . . . . . 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 6285 . . . . . . . 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 9607 . . . . . . . . . 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 22429 . . . . . . . . . . . . 13  |-  ( ph  ->  S__ [ A  ->  C ] D  _d x  =  -u S__ [ C  ->  A ] D  _d x )
8786oveq2d 6286 . . . . . . . . . . . 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 9912 . . . . . . . . . . . 12  |-  ( ph  ->  ( S__ [ C  ->  A ] D  _d x  +  -u S__ [ C  ->  A ] D  _d x )  =  0 )
8987, 88eqtrd 2495 . . . . . . . . . . 11  |-  ( ph  ->  ( S__ [ C  ->  A ] D  _d x  +  S__ [ A  ->  C ] D  _d x )  =  0 )
9089oveq2d 6286 . . . . . . . . . 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 9769 . . . . . . . . . 10  |-  ( ph  ->  ( S__ [ B  ->  C ] D  _d x  +  0 )  =  S__ [ B  ->  C ] D  _d x )
9285, 90, 913eqtrd 2499 . . . . . . . . 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 723 . . . . . . . 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 2496 . . . . . . 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 6286 . . . . . 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 723 . . . . . 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 2496 . . . . 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 716 . . . 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 22430 . . . . . . . . . . 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 6285 . . . . . . . . . 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 9607 . . . . . . . . . . . 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 22429 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S__ [ A  ->  B ] D  _d x  =  -u S__ [ B  ->  A ] D  _d x )
104103oveq2d 6286 . . . . . . . . . . . . . 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 9912 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S__ [ B  ->  A ] D  _d x  +  -u S__ [ B  ->  A ] D  _d x )  =  0 )
106104, 105eqtrd 2495 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S__ [ B  ->  A ] D  _d x  +  S__ [ A  ->  B ] D  _d x )  =  0 )
107106oveq2d 6286 . . . . . . . . . . . 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 9769 . . . . . . . . . . . 12  |-  ( ph  ->  ( S__ [ C  ->  B ] D  _d x  +  0 )  =  S__ [ C  ->  B ] D  _d x )
109102, 107, 1083eqtrd 2499 . . . . . . . . . . 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 723 . . . . . . . . . 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 2496 . . . . . . . . 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 6286 . . . . . . . 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 723 . . . . . . . 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 2496 . . . . . . 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 6285 . . . . . 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 723 . . . . . 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 2496 . . . . 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 712 . . . 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 9679 . . 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 9679 . 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 9679 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439    |-> cmpt 4497  (class class class)co 6270   RRcr 9480   0cc0 9481    + caddc 9484    <_ cle 9618   -ucneg 9797   (,)cioo 11532   [,]cicc 11535   L^1cibl 22192   S__cdit 22416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-ofr 6514  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-rlim 13394  df-sum 13591  df-rest 14912  df-topgen 14933  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-top 19566  df-bases 19568  df-topon 19569  df-cmp 20054  df-ovol 22042  df-vol 22043  df-mbf 22194  df-itg1 22195  df-itg2 22196  df-ibl 22197  df-itg 22198  df-0p 22243  df-ditg 22417
This theorem is referenced by:  itgsubstlem  22615
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