Theorem ditgsplit 23625

Description: This theorem is the raison d'être for the directed integral, because unlike itgspliticc 23603, 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

Step	Hyp	Ref	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 12238	. . . . 5 |- ( ( X e. RR /\ Y e. RR ) -> ( A e. ( X [,] Y ) <-> ( A e. RR /\ X <_ A /\ A <_ Y ) ) )
5	2, 3, 4	syl2anc 693	. . . 4 |- ( ph -> ( A e. ( X [,] Y ) <-> ( A e. RR /\ X <_ A /\ A <_ Y ) ) )
6	1, 5	mpbid 222	. . 3 |- ( ph -> ( A e. RR /\ X <_ A /\ A <_ Y ) )
7	6	simp1d 1073	. 2 |- ( ph -> A e. RR )
8		ditgsplit.b	. . . 4 |- ( ph -> B e. ( X [,] Y ) )
9		elicc2 12238	. . . . 5 |- ( ( X e. RR /\ Y e. RR ) -> ( B e. ( X [,] Y ) <-> ( B e. RR /\ X <_ B /\ B <_ Y ) ) )
10	2, 3, 9	syl2anc 693	. . . 4 |- ( ph -> ( B e. ( X [,] Y ) <-> ( B e. RR /\ X <_ B /\ B <_ Y ) ) )
11	8, 10	mpbid 222	. . 3 |- ( ph -> ( B e. RR /\ X <_ B /\ B <_ Y ) )
12	11	simp1d 1073	. 2 |- ( ph -> B e. RR )
13	7	adantr 481	. . 3 |- ( ( ph /\ A <_ B ) -> A e. RR )
14		ditgsplit.c	. . . . . 6 |- ( ph -> C e. ( X [,] Y ) )
15		elicc2 12238	. . . . . . 7 |- ( ( X e. RR /\ Y e. RR ) -> ( C e. ( X [,] Y ) <-> ( C e. RR /\ X <_ C /\ C <_ Y ) ) )
16	2, 3, 15	syl2anc 693	. . . . . 6 |- ( ph -> ( C e. ( X [,] Y ) <-> ( C e. RR /\ X <_ C /\ C <_ Y ) ) )
17	14, 16	mpbid 222	. . . . 5 |- ( ph -> ( C e. RR /\ X <_ C /\ C <_ Y ) )
18	17	simp1d 1073	. . . 4 |- ( ph -> C e. RR )
19	18	adantr 481	. . 3 |- ( ( ph /\ A <_ B ) -> C e. RR )
20	12	ad2antrr 762	. . . 4 |- ( ( ( ph /\ A <_ B ) /\ A <_ C ) -> B e. RR )
21	18	ad2antrr 762	. . . 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 251	. . . . . 6 |- ( ( A <_ B /\ B <_ C ) <-> ( A <_ B /\ B <_ C ) )
25	2, 3, 1, 8, 14, 22, 23, 24	ditgsplitlem 23624	. . . . 5 |- ( ( ( ph /\ A <_ B ) /\ B <_ C ) -> S__ [ A -> C ] D _d x = ( S__ [ A -> B ] D _d x + S__ [ B -> C ] D _d x ) )
26	25	adantlr 751	. . . 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 251	. . . . . . . 8 |- ( ( A <_ C /\ C <_ B ) <-> ( A <_ C /\ C <_ B ) )
28	2, 3, 1, 14, 8, 22, 23, 27	ditgsplitlem 23624	. . . . . . 7 |- ( ( ( ph /\ A <_ C ) /\ C <_ B ) -> S__ [ A -> B ] D _d x = ( S__ [ A -> C ] D _d x + S__ [ C -> B ] D _d x ) )
29	28	oveq1d 6665	. . . . . 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 ) )
30	2, 3, 1, 14, 22, 23	ditgcl 23622	. . . . . . . . 9 |- ( ph -> S__ [ A -> C ] D _d x e. CC )
31	2, 3, 14, 8, 22, 23	ditgcl 23622	. . . . . . . . 9 |- ( ph -> S__ [ C -> B ] D _d x e. CC )
32	2, 3, 8, 14, 22, 23	ditgcl 23622	. . . . . . . . 9 |- ( ph -> S__ [ B -> C ] D _d x e. CC )
33	30, 31, 32	addassd 10062	. . . . . . . 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 ) ) )
34	2, 3, 14, 8, 22, 23	ditgswap 23623	. . . . . . . . . . 11 |- ( ph -> S__ [ B -> C ] D _d x = -u S__ [ C -> B ] D _d x )
35	34	oveq2d 6666	. . . . . . . . . 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 ) )
36	31	negidd 10382	. . . . . . . . . 10 |- ( ph -> ( S__ [ C -> B ] D _d x + -u S__ [ C -> B ] D _d x ) = 0 )
37	35, 36	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( S__ [ C -> B ] D _d x + S__ [ B -> C ] D _d x ) = 0 )
38	37	oveq2d 6666	. . . . . . . 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 ) )
39	30	addid1d 10236	. . . . . . . 8 |- ( ph -> ( S__ [ A -> C ] D _d x + 0 ) = S__ [ A -> C ] D _d x )
40	33, 38, 39	3eqtrd 2660	. . . . . . 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 )
41	40	ad2antrr 762	. . . . . 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 )
42	29, 41	eqtr2d 2657	. . . . 5 |- ( ( ( ph /\ A <_ C ) /\ C <_ B ) -> S__ [ A -> C ] D _d x = ( S__ [ A -> B ] D _d x + S__ [ B -> C ] D _d x ) )
43	42	adantllr 755	. . . 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 ) )
44	20, 21, 26, 43	lecasei 10143	. . 3 |- ( ( ( ph /\ A <_ B ) /\ A <_ C ) -> S__ [ A -> C ] D _d x = ( S__ [ A -> B ] D _d x + S__ [ B -> C ] D _d x ) )
45	40	ad2antrr 762	. . . 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 466	. . . . . . . 8 |- ( ( A <_ B /\ C <_ A ) <-> ( C <_ A /\ A <_ B ) )
47	2, 3, 14, 1, 8, 22, 23, 46	ditgsplitlem 23624	. . . . . . 7 |- ( ( ( ph /\ A <_ B ) /\ C <_ A ) -> S__ [ C -> B ] D _d x = ( S__ [ C -> A ] D _d x + S__ [ A -> B ] D _d x ) )
48	47	oveq2d 6666	. . . . . 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 ) ) )
49	2, 3, 1, 14, 22, 23	ditgswap 23623	. . . . . . . . . . 11 |- ( ph -> S__ [ C -> A ] D _d x = -u S__ [ A -> C ] D _d x )
50	49	oveq2d 6666	. . . . . . . . . 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 ) )
51	30	negidd 10382	. . . . . . . . . 10 |- ( ph -> ( S__ [ A -> C ] D _d x + -u S__ [ A -> C ] D _d x ) = 0 )
52	50, 51	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( S__ [ A -> C ] D _d x + S__ [ C -> A ] D _d x ) = 0 )
53	52	oveq1d 6665	. . . . . . . 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 ) )
54	2, 3, 14, 1, 22, 23	ditgcl 23622	. . . . . . . . 9 |- ( ph -> S__ [ C -> A ] D _d x e. CC )
55	2, 3, 1, 8, 22, 23	ditgcl 23622	. . . . . . . . 9 |- ( ph -> S__ [ A -> B ] D _d x e. CC )
56	30, 54, 55	addassd 10062	. . . . . . . 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 ) ) )
57	55	addid2d 10237	. . . . . . . 8 |- ( ph -> ( 0 + S__ [ A -> B ] D _d x ) = S__ [ A -> B ] D _d x )
58	53, 56, 57	3eqtr3d 2664	. . . . . . 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 )
59	58	ad2antrr 762	. . . . . 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 )
60	48, 59	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ A <_ B ) /\ C <_ A ) -> ( S__ [ A -> C ] D _d x + S__ [ C -> B ] D _d x ) = S__ [ A -> B ] D _d x )
61	60	oveq1d 6665	. . . 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 ) )
62	45, 61	eqtr3d 2658	. . 3 |- ( ( ( ph /\ A <_ B ) /\ C <_ A ) -> S__ [ A -> C ] D _d x = ( S__ [ A -> B ] D _d x + S__ [ B -> C ] D _d x ) )
63	13, 19, 44, 62	lecasei 10143	. 2 |- ( ( ph /\ A <_ B ) -> S__ [ A -> C ] D _d x = ( S__ [ A -> B ] D _d x + S__ [ B -> C ] D _d x ) )
64	7	adantr 481	. . 3 |- ( ( ph /\ B <_ A ) -> A e. RR )
65	18	adantr 481	. . 3 |- ( ( ph /\ B <_ A ) -> C e. RR )
66		biid 251	. . . . . 6 |- ( ( B <_ A /\ A <_ C ) <-> ( B <_ A /\ A <_ C ) )
67	2, 3, 8, 1, 14, 22, 23, 66	ditgsplitlem 23624	. . . . 5 |- ( ( ( ph /\ B <_ A ) /\ A <_ C ) -> S__ [ B -> C ] D _d x = ( S__ [ B -> A ] D _d x + S__ [ A -> C ] D _d x ) )
68	67	oveq2d 6666	. . . 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 ) ) )
69	2, 3, 1, 8, 22, 23	ditgswap 23623	. . . . . . . . 9 |- ( ph -> S__ [ B -> A ] D _d x = -u S__ [ A -> B ] D _d x )
70	69	oveq2d 6666	. . . . . . . 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 ) )
71	55	negidd 10382	. . . . . . . 8 |- ( ph -> ( S__ [ A -> B ] D _d x + -u S__ [ A -> B ] D _d x ) = 0 )
72	70, 71	eqtrd 2656	. . . . . . 7 |- ( ph -> ( S__ [ A -> B ] D _d x + S__ [ B -> A ] D _d x ) = 0 )
73	72	oveq1d 6665	. . . . . 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 ) )
74	2, 3, 8, 1, 22, 23	ditgcl 23622	. . . . . . 7 |- ( ph -> S__ [ B -> A ] D _d x e. CC )
75	55, 74, 30	addassd 10062	. . . . . 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 ) ) )
76	30	addid2d 10237	. . . . . 6 |- ( ph -> ( 0 + S__ [ A -> C ] D _d x ) = S__ [ A -> C ] D _d x )
77	73, 75, 76	3eqtr3d 2664	. . . . 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 )
78	77	ad2antrr 762	. . . 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 )
79	68, 78	eqtr2d 2657	. . 3 |- ( ( ( ph /\ B <_ A ) /\ A <_ C ) -> S__ [ A -> C ] D _d x = ( S__ [ A -> B ] D _d x + S__ [ B -> C ] D _d x ) )
80	12	ad2antrr 762	. . . 4 |- ( ( ( ph /\ B <_ A ) /\ C <_ A ) -> B e. RR )
81	18	ad2antrr 762	. . . 4 |- ( ( ( ph /\ B <_ A ) /\ C <_ A ) -> C e. RR )
82		ancom 466	. . . . . . . . . 10 |- ( ( C <_ A /\ B <_ C ) <-> ( B <_ C /\ C <_ A ) )
83	2, 3, 8, 14, 1, 22, 23, 82	ditgsplitlem 23624	. . . . . . . . 9 |- ( ( ( ph /\ C <_ A ) /\ B <_ C ) -> S__ [ B -> A ] D _d x = ( S__ [ B -> C ] D _d x + S__ [ C -> A ] D _d x ) )
84	83	oveq1d 6665	. . . . . . . 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 ) )
85	32, 54, 30	addassd 10062	. . . . . . . . . 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 ) ) )
86	2, 3, 14, 1, 22, 23	ditgswap 23623	. . . . . . . . . . . . 13 |- ( ph -> S__ [ A -> C ] D _d x = -u S__ [ C -> A ] D _d x )
87	86	oveq2d 6666	. . . . . . . . . . . 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 ) )
88	54	negidd 10382	. . . . . . . . . . . 12 |- ( ph -> ( S__ [ C -> A ] D _d x + -u S__ [ C -> A ] D _d x ) = 0 )
89	87, 88	eqtrd 2656	. . . . . . . . . . 11 |- ( ph -> ( S__ [ C -> A ] D _d x + S__ [ A -> C ] D _d x ) = 0 )
90	89	oveq2d 6666	. . . . . . . . . 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 ) )
91	32	addid1d 10236	. . . . . . . . . 10 |- ( ph -> ( S__ [ B -> C ] D _d x + 0 ) = S__ [ B -> C ] D _d x )
92	85, 90, 91	3eqtrd 2660	. . . . . . . . 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 )
93	92	ad2antrr 762	. . . . . . . 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 )
94	84, 93	eqtr2d 2657	. . . . . . 7 |- ( ( ( ph /\ C <_ A ) /\ B <_ C ) -> S__ [ B -> C ] D _d x = ( S__ [ B -> A ] D _d x + S__ [ A -> C ] D _d x ) )
95	94	oveq2d 6666	. . . . . 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 ) ) )
96	77	ad2antrr 762	. . . . . 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 )
97	95, 96	eqtr2d 2657	. . . . 5 |- ( ( ( ph /\ C <_ A ) /\ B <_ C ) -> S__ [ A -> C ] D _d x = ( S__ [ A -> B ] D _d x + S__ [ B -> C ] D _d x ) )
98	97	adantllr 755	. . . 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 466	. . . . . . . . . . . 12 |- ( ( B <_ A /\ C <_ B ) <-> ( C <_ B /\ B <_ A ) )
100	2, 3, 14, 8, 1, 22, 23, 99	ditgsplitlem 23624	. . . . . . . . . . 11 |- ( ( ( ph /\ B <_ A ) /\ C <_ B ) -> S__ [ C -> A ] D _d x = ( S__ [ C -> B ] D _d x + S__ [ B -> A ] D _d x ) )
101	100	oveq1d 6665	. . . . . . . . . 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 ) )
102	31, 74, 55	addassd 10062	. . . . . . . . . . . 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 ) ) )
103	2, 3, 8, 1, 22, 23	ditgswap 23623	. . . . . . . . . . . . . . 15 |- ( ph -> S__ [ A -> B ] D _d x = -u S__ [ B -> A ] D _d x )
104	103	oveq2d 6666	. . . . . . . . . . . . . 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 ) )
105	74	negidd 10382	. . . . . . . . . . . . . 14 |- ( ph -> ( S__ [ B -> A ] D _d x + -u S__ [ B -> A ] D _d x ) = 0 )
106	104, 105	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ph -> ( S__ [ B -> A ] D _d x + S__ [ A -> B ] D _d x ) = 0 )
107	106	oveq2d 6666	. . . . . . . . . . . 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 ) )
108	31	addid1d 10236	. . . . . . . . . . . 12 |- ( ph -> ( S__ [ C -> B ] D _d x + 0 ) = S__ [ C -> B ] D _d x )
109	102, 107, 108	3eqtrd 2660	. . . . . . . . . . 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 )
110	109	ad2antrr 762	. . . . . . . . . 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 )
111	101, 110	eqtr2d 2657	. . . . . . . . 9 |- ( ( ( ph /\ B <_ A ) /\ C <_ B ) -> S__ [ C -> B ] D _d x = ( S__ [ C -> A ] D _d x + S__ [ A -> B ] D _d x ) )
112	111	oveq2d 6666	. . . . . . . 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 ) ) )
113	58	ad2antrr 762	. . . . . . . 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 )
114	112, 113	eqtr2d 2657	. . . . . . 7 |- ( ( ( ph /\ B <_ A ) /\ C <_ B ) -> S__ [ A -> B ] D _d x = ( S__ [ A -> C ] D _d x + S__ [ C -> B ] D _d x ) )
115	114	oveq1d 6665	. . . . . 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 ) )
116	40	ad2antrr 762	. . . . . 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 )
117	115, 116	eqtr2d 2657	. . . . 5 |- ( ( ( ph /\ B <_ A ) /\ C <_ B ) -> S__ [ A -> C ] D _d x = ( S__ [ A -> B ] D _d x + S__ [ B -> C ] D _d x ) )
118	117	adantlr 751	. . . 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 ) )
119	80, 81, 98, 118	lecasei 10143	. . 3 |- ( ( ( ph /\ B <_ A ) /\ C <_ A ) -> S__ [ A -> C ] D _d x = ( S__ [ A -> B ] D _d x + S__ [ B -> C ] D _d x ) )
120	64, 65, 79, 119	lecasei 10143	. 2 |- ( ( ph /\ B <_ A ) -> S__ [ A -> C ] D _d x = ( S__ [ A -> B ] D _d x + S__ [ B -> C ] D _d x ) )
121	7, 12, 63, 120	lecasei 10143	1 |- ( ph -> S__ [ A -> C ] D _d x = ( S__ [ A -> B ] D _d x + S__ [ B -> C ] D _d x ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   |-> cmpt 4729  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   <_ cle 10075   -ucneg 10267   (,)cioo 12175   [,]cicc 12178   L^1cibl 23386   S__cdit 23610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390  df-ibl 23391  df-itg 23392  df-0p 23437  df-ditg 23611

This theorem is referenced by:  itgsubstlem  23811