This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem fsumparts

Description: Summation by parts. (Contributed by Mario Carneiro, 13-Apr-2016)

Ref Expression
Hypotheses fsumparts.b 
|- ( k = j -> ( A = B /\ V = W ) )
fsumparts.c 
|- ( k = ( j + 1 ) -> ( A = C /\ V = X ) )
fsumparts.d 
|- ( k = M -> ( A = D /\ V = Y ) )
fsumparts.e 
|- ( k = N -> ( A = E /\ V = Z ) )
fsumparts.1 
|- ( ph -> N e. ( ZZ>= ` M ) )
fsumparts.2 
|- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
fsumparts.3 
|- ( ( ph /\ k e. ( M ... N ) ) -> V e. CC )
Assertion fsumparts 
|- ( ph -> sum_ j e. ( M ..^ N ) ( B x. ( X - W ) ) = ( ( ( E x. Z ) - ( D x. Y ) ) - sum_ j e. ( M ..^ N ) ( ( C - B ) x. X ) ) )

Proof

Step Hyp Ref Expression
1 fsumparts.b 
 |-  ( k = j -> ( A = B /\ V = W ) )
2 fsumparts.c 
 |-  ( k = ( j + 1 ) -> ( A = C /\ V = X ) )
3 fsumparts.d 
 |-  ( k = M -> ( A = D /\ V = Y ) )
4 fsumparts.e 
 |-  ( k = N -> ( A = E /\ V = Z ) )
5 fsumparts.1 
 |-  ( ph -> N e. ( ZZ>= ` M ) )
6 fsumparts.2 
 |-  ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
7 fsumparts.3 
 |-  ( ( ph /\ k e. ( M ... N ) ) -> V e. CC )
8 sum0 
 |-  sum_ j e. (/) ( B x. ( X - W ) ) = 0
9 0m0e0 
 |-  ( 0 - 0 ) = 0
10 8 9 eqtr4i 
 |-  sum_ j e. (/) ( B x. ( X - W ) ) = ( 0 - 0 )
11 simpr 
 |-  ( ( ph /\ N = M ) -> N = M )
12 11 oveq2d 
 |-  ( ( ph /\ N = M ) -> ( M ..^ N ) = ( M ..^ M ) )
13 fzo0 
 |-  ( M ..^ M ) = (/)
14 12 13 eqtrdi 
 |-  ( ( ph /\ N = M ) -> ( M ..^ N ) = (/) )
15 14 sumeq1d 
 |-  ( ( ph /\ N = M ) -> sum_ j e. ( M ..^ N ) ( B x. ( X - W ) ) = sum_ j e. (/) ( B x. ( X - W ) ) )
16 eluzfz1 
 |-  ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
17 5 16 syl 
 |-  ( ph -> M e. ( M ... N ) )
18 eqtr3 
 |-  ( ( k = M /\ N = M ) -> k = N )
19 oveq12 
 |-  ( ( A = E /\ V = Z ) -> ( A x. V ) = ( E x. Z ) )
20 18 4 19 3syl 
 |-  ( ( k = M /\ N = M ) -> ( A x. V ) = ( E x. Z ) )
21 oveq12 
 |-  ( ( A = D /\ V = Y ) -> ( A x. V ) = ( D x. Y ) )
22 3 21 syl 
 |-  ( k = M -> ( A x. V ) = ( D x. Y ) )
23 22 adantr 
 |-  ( ( k = M /\ N = M ) -> ( A x. V ) = ( D x. Y ) )
24 20 23 eqeq12d 
 |-  ( ( k = M /\ N = M ) -> ( ( A x. V ) = ( A x. V ) <-> ( E x. Z ) = ( D x. Y ) ) )
25 24 pm5.74da 
 |-  ( k = M -> ( ( N = M -> ( A x. V ) = ( A x. V ) ) <-> ( N = M -> ( E x. Z ) = ( D x. Y ) ) ) )
26 eqidd 
 |-  ( N = M -> ( A x. V ) = ( A x. V ) )
27 25 26 vtoclg 
 |-  ( M e. ( M ... N ) -> ( N = M -> ( E x. Z ) = ( D x. Y ) ) )
28 27 imp 
 |-  ( ( M e. ( M ... N ) /\ N = M ) -> ( E x. Z ) = ( D x. Y ) )
29 17 28 sylan 
 |-  ( ( ph /\ N = M ) -> ( E x. Z ) = ( D x. Y ) )
30 29 oveq1d 
 |-  ( ( ph /\ N = M ) -> ( ( E x. Z ) - ( D x. Y ) ) = ( ( D x. Y ) - ( D x. Y ) ) )
31 3 simpld 
 |-  ( k = M -> A = D )
32 31 eleq1d 
 |-  ( k = M -> ( A e. CC <-> D e. CC ) )
33 6 ralrimiva 
 |-  ( ph -> A. k e. ( M ... N ) A e. CC )
34 32 33 17 rspcdva 
 |-  ( ph -> D e. CC )
35 3 simprd 
 |-  ( k = M -> V = Y )
36 35 eleq1d 
 |-  ( k = M -> ( V e. CC <-> Y e. CC ) )
37 7 ralrimiva 
 |-  ( ph -> A. k e. ( M ... N ) V e. CC )
38 36 37 17 rspcdva 
 |-  ( ph -> Y e. CC )
39 34 38 mulcld 
 |-  ( ph -> ( D x. Y ) e. CC )
40 39 subidd 
 |-  ( ph -> ( ( D x. Y ) - ( D x. Y ) ) = 0 )
41 40 adantr 
 |-  ( ( ph /\ N = M ) -> ( ( D x. Y ) - ( D x. Y ) ) = 0 )
42 30 41 eqtrd 
 |-  ( ( ph /\ N = M ) -> ( ( E x. Z ) - ( D x. Y ) ) = 0 )
43 14 sumeq1d 
 |-  ( ( ph /\ N = M ) -> sum_ j e. ( M ..^ N ) ( ( C - B ) x. X ) = sum_ j e. (/) ( ( C - B ) x. X ) )
44 sum0 
 |-  sum_ j e. (/) ( ( C - B ) x. X ) = 0
45 43 44 eqtrdi 
 |-  ( ( ph /\ N = M ) -> sum_ j e. ( M ..^ N ) ( ( C - B ) x. X ) = 0 )
46 42 45 oveq12d 
 |-  ( ( ph /\ N = M ) -> ( ( ( E x. Z ) - ( D x. Y ) ) - sum_ j e. ( M ..^ N ) ( ( C - B ) x. X ) ) = ( 0 - 0 ) )
47 10 15 46 3eqtr4a 
 |-  ( ( ph /\ N = M ) -> sum_ j e. ( M ..^ N ) ( B x. ( X - W ) ) = ( ( ( E x. Z ) - ( D x. Y ) ) - sum_ j e. ( M ..^ N ) ( ( C - B ) x. X ) ) )
48 fzofi 
 |-  ( ( M + 1 ) ..^ N ) e. Fin
49 48 a1i 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( M + 1 ) ..^ N ) e. Fin )
50 eluzel2 
 |-  ( N e. ( ZZ>= ` M ) -> M e. ZZ )
51 5 50 syl 
 |-  ( ph -> M e. ZZ )
52 51 adantr 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> M e. ZZ )
53 uzid 
 |-  ( M e. ZZ -> M e. ( ZZ>= ` M ) )
54 peano2uz 
 |-  ( M e. ( ZZ>= ` M ) -> ( M + 1 ) e. ( ZZ>= ` M ) )
55 fzoss1 
 |-  ( ( M + 1 ) e. ( ZZ>= ` M ) -> ( ( M + 1 ) ..^ N ) C_ ( M ..^ N ) )
56 52 53 54 55 4syl 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( M + 1 ) ..^ N ) C_ ( M ..^ N ) )
57 56 sselda 
 |-  ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( ( M + 1 ) ..^ N ) ) -> k e. ( M ..^ N ) )
58 elfzofz 
 |-  ( k e. ( M ..^ N ) -> k e. ( M ... N ) )
59 6 7 mulcld 
 |-  ( ( ph /\ k e. ( M ... N ) ) -> ( A x. V ) e. CC )
60 58 59 sylan2 
 |-  ( ( ph /\ k e. ( M ..^ N ) ) -> ( A x. V ) e. CC )
61 60 adantlr 
 |-  ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( M ..^ N ) ) -> ( A x. V ) e. CC )
62 57 61 syldan 
 |-  ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( ( M + 1 ) ..^ N ) ) -> ( A x. V ) e. CC )
63 49 62 fsumcl 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) e. CC )
64 4 simpld 
 |-  ( k = N -> A = E )
65 64 eleq1d 
 |-  ( k = N -> ( A e. CC <-> E e. CC ) )
66 eluzfz2 
 |-  ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
67 5 66 syl 
 |-  ( ph -> N e. ( M ... N ) )
68 65 33 67 rspcdva 
 |-  ( ph -> E e. CC )
69 4 simprd 
 |-  ( k = N -> V = Z )
70 69 eleq1d 
 |-  ( k = N -> ( V e. CC <-> Z e. CC ) )
71 70 37 67 rspcdva 
 |-  ( ph -> Z e. CC )
72 68 71 mulcld 
 |-  ( ph -> ( E x. Z ) e. CC )
73 72 adantr 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( E x. Z ) e. CC )
74 simpr 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> N e. ( ZZ>= ` ( M + 1 ) ) )
75 fzp1ss 
 |-  ( M e. ZZ -> ( ( M + 1 ) ... N ) C_ ( M ... N ) )
76 52 75 syl 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( M + 1 ) ... N ) C_ ( M ... N ) )
77 76 sselda 
 |-  ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( ( M + 1 ) ... N ) ) -> k e. ( M ... N ) )
78 59 adantlr 
 |-  ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( M ... N ) ) -> ( A x. V ) e. CC )
79 77 78 syldan 
 |-  ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( ( M + 1 ) ... N ) ) -> ( A x. V ) e. CC )
80 4 19 syl 
 |-  ( k = N -> ( A x. V ) = ( E x. Z ) )
81 74 79 80 fsumm1 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ k e. ( ( M + 1 ) ... N ) ( A x. V ) = ( sum_ k e. ( ( M + 1 ) ... ( N - 1 ) ) ( A x. V ) + ( E x. Z ) ) )
82 eluzelz 
 |-  ( N e. ( ZZ>= ` M ) -> N e. ZZ )
83 5 82 syl 
 |-  ( ph -> N e. ZZ )
84 83 adantr 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> N e. ZZ )
85 fzoval 
 |-  ( N e. ZZ -> ( M ..^ N ) = ( M ... ( N - 1 ) ) )
86 84 85 syl 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( M ..^ N ) = ( M ... ( N - 1 ) ) )
87 52 zcnd 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> M e. CC )
88 ax-1cn 
 |-  1 e. CC
89 pncan 
 |-  ( ( M e. CC /\ 1 e. CC ) -> ( ( M + 1 ) - 1 ) = M )
90 87 88 89 sylancl 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( M + 1 ) - 1 ) = M )
91 90 oveq1d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( ( M + 1 ) - 1 ) ... ( N - 1 ) ) = ( M ... ( N - 1 ) ) )
92 86 91 eqtr4d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( M ..^ N ) = ( ( ( M + 1 ) - 1 ) ... ( N - 1 ) ) )
93 92 sumeq1d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ j e. ( M ..^ N ) ( C x. X ) = sum_ j e. ( ( ( M + 1 ) - 1 ) ... ( N - 1 ) ) ( C x. X ) )
94 1zzd 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> 1 e. ZZ )
95 52 peano2zd 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( M + 1 ) e. ZZ )
96 oveq12 
 |-  ( ( A = C /\ V = X ) -> ( A x. V ) = ( C x. X ) )
97 2 96 syl 
 |-  ( k = ( j + 1 ) -> ( A x. V ) = ( C x. X ) )
98 94 95 84 79 97 fsumshftm 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ k e. ( ( M + 1 ) ... N ) ( A x. V ) = sum_ j e. ( ( ( M + 1 ) - 1 ) ... ( N - 1 ) ) ( C x. X ) )
99 93 98 eqtr4d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ j e. ( M ..^ N ) ( C x. X ) = sum_ k e. ( ( M + 1 ) ... N ) ( A x. V ) )
100 fzoval 
 |-  ( N e. ZZ -> ( ( M + 1 ) ..^ N ) = ( ( M + 1 ) ... ( N - 1 ) ) )
101 84 100 syl 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( M + 1 ) ..^ N ) = ( ( M + 1 ) ... ( N - 1 ) ) )
102 101 sumeq1d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) = sum_ k e. ( ( M + 1 ) ... ( N - 1 ) ) ( A x. V ) )
103 102 oveq1d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) + ( E x. Z ) ) = ( sum_ k e. ( ( M + 1 ) ... ( N - 1 ) ) ( A x. V ) + ( E x. Z ) ) )
104 81 99 103 3eqtr4d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ j e. ( M ..^ N ) ( C x. X ) = ( sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) + ( E x. Z ) ) )
105 63 73 104 comraddd 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ j e. ( M ..^ N ) ( C x. X ) = ( ( E x. Z ) + sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) ) )
106 105 oveq1d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( sum_ j e. ( M ..^ N ) ( C x. X ) - sum_ j e. ( M ..^ N ) ( B x. X ) ) = ( ( ( E x. Z ) + sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) ) - sum_ j e. ( M ..^ N ) ( B x. X ) ) )
107 fzofzp1 
 |-  ( j e. ( M ..^ N ) -> ( j + 1 ) e. ( M ... N ) )
108 2 simpld 
 |-  ( k = ( j + 1 ) -> A = C )
109 108 eleq1d 
 |-  ( k = ( j + 1 ) -> ( A e. CC <-> C e. CC ) )
110 109 rspccva 
 |-  ( ( A. k e. ( M ... N ) A e. CC /\ ( j + 1 ) e. ( M ... N ) ) -> C e. CC )
111 33 107 110 syl2an 
 |-  ( ( ph /\ j e. ( M ..^ N ) ) -> C e. CC )
112 elfzofz 
 |-  ( j e. ( M ..^ N ) -> j e. ( M ... N ) )
113 1 simpld 
 |-  ( k = j -> A = B )
114 113 eleq1d 
 |-  ( k = j -> ( A e. CC <-> B e. CC ) )
115 114 rspccva 
 |-  ( ( A. k e. ( M ... N ) A e. CC /\ j e. ( M ... N ) ) -> B e. CC )
116 33 112 115 syl2an 
 |-  ( ( ph /\ j e. ( M ..^ N ) ) -> B e. CC )
117 2 simprd 
 |-  ( k = ( j + 1 ) -> V = X )
118 117 eleq1d 
 |-  ( k = ( j + 1 ) -> ( V e. CC <-> X e. CC ) )
119 118 rspccva 
 |-  ( ( A. k e. ( M ... N ) V e. CC /\ ( j + 1 ) e. ( M ... N ) ) -> X e. CC )
120 37 107 119 syl2an 
 |-  ( ( ph /\ j e. ( M ..^ N ) ) -> X e. CC )
121 111 116 120 subdird 
 |-  ( ( ph /\ j e. ( M ..^ N ) ) -> ( ( C - B ) x. X ) = ( ( C x. X ) - ( B x. X ) ) )
122 121 sumeq2dv 
 |-  ( ph -> sum_ j e. ( M ..^ N ) ( ( C - B ) x. X ) = sum_ j e. ( M ..^ N ) ( ( C x. X ) - ( B x. X ) ) )
123 fzofi 
 |-  ( M ..^ N ) e. Fin
124 123 a1i 
 |-  ( ph -> ( M ..^ N ) e. Fin )
125 111 120 mulcld 
 |-  ( ( ph /\ j e. ( M ..^ N ) ) -> ( C x. X ) e. CC )
126 116 120 mulcld 
 |-  ( ( ph /\ j e. ( M ..^ N ) ) -> ( B x. X ) e. CC )
127 124 125 126 fsumsub 
 |-  ( ph -> sum_ j e. ( M ..^ N ) ( ( C x. X ) - ( B x. X ) ) = ( sum_ j e. ( M ..^ N ) ( C x. X ) - sum_ j e. ( M ..^ N ) ( B x. X ) ) )
128 122 127 eqtrd 
 |-  ( ph -> sum_ j e. ( M ..^ N ) ( ( C - B ) x. X ) = ( sum_ j e. ( M ..^ N ) ( C x. X ) - sum_ j e. ( M ..^ N ) ( B x. X ) ) )
129 128 adantr 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ j e. ( M ..^ N ) ( ( C - B ) x. X ) = ( sum_ j e. ( M ..^ N ) ( C x. X ) - sum_ j e. ( M ..^ N ) ( B x. X ) ) )
130 124 126 fsumcl 
 |-  ( ph -> sum_ j e. ( M ..^ N ) ( B x. X ) e. CC )
131 130 adantr 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ j e. ( M ..^ N ) ( B x. X ) e. CC )
132 73 131 63 subsub3d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( E x. Z ) - ( sum_ j e. ( M ..^ N ) ( B x. X ) - sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) ) ) = ( ( ( E x. Z ) + sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) ) - sum_ j e. ( M ..^ N ) ( B x. X ) ) )
133 106 129 132 3eqtr4d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ j e. ( M ..^ N ) ( ( C - B ) x. X ) = ( ( E x. Z ) - ( sum_ j e. ( M ..^ N ) ( B x. X ) - sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) ) ) )
134 133 oveq2d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( ( E x. Z ) - ( D x. Y ) ) - sum_ j e. ( M ..^ N ) ( ( C - B ) x. X ) ) = ( ( ( E x. Z ) - ( D x. Y ) ) - ( ( E x. Z ) - ( sum_ j e. ( M ..^ N ) ( B x. X ) - sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) ) ) ) )
135 39 adantr 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( D x. Y ) e. CC )
136 131 63 subcld 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( sum_ j e. ( M ..^ N ) ( B x. X ) - sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) ) e. CC )
137 73 135 136 nnncan1d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( ( E x. Z ) - ( D x. Y ) ) - ( ( E x. Z ) - ( sum_ j e. ( M ..^ N ) ( B x. X ) - sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) ) ) ) = ( ( sum_ j e. ( M ..^ N ) ( B x. X ) - sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) ) - ( D x. Y ) ) )
138 63 135 addcomd 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) + ( D x. Y ) ) = ( ( D x. Y ) + sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) ) )
139 eluzp1m1 
 |-  ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
140 51 139 sylan 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
141 86 eleq2d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( k e. ( M ..^ N ) <-> k e. ( M ... ( N - 1 ) ) ) )
142 141 biimpar 
 |-  ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( M ... ( N - 1 ) ) ) -> k e. ( M ..^ N ) )
143 142 61 syldan 
 |-  ( ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) /\ k e. ( M ... ( N - 1 ) ) ) -> ( A x. V ) e. CC )
144 140 143 22 fsum1p 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ k e. ( M ... ( N - 1 ) ) ( A x. V ) = ( ( D x. Y ) + sum_ k e. ( ( M + 1 ) ... ( N - 1 ) ) ( A x. V ) ) )
145 86 sumeq1d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ k e. ( M ..^ N ) ( A x. V ) = sum_ k e. ( M ... ( N - 1 ) ) ( A x. V ) )
146 102 oveq2d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( D x. Y ) + sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) ) = ( ( D x. Y ) + sum_ k e. ( ( M + 1 ) ... ( N - 1 ) ) ( A x. V ) ) )
147 144 145 146 3eqtr4d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ k e. ( M ..^ N ) ( A x. V ) = ( ( D x. Y ) + sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) ) )
148 138 147 eqtr4d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) + ( D x. Y ) ) = sum_ k e. ( M ..^ N ) ( A x. V ) )
149 oveq12 
 |-  ( ( A = B /\ V = W ) -> ( A x. V ) = ( B x. W ) )
150 1 149 syl 
 |-  ( k = j -> ( A x. V ) = ( B x. W ) )
151 150 cbvsumv 
 |-  sum_ k e. ( M ..^ N ) ( A x. V ) = sum_ j e. ( M ..^ N ) ( B x. W )
152 148 151 eqtrdi 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) + ( D x. Y ) ) = sum_ j e. ( M ..^ N ) ( B x. W ) )
153 152 oveq2d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( sum_ j e. ( M ..^ N ) ( B x. X ) - ( sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) + ( D x. Y ) ) ) = ( sum_ j e. ( M ..^ N ) ( B x. X ) - sum_ j e. ( M ..^ N ) ( B x. W ) ) )
154 131 63 135 subsub4d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( sum_ j e. ( M ..^ N ) ( B x. X ) - sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) ) - ( D x. Y ) ) = ( sum_ j e. ( M ..^ N ) ( B x. X ) - ( sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) + ( D x. Y ) ) ) )
155 1 simprd 
 |-  ( k = j -> V = W )
156 155 eleq1d 
 |-  ( k = j -> ( V e. CC <-> W e. CC ) )
157 156 rspccva 
 |-  ( ( A. k e. ( M ... N ) V e. CC /\ j e. ( M ... N ) ) -> W e. CC )
158 37 112 157 syl2an 
 |-  ( ( ph /\ j e. ( M ..^ N ) ) -> W e. CC )
159 116 120 158 subdid 
 |-  ( ( ph /\ j e. ( M ..^ N ) ) -> ( B x. ( X - W ) ) = ( ( B x. X ) - ( B x. W ) ) )
160 159 sumeq2dv 
 |-  ( ph -> sum_ j e. ( M ..^ N ) ( B x. ( X - W ) ) = sum_ j e. ( M ..^ N ) ( ( B x. X ) - ( B x. W ) ) )
161 116 158 mulcld 
 |-  ( ( ph /\ j e. ( M ..^ N ) ) -> ( B x. W ) e. CC )
162 124 126 161 fsumsub 
 |-  ( ph -> sum_ j e. ( M ..^ N ) ( ( B x. X ) - ( B x. W ) ) = ( sum_ j e. ( M ..^ N ) ( B x. X ) - sum_ j e. ( M ..^ N ) ( B x. W ) ) )
163 160 162 eqtrd 
 |-  ( ph -> sum_ j e. ( M ..^ N ) ( B x. ( X - W ) ) = ( sum_ j e. ( M ..^ N ) ( B x. X ) - sum_ j e. ( M ..^ N ) ( B x. W ) ) )
164 163 adantr 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ j e. ( M ..^ N ) ( B x. ( X - W ) ) = ( sum_ j e. ( M ..^ N ) ( B x. X ) - sum_ j e. ( M ..^ N ) ( B x. W ) ) )
165 153 154 164 3eqtr4d 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( sum_ j e. ( M ..^ N ) ( B x. X ) - sum_ k e. ( ( M + 1 ) ..^ N ) ( A x. V ) ) - ( D x. Y ) ) = sum_ j e. ( M ..^ N ) ( B x. ( X - W ) ) )
166 134 137 165 3eqtrrd 
 |-  ( ( ph /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> sum_ j e. ( M ..^ N ) ( B x. ( X - W ) ) = ( ( ( E x. Z ) - ( D x. Y ) ) - sum_ j e. ( M ..^ N ) ( ( C - B ) x. X ) ) )
167 uzp1 
 |-  ( N e. ( ZZ>= ` M ) -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
168 5 167 syl 
 |-  ( ph -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
169 47 166 168 mpjaodan 
 |-  ( ph -> sum_ j e. ( M ..^ N ) ( B x. ( X - W ) ) = ( ( ( E x. Z ) - ( D x. Y ) ) - sum_ j e. ( M ..^ N ) ( ( C - B ) x. X ) ) )