Step |
Hyp |
Ref |
Expression |
1 |
|
binomfallfaclem.1 |
|
2 |
|
binomfallfaclem.2 |
|
3 |
|
binomfallfaclem.3 |
|
4 |
|
binomfallfaclem.4 |
|
5 |
|
elfzelz |
|
6 |
|
bccl |
|
7 |
3 5 6
|
syl2an |
|
8 |
7
|
nn0cnd |
|
9 |
|
fznn0sub |
|
10 |
|
fallfaccl |
|
11 |
1 9 10
|
syl2an |
|
12 |
|
elfznn0 |
|
13 |
|
fallfaccl |
|
14 |
2 12 13
|
syl2an |
|
15 |
11 14
|
mulcld |
|
16 |
1 2
|
addcld |
|
17 |
3
|
nn0cnd |
|
18 |
16 17
|
subcld |
|
19 |
18
|
adantr |
|
20 |
8 15 19
|
mulassd |
|
21 |
9
|
nn0cnd |
|
22 |
|
subcl |
|
23 |
1 21 22
|
syl2an |
|
24 |
12
|
nn0cnd |
|
25 |
|
subcl |
|
26 |
2 24 25
|
syl2an |
|
27 |
15 23 26
|
adddid |
|
28 |
1
|
adantr |
|
29 |
17
|
adantr |
|
30 |
28 29
|
subcld |
|
31 |
24
|
adantl |
|
32 |
2
|
adantr |
|
33 |
30 31 32
|
ppncand |
|
34 |
28 29 31
|
subsubd |
|
35 |
34
|
oveq1d |
|
36 |
28 32 29
|
addsubd |
|
37 |
33 35 36
|
3eqtr4d |
|
38 |
37
|
oveq2d |
|
39 |
11 14 23
|
mul32d |
|
40 |
|
1cnd |
|
41 |
29 40 31
|
addsubd |
|
42 |
41
|
oveq2d |
|
43 |
|
fallfacp1 |
|
44 |
1 9 43
|
syl2an |
|
45 |
42 44
|
eqtrd |
|
46 |
45
|
oveq1d |
|
47 |
39 46
|
eqtr4d |
|
48 |
11 14 26
|
mulassd |
|
49 |
|
fallfacp1 |
|
50 |
2 12 49
|
syl2an |
|
51 |
50
|
oveq2d |
|
52 |
48 51
|
eqtr4d |
|
53 |
47 52
|
oveq12d |
|
54 |
27 38 53
|
3eqtr3d |
|
55 |
54
|
oveq2d |
|
56 |
3
|
nn0zd |
|
57 |
|
uzid |
|
58 |
|
peano2uz |
|
59 |
|
fzss2 |
|
60 |
56 57 58 59
|
4syl |
|
61 |
60
|
sselda |
|
62 |
|
fznn0sub |
|
63 |
|
fallfaccl |
|
64 |
1 62 63
|
syl2an |
|
65 |
61 64
|
syldan |
|
66 |
65 14
|
mulcld |
|
67 |
|
peano2nn0 |
|
68 |
12 67
|
syl |
|
69 |
|
fallfaccl |
|
70 |
2 68 69
|
syl2an |
|
71 |
11 70
|
mulcld |
|
72 |
8 66 71
|
adddid |
|
73 |
20 55 72
|
3eqtrd |
|
74 |
73
|
sumeq2dv |
|
75 |
74
|
adantr |
|
76 |
16 3
|
fallfacp1d |
|
77 |
4
|
oveq1d |
|
78 |
76 77
|
sylan9eq |
|
79 |
|
fzfid |
|
80 |
8 15
|
mulcld |
|
81 |
79 18 80
|
fsummulc1 |
|
82 |
81
|
adantr |
|
83 |
78 82
|
eqtrd |
|
84 |
|
elfzelz |
|
85 |
|
bcpasc |
|
86 |
3 84 85
|
syl2an |
|
87 |
86
|
oveq1d |
|
88 |
3 84 6
|
syl2an |
|
89 |
88
|
nn0cnd |
|
90 |
|
peano2zm |
|
91 |
84 90
|
syl |
|
92 |
|
bccl |
|
93 |
3 91 92
|
syl2an |
|
94 |
93
|
nn0cnd |
|
95 |
|
elfznn0 |
|
96 |
2 95 13
|
syl2an |
|
97 |
64 96
|
mulcld |
|
98 |
89 94 97
|
adddird |
|
99 |
87 98
|
eqtr3d |
|
100 |
99
|
sumeq2dv |
|
101 |
|
nn0uz |
|
102 |
3 101
|
eleqtrdi |
|
103 |
89 97
|
mulcld |
|
104 |
|
oveq2 |
|
105 |
|
oveq2 |
|
106 |
105
|
oveq2d |
|
107 |
|
oveq2 |
|
108 |
106 107
|
oveq12d |
|
109 |
104 108
|
oveq12d |
|
110 |
102 103 109
|
fsump1 |
|
111 |
|
peano2nn0 |
|
112 |
3 111
|
syl |
|
113 |
112
|
nn0zd |
|
114 |
3
|
nn0red |
|
115 |
114
|
ltp1d |
|
116 |
115
|
olcd |
|
117 |
|
bcval4 |
|
118 |
3 113 116 117
|
syl3anc |
|
119 |
118
|
oveq1d |
|
120 |
112
|
nn0cnd |
|
121 |
120
|
subidd |
|
122 |
121
|
oveq2d |
|
123 |
|
0nn0 |
|
124 |
|
fallfaccl |
|
125 |
1 123 124
|
sylancl |
|
126 |
122 125
|
eqeltrd |
|
127 |
|
fallfaccl |
|
128 |
2 112 127
|
syl2anc |
|
129 |
126 128
|
mulcld |
|
130 |
129
|
mul02d |
|
131 |
119 130
|
eqtrd |
|
132 |
131
|
oveq2d |
|
133 |
61 103
|
syldan |
|
134 |
79 133
|
fsumcl |
|
135 |
134
|
addid1d |
|
136 |
110 132 135
|
3eqtrd |
|
137 |
112 101
|
eleqtrdi |
|
138 |
94 97
|
mulcld |
|
139 |
|
oveq1 |
|
140 |
|
df-neg |
|
141 |
139 140
|
eqtr4di |
|
142 |
141
|
oveq2d |
|
143 |
|
oveq2 |
|
144 |
143
|
oveq2d |
|
145 |
|
oveq2 |
|
146 |
144 145
|
oveq12d |
|
147 |
142 146
|
oveq12d |
|
148 |
137 138 147
|
fsum1p |
|
149 |
|
neg1z |
|
150 |
|
neg1lt0 |
|
151 |
150
|
orci |
|
152 |
|
bcval4 |
|
153 |
149 151 152
|
mp3an23 |
|
154 |
3 153
|
syl |
|
155 |
154
|
oveq1d |
|
156 |
120
|
subid1d |
|
157 |
156
|
oveq2d |
|
158 |
|
fallfaccl |
|
159 |
1 112 158
|
syl2anc |
|
160 |
157 159
|
eqeltrd |
|
161 |
|
fallfaccl |
|
162 |
2 123 161
|
sylancl |
|
163 |
160 162
|
mulcld |
|
164 |
163
|
mul02d |
|
165 |
155 164
|
eqtrd |
|
166 |
|
1zzd |
|
167 |
|
0zd |
|
168 |
1 2 3
|
binomfallfaclem1 |
|
169 |
|
oveq2 |
|
170 |
|
oveq2 |
|
171 |
170
|
oveq2d |
|
172 |
|
oveq1 |
|
173 |
172
|
oveq2d |
|
174 |
171 173
|
oveq12d |
|
175 |
169 174
|
oveq12d |
|
176 |
166 167 56 168 175
|
fsumshft |
|
177 |
17
|
adantr |
|
178 |
|
elfzelz |
|
179 |
178
|
adantl |
|
180 |
179
|
zcnd |
|
181 |
|
1cnd |
|
182 |
177 180 181
|
subsub3d |
|
183 |
182
|
oveq2d |
|
184 |
180 181
|
npcand |
|
185 |
184
|
oveq2d |
|
186 |
183 185
|
oveq12d |
|
187 |
186
|
oveq2d |
|
188 |
187
|
sumeq2dv |
|
189 |
176 188
|
eqtr2d |
|
190 |
|
oveq2 |
|
191 |
|
oveq2 |
|
192 |
191
|
oveq2d |
|
193 |
|
oveq1 |
|
194 |
193
|
oveq2d |
|
195 |
192 194
|
oveq12d |
|
196 |
190 195
|
oveq12d |
|
197 |
196
|
cbvsumv |
|
198 |
189 197
|
eqtr4di |
|
199 |
165 198
|
oveq12d |
|
200 |
1 2 3
|
binomfallfaclem1 |
|
201 |
79 200
|
fsumcl |
|
202 |
201
|
addid2d |
|
203 |
148 199 202
|
3eqtrd |
|
204 |
136 203
|
oveq12d |
|
205 |
|
fzfid |
|
206 |
205 103 138
|
fsumadd |
|
207 |
79 133 200
|
fsumadd |
|
208 |
204 206 207
|
3eqtr4d |
|
209 |
100 208
|
eqtrd |
|
210 |
209
|
adantr |
|
211 |
75 83 210
|
3eqtr4d |
|