Metamath Proof Explorer


Theorem dvnprodlem2

Description: Induction step for dvnprodlem2 . (Contributed by Glauco Siliprandi, 5-Apr-2020)

Ref Expression
Hypotheses dvnprodlem2.s φ S
dvnprodlem2.x φ X TopOpen fld 𝑡 S
dvnprodlem2.t φ T Fin
dvnprodlem2.h φ t T H t : X
dvnprodlem2.n φ N 0
dvnprodlem2.dvnh φ t T j 0 N S D n H t j : X
dvnprodlem2.c C = s 𝒫 T n 0 c 0 n s | t s c t = n
dvnprodlem2.r φ R T
dvnprodlem2.z φ Z T R
dvnprodlem2.ind φ k 0 N S D n x X t R H t x k = x X c C R k k ! t R c t ! t R S D n H t c t x
dvnprodlem2.j φ J 0 N
dvnprodlem2.d D = c C R Z J J c Z c R
Assertion dvnprodlem2 φ S D n x X t R Z H t x J = x X c C R Z J J ! t R Z c t ! t R Z S D n H t c t x

Proof

Step Hyp Ref Expression
1 dvnprodlem2.s φ S
2 dvnprodlem2.x φ X TopOpen fld 𝑡 S
3 dvnprodlem2.t φ T Fin
4 dvnprodlem2.h φ t T H t : X
5 dvnprodlem2.n φ N 0
6 dvnprodlem2.dvnh φ t T j 0 N S D n H t j : X
7 dvnprodlem2.c C = s 𝒫 T n 0 c 0 n s | t s c t = n
8 dvnprodlem2.r φ R T
9 dvnprodlem2.z φ Z T R
10 dvnprodlem2.ind φ k 0 N S D n x X t R H t x k = x X c C R k k ! t R c t ! t R S D n H t c t x
11 dvnprodlem2.j φ J 0 N
12 dvnprodlem2.d D = c C R Z J J c Z c R
13 nfv t φ x X
14 nfcv _ t H Z x
15 ssfi T Fin R T R Fin
16 3 8 15 syl2anc φ R Fin
17 16 adantr φ x X R Fin
18 9 adantr φ x X Z T R
19 9 eldifbd φ ¬ Z R
20 19 adantr φ x X ¬ Z R
21 simpl φ t R φ
22 8 sselda φ t R t T
23 21 22 4 syl2anc φ t R H t : X
24 23 adantlr φ x X t R H t : X
25 simplr φ x X t R x X
26 24 25 ffvelrnd φ x X t R H t x
27 fveq2 t = Z H t = H Z
28 27 fveq1d t = Z H t x = H Z x
29 id φ φ
30 eldifi Z T R Z T
31 9 30 syl φ Z T
32 simpr φ Z T Z T
33 id φ Z T φ Z T
34 eleq1 t = Z t T Z T
35 34 anbi2d t = Z φ t T φ Z T
36 27 feq1d t = Z H t : X H Z : X
37 35 36 imbi12d t = Z φ t T H t : X φ Z T H Z : X
38 37 4 vtoclg Z T φ Z T H Z : X
39 32 33 38 sylc φ Z T H Z : X
40 29 31 39 syl2anc φ H Z : X
41 40 adantr φ x X H Z : X
42 simpr φ x X x X
43 41 42 ffvelrnd φ x X H Z x
44 13 14 17 18 20 26 28 43 fprodsplitsn φ x X t R Z H t x = t R H t x H Z x
45 44 mpteq2dva φ x X t R Z H t x = x X t R H t x H Z x
46 45 oveq2d φ S D n x X t R Z H t x = S D n x X t R H t x H Z x
47 46 fveq1d φ S D n x X t R Z H t x J = S D n x X t R H t x H Z x J
48 13 17 26 fprodclf φ x X t R H t x
49 elfznn0 J 0 N J 0
50 11 49 syl φ J 0
51 eqid x X t R H t x = x X t R H t x
52 eqid x X H Z x = x X H Z x
53 oveq2 s = R 0 n s = 0 n R
54 rabeq 0 n s = 0 n R c 0 n s | t s c t = n = c 0 n R | t s c t = n
55 53 54 syl s = R c 0 n s | t s c t = n = c 0 n R | t s c t = n
56 sumeq1 s = R t s c t = t R c t
57 56 eqeq1d s = R t s c t = n t R c t = n
58 57 rabbidv s = R c 0 n R | t s c t = n = c 0 n R | t R c t = n
59 55 58 eqtrd s = R c 0 n s | t s c t = n = c 0 n R | t R c t = n
60 59 mpteq2dv s = R n 0 c 0 n s | t s c t = n = n 0 c 0 n R | t R c t = n
61 ssexg R T T Fin R V
62 8 3 61 syl2anc φ R V
63 elpwg R V R 𝒫 T R T
64 62 63 syl φ R 𝒫 T R T
65 8 64 mpbird φ R 𝒫 T
66 65 adantr φ k 0 J R 𝒫 T
67 nn0ex 0 V
68 67 mptex n 0 c 0 n R | t R c t = n V
69 68 a1i φ k 0 J n 0 c 0 n R | t R c t = n V
70 7 60 66 69 fvmptd3 φ k 0 J C R = n 0 c 0 n R | t R c t = n
71 oveq2 n = k 0 n = 0 k
72 71 oveq1d n = k 0 n R = 0 k R
73 rabeq 0 n R = 0 k R c 0 n R | t R c t = n = c 0 k R | t R c t = n
74 72 73 syl n = k c 0 n R | t R c t = n = c 0 k R | t R c t = n
75 eqeq2 n = k t R c t = n t R c t = k
76 75 rabbidv n = k c 0 k R | t R c t = n = c 0 k R | t R c t = k
77 74 76 eqtrd n = k c 0 n R | t R c t = n = c 0 k R | t R c t = k
78 77 adantl φ k 0 J n = k c 0 n R | t R c t = n = c 0 k R | t R c t = k
79 elfznn0 k 0 J k 0
80 79 adantl φ k 0 J k 0
81 fzfid φ 0 k Fin
82 mapfi 0 k Fin R Fin 0 k R Fin
83 81 16 82 syl2anc φ 0 k R Fin
84 83 adantr φ k 0 J 0 k R Fin
85 ssrab2 c 0 k R | t R c t = k 0 k R
86 85 a1i φ k 0 J c 0 k R | t R c t = k 0 k R
87 84 86 ssexd φ k 0 J c 0 k R | t R c t = k V
88 70 78 80 87 fvmptd φ k 0 J C R k = c 0 k R | t R c t = k
89 ssfi 0 k R Fin c 0 k R | t R c t = k 0 k R c 0 k R | t R c t = k Fin
90 83 85 89 sylancl φ c 0 k R | t R c t = k Fin
91 90 adantr φ k 0 J c 0 k R | t R c t = k Fin
92 88 91 eqeltrd φ k 0 J C R k Fin
93 92 adantr φ k 0 J x X C R k Fin
94 79 faccld k 0 J k !
95 94 nncnd k 0 J k !
96 95 ad2antlr φ k 0 J c C R k k !
97 16 adantr φ c C R k R Fin
98 97 adantlr φ k 0 J c C R k R Fin
99 elfznn0 z 0 k z 0
100 99 ssriv 0 k 0
101 100 a1i φ k 0 J c C R k t R 0 k 0
102 simpr φ k 0 J c C R k c C R k
103 88 eleq2d φ k 0 J c C R k c c 0 k R | t R c t = k
104 103 adantr φ k 0 J c C R k c C R k c c 0 k R | t R c t = k
105 102 104 mpbid φ k 0 J c C R k c c 0 k R | t R c t = k
106 85 sseli c c 0 k R | t R c t = k c 0 k R
107 105 106 syl φ k 0 J c C R k c 0 k R
108 elmapi c 0 k R c : R 0 k
109 107 108 syl φ k 0 J c C R k c : R 0 k
110 109 adantr φ k 0 J c C R k t R c : R 0 k
111 simpr φ k 0 J c C R k t R t R
112 110 111 ffvelrnd φ k 0 J c C R k t R c t 0 k
113 101 112 sseldd φ k 0 J c C R k t R c t 0
114 113 faccld φ k 0 J c C R k t R c t !
115 114 nncnd φ k 0 J c C R k t R c t !
116 98 115 fprodcl φ k 0 J c C R k t R c t !
117 114 nnne0d φ k 0 J c C R k t R c t ! 0
118 98 115 117 fprodn0 φ k 0 J c C R k t R c t ! 0
119 96 116 118 divcld φ k 0 J c C R k k ! t R c t !
120 119 adantlr φ k 0 J x X c C R k k ! t R c t !
121 98 adantlr φ k 0 J x X c C R k R Fin
122 29 ad4antr φ k 0 J x X c C R k t R φ
123 122 22 sylancom φ k 0 J x X c C R k t R t T
124 elfzuz3 k 0 J J k
125 fzss2 J k 0 k 0 J
126 124 125 syl k 0 J 0 k 0 J
127 126 adantl φ k 0 J 0 k 0 J
128 50 nn0zd φ J
129 5 nn0zd φ N
130 elfzle2 J 0 N J N
131 11 130 syl φ J N
132 128 129 131 3jca φ J N J N
133 eluz2 N J J N J N
134 132 133 sylibr φ N J
135 fzss2 N J 0 J 0 N
136 134 135 syl φ 0 J 0 N
137 136 adantr φ k 0 J 0 J 0 N
138 127 137 sstrd φ k 0 J 0 k 0 N
139 138 ad2antrr φ k 0 J c C R k t R 0 k 0 N
140 139 112 sseldd φ k 0 J c C R k t R c t 0 N
141 140 adantllr φ k 0 J x X c C R k t R c t 0 N
142 fvex c t V
143 eleq1 j = c t j 0 N c t 0 N
144 143 3anbi3d j = c t φ t T j 0 N φ t T c t 0 N
145 fveq2 j = c t S D n H t j = S D n H t c t
146 145 feq1d j = c t S D n H t j : X S D n H t c t : X
147 144 146 imbi12d j = c t φ t T j 0 N S D n H t j : X φ t T c t 0 N S D n H t c t : X
148 142 147 6 vtocl φ t T c t 0 N S D n H t c t : X
149 122 123 141 148 syl3anc φ k 0 J x X c C R k t R S D n H t c t : X
150 simpllr φ k 0 J x X c C R k t R x X
151 149 150 ffvelrnd φ k 0 J x X c C R k t R S D n H t c t x
152 121 151 fprodcl φ k 0 J x X c C R k t R S D n H t c t x
153 120 152 mulcld φ k 0 J x X c C R k k ! t R c t ! t R S D n H t c t x
154 93 153 fsumcl φ k 0 J x X c C R k k ! t R c t ! t R S D n H t c t x
155 154 fmpttd φ k 0 J x X c C R k k ! t R c t ! t R S D n H t c t x : X
156 10 adantr φ k 0 J k 0 N S D n x X t R H t x k = x X c C R k k ! t R c t ! t R S D n H t c t x
157 0zd φ k 0 J 0
158 129 adantr φ k 0 J N
159 elfzelz k 0 J k
160 159 adantl φ k 0 J k
161 elfzle1 k 0 J 0 k
162 161 adantl φ k 0 J 0 k
163 160 zred φ k 0 J k
164 50 nn0red φ J
165 164 adantr φ k 0 J J
166 158 zred φ k 0 J N
167 elfzle2 k 0 J k J
168 167 adantl φ k 0 J k J
169 131 adantr φ k 0 J J N
170 163 165 166 168 169 letrd φ k 0 J k N
171 157 158 160 162 170 elfzd φ k 0 J k 0 N
172 rspa k 0 N S D n x X t R H t x k = x X c C R k k ! t R c t ! t R S D n H t c t x k 0 N S D n x X t R H t x k = x X c C R k k ! t R c t ! t R S D n H t c t x
173 156 171 172 syl2anc φ k 0 J S D n x X t R H t x k = x X c C R k k ! t R c t ! t R S D n H t c t x
174 173 feq1d φ k 0 J S D n x X t R H t x k : X x X c C R k k ! t R c t ! t R S D n H t c t x : X
175 155 174 mpbird φ k 0 J S D n x X t R H t x k : X
176 31 adantr φ k 0 J Z T
177 simpl φ k 0 J φ
178 177 176 171 3jca φ k 0 J φ Z T k 0 N
179 34 3anbi2d t = Z φ t T k 0 N φ Z T k 0 N
180 27 oveq2d t = Z S D n H t = S D n H Z
181 180 fveq1d t = Z S D n H t k = S D n H Z k
182 181 feq1d t = Z S D n H t k : X S D n H Z k : X
183 179 182 imbi12d t = Z φ t T k 0 N S D n H t k : X φ Z T k 0 N S D n H Z k : X
184 eleq1 j = k j 0 N k 0 N
185 184 3anbi3d j = k φ t T j 0 N φ t T k 0 N
186 fveq2 j = k S D n H t j = S D n H t k
187 186 feq1d j = k S D n H t j : X S D n H t k : X
188 185 187 imbi12d j = k φ t T j 0 N S D n H t j : X φ t T k 0 N S D n H t k : X
189 188 6 chvarvv φ t T k 0 N S D n H t k : X
190 183 189 vtoclg Z T φ Z T k 0 N S D n H Z k : X
191 176 178 190 sylc φ k 0 J S D n H Z k : X
192 40 feqmptd φ H Z = x X H Z x
193 192 eqcomd φ x X H Z x = H Z
194 193 oveq2d φ S D n x X H Z x = S D n H Z
195 194 fveq1d φ S D n x X H Z x k = S D n H Z k
196 195 adantr φ k 0 J S D n x X H Z x k = S D n H Z k
197 196 feq1d φ k 0 J S D n x X H Z x k : X S D n H Z k : X
198 191 197 mpbird φ k 0 J S D n x X H Z x k : X
199 fveq2 y = x H t y = H t x
200 199 prodeq2ad y = x t R H t y = t R H t x
201 200 cbvmptv y X t R H t y = x X t R H t x
202 201 oveq2i S D n y X t R H t y = S D n x X t R H t x
203 202 fveq1i S D n y X t R H t y k = S D n x X t R H t x k
204 203 mpteq2i k 0 J S D n y X t R H t y k = k 0 J S D n x X t R H t x k
205 fveq2 y = x H Z y = H Z x
206 205 cbvmptv y X H Z y = x X H Z x
207 206 oveq2i S D n y X H Z y = S D n x X H Z x
208 207 fveq1i S D n y X H Z y k = S D n x X H Z x k
209 208 mpteq2i k 0 J S D n y X H Z y k = k 0 J S D n x X H Z x k
210 1 2 48 43 50 51 52 175 198 204 209 dvnmul φ S D n x X t R H t x H Z x J = x X k = 0 J ( J k) k 0 J S D n y X t R H t y k k x k 0 J S D n y X H Z y k J k x
211 203 a1i φ k 0 J S D n y X t R H t y k = S D n x X t R H t x k
212 10 r19.21bi φ k 0 N S D n x X t R H t x k = x X c C R k k ! t R c t ! t R S D n H t c t x
213 177 171 212 syl2anc φ k 0 J S D n x X t R H t x k = x X c C R k k ! t R c t ! t R S D n H t c t x
214 211 213 eqtrd φ k 0 J S D n y X t R H t y k = x X c C R k k ! t R c t ! t R S D n H t c t x
215 214 mpteq2dva φ k 0 J S D n y X t R H t y k = k 0 J x X c C R k k ! t R c t ! t R S D n H t c t x
216 mptexg X TopOpen fld 𝑡 S x X c C R k k ! t R c t ! t R S D n H t c t x V
217 2 216 syl φ x X c C R k k ! t R c t ! t R S D n H t c t x V
218 217 adantr φ k 0 J x X c C R k k ! t R c t ! t R S D n H t c t x V
219 215 218 fvmpt2d φ k 0 J k 0 J S D n y X t R H t y k k = x X c C R k k ! t R c t ! t R S D n H t c t x
220 219 adantlr φ x X k 0 J k 0 J S D n y X t R H t y k k = x X c C R k k ! t R c t ! t R S D n H t c t x
221 220 fveq1d φ x X k 0 J k 0 J S D n y X t R H t y k k x = x X c C R k k ! t R c t ! t R S D n H t c t x x
222 42 adantr φ x X k 0 J x X
223 154 an32s φ x X k 0 J c C R k k ! t R c t ! t R S D n H t c t x
224 eqid x X c C R k k ! t R c t ! t R S D n H t c t x = x X c C R k k ! t R c t ! t R S D n H t c t x
225 224 fvmpt2 x X c C R k k ! t R c t ! t R S D n H t c t x x X c C R k k ! t R c t ! t R S D n H t c t x x = c C R k k ! t R c t ! t R S D n H t c t x
226 222 223 225 syl2anc φ x X k 0 J x X c C R k k ! t R c t ! t R S D n H t c t x x = c C R k k ! t R c t ! t R S D n H t c t x
227 221 226 eqtrd φ x X k 0 J k 0 J S D n y X t R H t y k k x = c C R k k ! t R c t ! t R S D n H t c t x
228 fveq2 k = j S D n y X H Z y k = S D n y X H Z y j
229 228 cbvmptv k 0 J S D n y X H Z y k = j 0 J S D n y X H Z y j
230 229 a1i φ k 0 J S D n y X H Z y k = j 0 J S D n y X H Z y j
231 207 194 syl5eq φ S D n y X H Z y = S D n H Z
232 231 fveq1d φ S D n y X H Z y j = S D n H Z j
233 232 mpteq2dv φ j 0 J S D n y X H Z y j = j 0 J S D n H Z j
234 230 233 eqtrd φ k 0 J S D n y X H Z y k = j 0 J S D n H Z j
235 234 adantr φ k 0 J k 0 J S D n y X H Z y k = j 0 J S D n H Z j
236 fveq2 j = J k S D n H Z j = S D n H Z J k
237 236 adantl φ k 0 J j = J k S D n H Z j = S D n H Z J k
238 0zd k 0 J 0
239 elfzel2 k 0 J J
240 239 159 zsubcld k 0 J J k
241 238 239 240 3jca k 0 J 0 J J k
242 239 zred k 0 J J
243 79 nn0red k 0 J k
244 242 243 subge0d k 0 J 0 J k k J
245 167 244 mpbird k 0 J 0 J k
246 242 243 subge02d k 0 J 0 k J k J
247 161 246 mpbid k 0 J J k J
248 241 245 247 jca32 k 0 J 0 J J k 0 J k J k J
249 248 adantl φ k 0 J 0 J J k 0 J k J k J
250 elfz2 J k 0 J 0 J J k 0 J k J k J
251 249 250 sylibr φ k 0 J J k 0 J
252 fvex S D n H Z J k V
253 252 a1i φ k 0 J S D n H Z J k V
254 235 237 251 253 fvmptd φ k 0 J k 0 J S D n y X H Z y k J k = S D n H Z J k
255 254 adantlr φ x X k 0 J k 0 J S D n y X H Z y k J k = S D n H Z J k
256 255 fveq1d φ x X k 0 J k 0 J S D n y X H Z y k J k x = S D n H Z J k x
257 227 256 oveq12d φ x X k 0 J k 0 J S D n y X t R H t y k k x k 0 J S D n y X H Z y k J k x = c C R k k ! t R c t ! t R S D n H t c t x S D n H Z J k x
258 257 oveq2d φ x X k 0 J ( J k) k 0 J S D n y X t R H t y k k x k 0 J S D n y X H Z y k J k x = ( J k) c C R k k ! t R c t ! t R S D n H t c t x S D n H Z J k x
259 92 adantlr φ x X k 0 J C R k Fin
260 ovex J k V
261 eleq1 j = J k j 0 J J k 0 J
262 261 anbi2d j = J k φ j 0 J φ J k 0 J
263 236 feq1d j = J k S D n H Z j : X S D n H Z J k : X
264 262 263 imbi12d j = J k φ j 0 J S D n H Z j : X φ J k 0 J S D n H Z J k : X
265 eleq1 k = j k 0 J j 0 J
266 265 anbi2d k = j φ k 0 J φ j 0 J
267 fveq2 k = j S D n H Z k = S D n H Z j
268 267 feq1d k = j S D n H Z k : X S D n H Z j : X
269 266 268 imbi12d k = j φ k 0 J S D n H Z k : X φ j 0 J S D n H Z j : X
270 269 191 chvarvv φ j 0 J S D n H Z j : X
271 260 264 270 vtocl φ J k 0 J S D n H Z J k : X
272 177 251 271 syl2anc φ k 0 J S D n H Z J k : X
273 272 adantlr φ x X k 0 J S D n H Z J k : X
274 273 222 ffvelrnd φ x X k 0 J S D n H Z J k x
275 anass φ k 0 J x X φ k 0 J x X
276 ancom k 0 J x X x X k 0 J
277 276 anbi2i φ k 0 J x X φ x X k 0 J
278 anass φ x X k 0 J φ x X k 0 J
279 278 bicomi φ x X k 0 J φ x X k 0 J
280 277 279 bitri φ k 0 J x X φ x X k 0 J
281 275 280 bitri φ k 0 J x X φ x X k 0 J
282 281 anbi1i φ k 0 J x X c C R k φ x X k 0 J c C R k
283 282 imbi1i φ k 0 J x X c C R k k ! t R c t ! t R S D n H t c t x φ x X k 0 J c C R k k ! t R c t ! t R S D n H t c t x
284 153 283 mpbi φ x X k 0 J c C R k k ! t R c t ! t R S D n H t c t x
285 259 274 284 fsummulc1 φ x X k 0 J c C R k k ! t R c t ! t R S D n H t c t x S D n H Z J k x = c C R k k ! t R c t ! t R S D n H t c t x S D n H Z J k x
286 285 oveq2d φ x X k 0 J ( J k) c C R k k ! t R c t ! t R S D n H t c t x S D n H Z J k x = ( J k) c C R k k ! t R c t ! t R S D n H t c t x S D n H Z J k x
287 177 50 syl φ k 0 J J 0
288 287 160 bccld φ k 0 J ( J k) 0
289 288 nn0cnd φ k 0 J ( J k)
290 289 adantlr φ x X k 0 J ( J k)
291 274 adantr φ x X k 0 J c C R k S D n H Z J k x
292 284 291 mulcld φ x X k 0 J c C R k k ! t R c t ! t R S D n H t c t x S D n H Z J k x
293 259 290 292 fsummulc2 φ x X k 0 J ( J k) c C R k k ! t R c t ! t R S D n H t c t x S D n H Z J k x = c C R k ( J k) k ! t R c t ! t R S D n H t c t x S D n H Z J k x
294 258 286 293 3eqtrd φ x X k 0 J ( J k) k 0 J S D n y X t R H t y k k x k 0 J S D n y X H Z y k J k x = c C R k ( J k) k ! t R c t ! t R S D n H t c t x S D n H Z J k x
295 294 sumeq2dv φ x X k = 0 J ( J k) k 0 J S D n y X t R H t y k k x k 0 J S D n y X H Z y k J k x = k = 0 J c C R k ( J k) k ! t R c t ! t R S D n H t c t x S D n H Z J k x
296 vex k V
297 vex c V
298 296 297 op1std p = k c 1 st p = k
299 298 oveq2d p = k c ( J 1 st p ) = ( J k)
300 298 fveq2d p = k c 1 st p ! = k !
301 296 297 op2ndd p = k c 2 nd p = c
302 301 fveq1d p = k c 2 nd p t = c t
303 302 fveq2d p = k c 2 nd p t ! = c t !
304 303 prodeq2ad p = k c t R 2 nd p t ! = t R c t !
305 300 304 oveq12d p = k c 1 st p ! t R 2 nd p t ! = k ! t R c t !
306 302 fveq2d p = k c S D n H t 2 nd p t = S D n H t c t
307 306 fveq1d p = k c S D n H t 2 nd p t x = S D n H t c t x
308 307 prodeq2ad p = k c t R S D n H t 2 nd p t x = t R S D n H t c t x
309 305 308 oveq12d p = k c 1 st p ! t R 2 nd p t ! t R S D n H t 2 nd p t x = k ! t R c t ! t R S D n H t c t x
310 298 oveq2d p = k c J 1 st p = J k
311 310 fveq2d p = k c S D n H Z J 1 st p = S D n H Z J k
312 311 fveq1d p = k c S D n H Z J 1 st p x = S D n H Z J k x
313 309 312 oveq12d p = k c 1 st p ! t R 2 nd p t ! t R S D n H t 2 nd p t x S D n H Z J 1 st p x = k ! t R c t ! t R S D n H t c t x S D n H Z J k x
314 299 313 oveq12d p = k c ( J 1 st p ) 1 st p ! t R 2 nd p t ! t R S D n H t 2 nd p t x S D n H Z J 1 st p x = ( J k) k ! t R c t ! t R S D n H t c t x S D n H Z J k x
315 fzfid φ x X 0 J Fin
316 290 adantrr φ x X k 0 J c C R k ( J k)
317 292 anasss φ x X k 0 J c C R k k ! t R c t ! t R S D n H t c t x S D n H Z J k x
318 316 317 mulcld φ x X k 0 J c C R k ( J k) k ! t R c t ! t R S D n H t c t x S D n H Z J k x
319 314 315 259 318 fsum2d φ x X k = 0 J c C R k ( J k) k ! t R c t ! t R S D n H t c t x S D n H Z J k x = p k = 0 J k × C R k ( J 1 st p ) 1 st p ! t R 2 nd p t ! t R S D n H t 2 nd p t x S D n H Z J 1 st p x
320 ovex J c Z V
321 297 resex c R V
322 320 321 op1std p = J c Z c R 1 st p = J c Z
323 322 oveq2d p = J c Z c R ( J 1 st p ) = ( J J c Z )
324 322 fveq2d p = J c Z c R 1 st p ! = J c Z !
325 320 321 op2ndd p = J c Z c R 2 nd p = c R
326 325 fveq1d p = J c Z c R 2 nd p t = c R t
327 326 fveq2d p = J c Z c R 2 nd p t ! = c R t !
328 327 prodeq2ad p = J c Z c R t R 2 nd p t ! = t R c R t !
329 324 328 oveq12d p = J c Z c R 1 st p ! t R 2 nd p t ! = J c Z ! t R c R t !
330 326 fveq2d p = J c Z c R S D n H t 2 nd p t = S D n H t c R t
331 330 fveq1d p = J c Z c R S D n H t 2 nd p t x = S D n H t c R t x
332 331 prodeq2ad p = J c Z c R t R S D n H t 2 nd p t x = t R S D n H t c R t x
333 329 332 oveq12d p = J c Z c R 1 st p ! t R 2 nd p t ! t R S D n H t 2 nd p t x = J c Z ! t R c R t ! t R S D n H t c R t x
334 322 oveq2d p = J c Z c R J 1 st p = J J c Z
335 334 fveq2d p = J c Z c R S D n H Z J 1 st p = S D n H Z J J c Z
336 335 fveq1d p = J c Z c R S D n H Z J 1 st p x = S D n H Z J J c Z x
337 333 336 oveq12d p = J c Z c R 1 st p ! t R 2 nd p t ! t R S D n H t 2 nd p t x S D n H Z J 1 st p x = J c Z ! t R c R t ! t R S D n H t c R t x S D n H Z J J c Z x
338 323 337 oveq12d p = J c Z c R ( J 1 st p ) 1 st p ! t R 2 nd p t ! t R S D n H t 2 nd p t x S D n H Z J 1 st p x = ( J J c Z ) J c Z ! t R c R t ! t R S D n H t c R t x S D n H Z J J c Z x
339 oveq2 s = R Z 0 n s = 0 n R Z
340 rabeq 0 n s = 0 n R Z c 0 n s | t s c t = n = c 0 n R Z | t s c t = n
341 339 340 syl s = R Z c 0 n s | t s c t = n = c 0 n R Z | t s c t = n
342 sumeq1 s = R Z t s c t = t R Z c t
343 342 eqeq1d s = R Z t s c t = n t R Z c t = n
344 343 rabbidv s = R Z c 0 n R Z | t s c t = n = c 0 n R Z | t R Z c t = n
345 341 344 eqtrd s = R Z c 0 n s | t s c t = n = c 0 n R Z | t R Z c t = n
346 345 mpteq2dv s = R Z n 0 c 0 n s | t s c t = n = n 0 c 0 n R Z | t R Z c t = n
347 31 snssd φ Z T
348 8 347 unssd φ R Z T
349 3 348 ssexd φ R Z V
350 elpwg R Z V R Z 𝒫 T R Z T
351 349 350 syl φ R Z 𝒫 T R Z T
352 348 351 mpbird φ R Z 𝒫 T
353 67 mptex n 0 c 0 n R Z | t R Z c t = n V
354 353 a1i φ n 0 c 0 n R Z | t R Z c t = n V
355 7 346 352 354 fvmptd3 φ C R Z = n 0 c 0 n R Z | t R Z c t = n
356 oveq2 n = J 0 n = 0 J
357 356 oveq1d n = J 0 n R Z = 0 J R Z
358 rabeq 0 n R Z = 0 J R Z c 0 n R Z | t R Z c t = n = c 0 J R Z | t R Z c t = n
359 357 358 syl n = J c 0 n R Z | t R Z c t = n = c 0 J R Z | t R Z c t = n
360 eqeq2 n = J t R Z c t = n t R Z c t = J
361 360 rabbidv n = J c 0 J R Z | t R Z c t = n = c 0 J R Z | t R Z c t = J
362 359 361 eqtrd n = J c 0 n R Z | t R Z c t = n = c 0 J R Z | t R Z c t = J
363 362 adantl φ n = J c 0 n R Z | t R Z c t = n = c 0 J R Z | t R Z c t = J
364 ovex 0 J R Z V
365 364 rabex c 0 J R Z | t R Z c t = J V
366 365 a1i φ c 0 J R Z | t R Z c t = J V
367 355 363 50 366 fvmptd φ C R Z J = c 0 J R Z | t R Z c t = J
368 fzfid φ 0 J Fin
369 snfi Z Fin
370 369 a1i φ Z Fin
371 unfi R Fin Z Fin R Z Fin
372 16 370 371 syl2anc φ R Z Fin
373 mapfi 0 J Fin R Z Fin 0 J R Z Fin
374 368 372 373 syl2anc φ 0 J R Z Fin
375 ssrab2 c 0 J R Z | t R Z c t = J 0 J R Z
376 375 a1i φ c 0 J R Z | t R Z c t = J 0 J R Z
377 ssfi 0 J R Z Fin c 0 J R Z | t R Z c t = J 0 J R Z c 0 J R Z | t R Z c t = J Fin
378 374 376 377 syl2anc φ c 0 J R Z | t R Z c t = J Fin
379 367 378 eqeltrd φ C R Z J Fin
380 379 adantr φ x X C R Z J Fin
381 7 50 12 3 31 19 348 dvnprodlem1 φ D : C R Z J 1-1 onto k = 0 J k × C R k
382 381 adantr φ x X D : C R Z J 1-1 onto k = 0 J k × C R k
383 simpr φ c C R Z J c C R Z J
384 opex J c Z c R V
385 384 a1i φ c C R Z J J c Z c R V
386 12 fvmpt2 c C R Z J J c Z c R V D c = J c Z c R
387 383 385 386 syl2anc φ c C R Z J D c = J c Z c R
388 387 adantlr φ x X c C R Z J D c = J c Z c R
389 50 adantr φ p k = 0 J k × C R k J 0
390 eliun p k = 0 J k × C R k k 0 J p k × C R k
391 390 biimpi p k = 0 J k × C R k k 0 J p k × C R k
392 391 adantl φ p k = 0 J k × C R k k 0 J p k × C R k
393 nfv k φ
394 nfcv _ k p
395 nfiu1 _ k k = 0 J k × C R k
396 394 395 nfel k p k = 0 J k × C R k
397 393 396 nfan k φ p k = 0 J k × C R k
398 nfv k 1 st p 0 J
399 xp1st p k × C R k 1 st p k
400 elsni 1 st p k 1 st p = k
401 399 400 syl p k × C R k 1 st p = k
402 401 adantl k 0 J p k × C R k 1 st p = k
403 simpl k 0 J p k × C R k k 0 J
404 402 403 eqeltrd k 0 J p k × C R k 1 st p 0 J
405 404 ex k 0 J p k × C R k 1 st p 0 J
406 405 a1i φ p k = 0 J k × C R k k 0 J p k × C R k 1 st p 0 J
407 397 398 406 rexlimd φ p k = 0 J k × C R k k 0 J p k × C R k 1 st p 0 J
408 392 407 mpd φ p k = 0 J k × C R k 1 st p 0 J
409 elfzelz 1 st p 0 J 1 st p
410 408 409 syl φ p k = 0 J k × C R k 1 st p
411 389 410 bccld φ p k = 0 J k × C R k ( J 1 st p ) 0
412 411 nn0cnd φ p k = 0 J k × C R k ( J 1 st p )
413 412 adantlr φ x X p k = 0 J k × C R k ( J 1 st p )
414 elfznn0 1 st p 0 J 1 st p 0
415 408 414 syl φ p k = 0 J k × C R k 1 st p 0
416 415 faccld φ p k = 0 J k × C R k 1 st p !
417 416 nncnd φ p k = 0 J k × C R k 1 st p !
418 417 adantlr φ x X p k = 0 J k × C R k 1 st p !
419 16 adantr φ p k = 0 J k × C R k R Fin
420 nfv k 2 nd p : R 0 J
421 88 86 eqsstrd φ k 0 J C R k 0 k R
422 ovex 0 J V
423 422 a1i k 0 J 0 J V
424 mapss 0 J V 0 k 0 J 0 k R 0 J R
425 423 126 424 syl2anc k 0 J 0 k R 0 J R
426 425 adantl φ k 0 J 0 k R 0 J R
427 421 426 sstrd φ k 0 J C R k 0 J R
428 427 3adant3 φ k 0 J p k × C R k C R k 0 J R
429 xp2nd p k × C R k 2 nd p C R k
430 429 3ad2ant3 φ k 0 J p k × C R k 2 nd p C R k
431 428 430 sseldd φ k 0 J p k × C R k 2 nd p 0 J R
432 elmapi 2 nd p 0 J R 2 nd p : R 0 J
433 431 432 syl φ k 0 J p k × C R k 2 nd p : R 0 J
434 433 3exp φ k 0 J p k × C R k 2 nd p : R 0 J
435 434 adantr φ p k = 0 J k × C R k k 0 J p k × C R k 2 nd p : R 0 J
436 397 420 435 rexlimd φ p k = 0 J k × C R k k 0 J p k × C R k 2 nd p : R 0 J
437 392 436 mpd φ p k = 0 J k × C R k 2 nd p : R 0 J
438 437 ffvelrnda φ p k = 0 J k × C R k t R 2 nd p t 0 J
439 elfznn0 2 nd p t 0 J 2 nd p t 0
440 439 faccld 2 nd p t 0 J 2 nd p t !
441 440 nncnd 2 nd p t 0 J 2 nd p t !
442 438 441 syl φ p k = 0 J k × C R k t R 2 nd p t !
443 419 442 fprodcl φ p k = 0 J k × C R k t R 2 nd p t !
444 443 adantlr φ x X p k = 0 J k × C R k t R 2 nd p t !
445 438 440 syl φ p k = 0 J k × C R k t R 2 nd p t !
446 nnne0 2 nd p t ! 2 nd p t ! 0
447 445 446 syl φ p k = 0 J k × C R k t R 2 nd p t ! 0
448 419 442 447 fprodn0 φ p k = 0 J k × C R k t R 2 nd p t ! 0
449 448 adantlr φ x X p k = 0 J k × C R k t R 2 nd p t ! 0
450 418 444 449 divcld φ x X p k = 0 J k × C R k 1 st p ! t R 2 nd p t !
451 17 adantr φ x X p k = 0 J k × C R k R Fin
452 simpll φ p k = 0 J k × C R k t R φ
453 452 22 sylancom φ p k = 0 J k × C R k t R t T
454 452 136 syl φ p k = 0 J k × C R k t R 0 J 0 N
455 454 438 sseldd φ p k = 0 J k × C R k t R 2 nd p t 0 N
456 452 453 455 3jca φ p k = 0 J k × C R k t R φ t T 2 nd p t 0 N
457 eleq1 j = 2 nd p t j 0 N 2 nd p t 0 N
458 457 3anbi3d j = 2 nd p t φ t T j 0 N φ t T 2 nd p t 0 N
459 fveq2 j = 2 nd p t S D n H t j = S D n H t 2 nd p t
460 459 feq1d j = 2 nd p t S D n H t j : X S D n H t 2 nd p t : X
461 458 460 imbi12d j = 2 nd p t φ t T j 0 N S D n H t j : X φ t T 2 nd p t 0 N S D n H t 2 nd p t : X
462 461 6 vtoclg 2 nd p t 0 J φ t T 2 nd p t 0 N S D n H t 2 nd p t : X
463 438 456 462 sylc φ p k = 0 J k × C R k t R S D n H t 2 nd p t : X
464 463 adantllr φ x X p k = 0 J k × C R k t R S D n H t 2 nd p t : X
465 25 adantlr φ x X p k = 0 J k × C R k t R x X
466 464 465 ffvelrnd φ x X p k = 0 J k × C R k t R S D n H t 2 nd p t x
467 451 466 fprodcl φ x X p k = 0 J k × C R k t R S D n H t 2 nd p t x
468 450 467 mulcld φ x X p k = 0 J k × C R k 1 st p ! t R 2 nd p t ! t R S D n H t 2 nd p t x
469 nfv k J 1 st p 0 J
470 simp1 φ k 0 J p k × C R k φ
471 404 3adant1 φ k 0 J p k × C R k 1 st p 0 J
472 fznn0sub2 1 st p 0 J J 1 st p 0 J
473 472 adantl φ 1 st p 0 J J 1 st p 0 J
474 470 471 473 syl2anc φ k 0 J p k × C R k J 1 st p 0 J
475 474 3exp φ k 0 J p k × C R k J 1 st p 0 J
476 475 adantr φ p k = 0 J k × C R k k 0 J p k × C R k J 1 st p 0 J
477 397 469 476 rexlimd φ p k = 0 J k × C R k k 0 J p k × C R k J 1 st p 0 J
478 392 477 mpd φ p k = 0 J k × C R k J 1 st p 0 J
479 simpl φ p k = 0 J k × C R k φ
480 479 31 syl φ p k = 0 J k × C R k Z T
481 479 136 syl φ p k = 0 J k × C R k 0 J 0 N
482 481 478 sseldd φ p k = 0 J k × C R k J 1 st p 0 N
483 479 480 482 3jca φ p k = 0 J k × C R k φ Z T J 1 st p 0 N
484 eleq1 j = J 1 st p j 0 N J 1 st p 0 N
485 484 3anbi3d j = J 1 st p φ Z T j 0 N φ Z T J 1 st p 0 N
486 fveq2 j = J 1 st p S D n H Z j = S D n H Z J 1 st p
487 486 feq1d j = J 1 st p S D n H Z j : X S D n H Z J 1 st p : X
488 485 487 imbi12d j = J 1 st p φ Z T j 0 N S D n H Z j : X φ Z T J 1 st p 0 N S D n H Z J 1 st p : X
489 simp2 φ Z T j 0 N Z T
490 id φ Z T j 0 N φ Z T j 0 N
491 34 3anbi2d t = Z φ t T j 0 N φ Z T j 0 N
492 180 fveq1d t = Z S D n H t j = S D n H Z j
493 492 feq1d t = Z S D n H t j : X S D n H Z j : X
494 491 493 imbi12d t = Z φ t T j 0 N S D n H t j : X φ Z T j 0 N S D n H Z j : X
495 494 6 vtoclg Z T φ Z T j 0 N S D n H Z j : X
496 489 490 495 sylc φ Z T j 0 N S D n H Z j : X
497 488 496 vtoclg J 1 st p 0 J φ Z T J 1 st p 0 N S D n H Z J 1 st p : X
498 478 483 497 sylc φ p k = 0 J k × C R k S D n H Z J 1 st p : X
499 498 adantlr φ x X p k = 0 J k × C R k S D n H Z J 1 st p : X
500 42 adantr φ x X p k = 0 J k × C R k x X
501 499 500 ffvelrnd φ x X p k = 0 J k × C R k S D n H Z J 1 st p x
502 468 501 mulcld φ x X p k = 0 J k × C R k 1 st p ! t R 2 nd p t ! t R S D n H t 2 nd p t x S D n H Z J 1 st p x
503 413 502 mulcld φ x X p k = 0 J k × C R k ( J 1 st p ) 1 st p ! t R 2 nd p t ! t R S D n H t 2 nd p t x S D n H Z J 1 st p x
504 338 380 382 388 503 fsumf1o φ x X p k = 0 J k × C R k ( J 1 st p ) 1 st p ! t R 2 nd p t ! t R S D n H t 2 nd p t x S D n H Z J 1 st p x = c C R Z J ( J J c Z ) J c Z ! t R c R t ! t R S D n H t c R t x S D n H Z J J c Z x
505 simpl φ c C R Z J φ
506 367 adantr φ c C R Z J C R Z J = c 0 J R Z | t R Z c t = J
507 383 506 eleqtrd φ c C R Z J c c 0 J R Z | t R Z c t = J
508 375 sseli c c 0 J R Z | t R Z c t = J c 0 J R Z
509 507 508 syl φ c C R Z J c 0 J R Z
510 elmapi c 0 J R Z c : R Z 0 J
511 509 510 syl φ c C R Z J c : R Z 0 J
512 snidg Z T Z Z
513 31 512 syl φ Z Z
514 elun2 Z Z Z R Z
515 513 514 syl φ Z R Z
516 515 adantr φ c C R Z J Z R Z
517 511 516 ffvelrnd φ c C R Z J c Z 0 J
518 0zd φ c Z 0 J 0
519 128 adantr φ c Z 0 J J
520 fzssz 0 J
521 520 sseli c Z 0 J c Z
522 521 adantl φ c Z 0 J c Z
523 519 522 zsubcld φ c Z 0 J J c Z
524 elfzle2 c Z 0 J c Z J
525 524 adantl φ c Z 0 J c Z J
526 164 adantr φ c Z 0 J J
527 522 zred φ c Z 0 J c Z
528 526 527 subge0d φ c Z 0 J 0 J c Z c Z J
529 525 528 mpbird φ c Z 0 J 0 J c Z
530 elfzle1 c Z 0 J 0 c Z
531 530 adantl φ c Z 0 J 0 c Z
532 526 527 subge02d φ c Z 0 J 0 c Z J c Z J
533 531 532 mpbid φ c Z 0 J J c Z J
534 518 519 523 529 533 elfzd φ c Z 0 J J c Z 0 J
535 505 517 534 syl2anc φ c C R Z J J c Z 0 J
536 bcval2 J c Z 0 J ( J J c Z ) = J ! J J c Z ! J c Z !
537 535 536 syl φ c C R Z J ( J J c Z ) = J ! J J c Z ! J c Z !
538 164 recnd φ J
539 538 adantr φ c C R Z J J
540 zsscn
541 520 540 sstri 0 J
542 541 a1i φ c C R Z J 0 J
543 542 517 sseldd φ c C R Z J c Z
544 539 543 nncand φ c C R Z J J J c Z = c Z
545 544 fveq2d φ c C R Z J J J c Z ! = c Z !
546 545 oveq1d φ c C R Z J J J c Z ! J c Z ! = c Z ! J c Z !
547 546 oveq2d φ c C R Z J J ! J J c Z ! J c Z ! = J ! c Z ! J c Z !
548 50 faccld φ J !
549 548 nncnd φ J !
550 549 adantr φ c C R Z J J !
551 elfznn0 c Z 0 J c Z 0
552 517 551 syl φ c C R Z J c Z 0
553 552 faccld φ c C R Z J c Z !
554 553 nncnd φ c C R Z J c Z !
555 elfznn0 J c Z 0 J J c Z 0
556 535 555 syl φ c C R Z J J c Z 0
557 556 faccld φ c C R Z J J c Z !
558 557 nncnd φ c C R Z J J c Z !
559 553 nnne0d φ c C R Z J c Z ! 0
560 557 nnne0d φ c C R Z J J c Z ! 0
561 550 554 558 559 560 divdiv1d φ c C R Z J J ! c Z ! J c Z ! = J ! c Z ! J c Z !
562 561 eqcomd φ c C R Z J J ! c Z ! J c Z ! = J ! c Z ! J c Z !
563 537 547 562 3eqtrd φ c C R Z J ( J J c Z ) = J ! c Z ! J c Z !
564 563 adantlr φ x X c C R Z J ( J J c Z ) = J ! c Z ! J c Z !
565 fvres t R c R t = c t
566 565 fveq2d t R c R t ! = c t !
567 566 prodeq2i t R c R t ! = t R c t !
568 567 oveq2i J c Z ! t R c R t ! = J c Z ! t R c t !
569 565 fveq2d t R S D n H t c R t = S D n H t c t
570 569 fveq1d t R S D n H t c R t x = S D n H t c t x
571 570 prodeq2i t R S D n H t c R t x = t R S D n H t c t x
572 568 571 oveq12i J c Z ! t R c R t ! t R S D n H t c R t x = J c Z ! t R c t ! t R S D n H t c t x
573 572 a1i φ c C R Z J J c Z ! t R c R t ! t R S D n H t c R t x = J c Z ! t R c t ! t R S D n H t c t x
574 573 adantlr φ x X c C R Z J J c Z ! t R c R t ! t R S D n H t c R t x = J c Z ! t R c t ! t R S D n H t c t x
575 558 adantlr φ x X c C R Z J J c Z !
576 505 16 syl φ c C R Z J R Fin
577 79 ssriv 0 J 0
578 577 a1i φ c C R Z J t R 0 J 0
579 511 adantr φ c C R Z J t R c : R Z 0 J
580 elun1 t R t R Z
581 580 adantl φ c C R Z J t R t R Z
582 579 581 ffvelrnd φ c C R Z J t R c t 0 J
583 578 582 sseldd φ c C R Z J t R c t 0
584 583 faccld φ c C R Z J t R c t !
585 584 nncnd φ c C R Z J t R c t !
586 576 585 fprodcl φ c C R Z J t R c t !
587 586 adantlr φ x X c C R Z J t R c t !
588 17 adantr φ x X c C R Z J R Fin
589 505 adantr φ c C R Z J t R φ
590 505 22 sylan φ c C R Z J t R t T
591 589 136 syl φ c C R Z J t R 0 J 0 N
592 591 582 sseldd φ c C R Z J t R c t 0 N
593 589 590 592 148 syl3anc φ c C R Z J t R S D n H t c t : X
594 593 adantllr φ x X c C R Z J t R S D n H t c t : X
595 25 adantlr φ x X c C R Z J t R x X
596 594 595 ffvelrnd φ x X c C R Z J t R S D n H t c t x
597 588 596 fprodcl φ x X c C R Z J t R S D n H t c t x
598 576 584 fprodnncl φ c C R Z J t R c t !
599 nnne0 t R c t ! t R c t ! 0
600 598 599 syl φ c C R Z J t R c t ! 0
601 600 adantlr φ x X c C R Z J t R c t ! 0
602 575 587 597 601 div32d φ x X c C R Z J J c Z ! t R c t ! t R S D n H t c t x = J c Z ! t R S D n H t c t x t R c t !
603 574 602 eqtrd φ x X c C R Z J J c Z ! t R c R t ! t R S D n H t c R t x = J c Z ! t R S D n H t c t x t R c t !
604 544 fveq2d φ c C R Z J S D n H Z J J c Z = S D n H Z c Z
605 604 fveq1d φ c C R Z J S D n H Z J J c Z x = S D n H Z c Z x
606 605 adantlr φ x X c C R Z J S D n H Z J J c Z x = S D n H Z c Z x
607 603 606 oveq12d φ x X c C R Z J J c Z ! t R c R t ! t R S D n H t c R t x S D n H Z J J c Z x = J c Z ! t R S D n H t c t x t R c t ! S D n H Z c Z x
608 597 587 601 divcld φ x X c C R Z J t R S D n H t c t x t R c t !
609 505 31 syl φ c C R Z J Z T
610 505 136 syl φ c C R Z J 0 J 0 N
611 610 517 sseldd φ c C R Z J c Z 0 N
612 505 609 611 3jca φ c C R Z J φ Z T c Z 0 N
613 eleq1 j = c Z j 0 N c Z 0 N
614 613 3anbi3d j = c Z φ Z T j 0 N φ Z T c Z 0 N
615 fveq2 j = c Z S D n H Z j = S D n H Z c Z
616 615 feq1d j = c Z S D n H Z j : X S D n H Z c Z : X
617 614 616 imbi12d j = c Z φ Z T j 0 N S D n H Z j : X φ Z T c Z 0 N S D n H Z c Z : X
618 617 496 vtoclg c Z 0 J φ Z T c Z 0 N S D n H Z c Z : X
619 517 612 618 sylc φ c C R Z J S D n H Z c Z : X
620 619 adantlr φ x X c C R Z J S D n H Z c Z : X
621 42 adantr φ x X c C R Z J x X
622 620 621 ffvelrnd φ x X c C R Z J S D n H Z c Z x
623 575 608 622 mulassd φ x X c C R Z J J c Z ! t R S D n H t c t x t R c t ! S D n H Z c Z x = J c Z ! t R S D n H t c t x t R c t ! S D n H Z c Z x
624 607 623 eqtrd φ x X c C R Z J J c Z ! t R c R t ! t R S D n H t c R t x S D n H Z J J c Z x = J c Z ! t R S D n H t c t x t R c t ! S D n H Z c Z x
625 564 624 oveq12d φ x X c C R Z J ( J J c Z ) J c Z ! t R c R t ! t R S D n H t c R t x S D n H Z J J c Z x = J ! c Z ! J c Z ! J c Z ! t R S D n H t c t x t R c t ! S D n H Z c Z x
626 549 ad2antrr φ x X c C R Z J J !
627 554 adantlr φ x X c C R Z J c Z !
628 559 adantlr φ x X c C R Z J c Z ! 0
629 626 627 628 divcld φ x X c C R Z J J ! c Z !
630 608 622 mulcld φ x X c C R Z J t R S D n H t c t x t R c t ! S D n H Z c Z x
631 560 adantlr φ x X c C R Z J J c Z ! 0
632 629 575 630 631 dmmcand φ x X c C R Z J J ! c Z ! J c Z ! J c Z ! t R S D n H t c t x t R c t ! S D n H Z c Z x = J ! c Z ! t R S D n H t c t x t R c t ! S D n H Z c Z x
633 597 622 587 601 div23d φ x X c C R Z J t R S D n H t c t x S D n H Z c Z x t R c t ! = t R S D n H t c t x t R c t ! S D n H Z c Z x
634 633 eqcomd φ x X c C R Z J t R S D n H t c t x t R c t ! S D n H Z c Z x = t R S D n H t c t x S D n H Z c Z x t R c t !
635 nfv t φ x X c C R Z J
636 nfcv _ t S D n H Z c Z x
637 609 adantlr φ x X c C R Z J Z T
638 20 adantr φ x X c C R Z J ¬ Z R
639 fveq2 t = Z c t = c Z
640 180 639 fveq12d t = Z S D n H t c t = S D n H Z c Z
641 640 fveq1d t = Z S D n H t c t x = S D n H Z c Z x
642 635 636 588 637 638 596 641 622 fprodsplitsn φ x X c C R Z J t R Z S D n H t c t x = t R S D n H t c t x S D n H Z c Z x
643 642 eqcomd φ x X c C R Z J t R S D n H t c t x S D n H Z c Z x = t R Z S D n H t c t x
644 643 oveq1d φ x X c C R Z J t R S D n H t c t x S D n H Z c Z x t R c t ! = t R Z S D n H t c t x t R c t !
645 634 644 eqtrd φ x X c C R Z J t R S D n H t c t x t R c t ! S D n H Z c Z x = t R Z S D n H t c t x t R c t !
646 645 oveq2d φ x X c C R Z J J ! c Z ! t R S D n H t c t x t R c t ! S D n H Z c Z x = J ! c Z ! t R Z S D n H t c t x t R c t !
647 588 369 371 sylancl φ x X c C R Z J R Z Fin
648 505 adantr φ c C R Z J t R Z φ
649 348 sselda φ t R Z t T
650 649 adantlr φ c C R Z J t R Z t T
651 511 610 fssd φ c C R Z J c : R Z 0 N
652 651 ffvelrnda φ c C R Z J t R Z c t 0 N
653 648 650 652 148 syl3anc φ c C R Z J t R Z S D n H t c t : X
654 653 adantllr φ x X c C R Z J t R Z S D n H t c t : X
655 621 adantr φ x X c C R Z J t R Z x X
656 654 655 ffvelrnd φ x X c C R Z J t R Z S D n H t c t x
657 647 656 fprodcl φ x X c C R Z J t R Z S D n H t c t x
658 626 627 657 587 628 601 divmuldivd φ x X c C R Z J J ! c Z ! t R Z S D n H t c t x t R c t ! = J ! t R Z S D n H t c t x c Z ! t R c t !
659 554 586 mulcomd φ c C R Z J c Z ! t R c t ! = t R c t ! c Z !
660 nfv t φ c C R Z J
661 nfcv _ t c Z !
662 505 19 syl φ c C R Z J ¬ Z R
663 639 fveq2d t = Z c t ! = c Z !
664 660 661 576 609 662 585 663 554 fprodsplitsn φ c C R Z J t R Z c t ! = t R c t ! c Z !
665 664 eqcomd φ c C R Z J t R c t ! c Z ! = t R Z c t !
666 659 665 eqtrd φ c C R Z J c Z ! t R c t ! = t R Z c t !
667 666 oveq2d φ c C R Z J J ! t R Z S D n H t c t x c Z ! t R c t ! = J ! t R Z S D n H t c t x t R Z c t !
668 667 adantlr φ x X c C R Z J J ! t R Z S D n H t c t x c Z ! t R c t ! = J ! t R Z S D n H t c t x t R Z c t !
669 505 372 syl φ c C R Z J R Z Fin
670 577 a1i φ c C R Z J t R Z 0 J 0
671 511 ffvelrnda φ c C R Z J t R Z c t 0 J
672 670 671 sseldd φ c C R Z J t R Z c t 0
673 672 faccld φ c C R Z J t R Z c t !
674 673 nncnd φ c C R Z J t R Z c t !
675 669 674 fprodcl φ c C R Z J t R Z c t !
676 675 adantlr φ x X c C R Z J t R Z c t !
677 673 nnne0d φ c C R Z J t R Z c t ! 0
678 669 674 677 fprodn0 φ c C R Z J t R Z c t ! 0
679 678 adantlr φ x X c C R Z J t R Z c t ! 0
680 626 657 676 679 div23d φ x X c C R Z J J ! t R Z S D n H t c t x t R Z c t ! = J ! t R Z c t ! t R Z S D n H t c t x
681 eqidd φ x X c C R Z J J ! t R Z c t ! t R Z S D n H t c t x = J ! t R Z c t ! t R Z S D n H t c t x
682 668 680 681 3eqtrd φ x X c C R Z J J ! t R Z S D n H t c t x c Z ! t R c t ! = J ! t R Z c t ! t R Z S D n H t c t x
683 646 658 682 3eqtrd φ x X c C R Z J J ! c Z ! t R S D n H t c t x t R c t ! S D n H Z c Z x = J ! t R Z c t ! t R Z S D n H t c t x
684 625 632 683 3eqtrd φ x X c C R Z J ( J J c Z ) J c Z ! t R c R t ! t R S D n H t c R t x S D n H Z J J c Z x = J ! t R Z c t ! t R Z S D n H t c t x
685 684 sumeq2dv φ x X c C R Z J ( J J c Z ) J c Z ! t R c R t ! t R S D n H t c R t x S D n H Z J J c Z x = c C R Z J J ! t R Z c t ! t R Z S D n H t c t x
686 504 685 eqtrd φ x X p k = 0 J k × C R k ( J 1 st p ) 1 st p ! t R 2 nd p t ! t R S D n H t 2 nd p t x S D n H Z J 1 st p x = c C R Z J J ! t R Z c t ! t R Z S D n H t c t x
687 295 319 686 3eqtrd φ x X k = 0 J ( J k) k 0 J S D n y X t R H t y k k x k 0 J S D n y X H Z y k J k x = c C R Z J J ! t R Z c t ! t R Z S D n H t c t x
688 687 mpteq2dva φ x X k = 0 J ( J k) k 0 J S D n y X t R H t y k k x k 0 J S D n y X H Z y k J k x = x X c C R Z J J ! t R Z c t ! t R Z S D n H t c t x
689 47 210 688 3eqtrd φ S D n x X t R Z H t x J = x X c C R Z J J ! t R Z c t ! t R Z S D n H t c t x