Metamath Proof Explorer


Theorem fourierdlem12

Description: A point of a partition is not an element of any open interval determined by the partition. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses fourierdlem12.1 P = m p 0 m | p 0 = A p m = B i 0 ..^ m p i < p i + 1
fourierdlem12.2 φ M
fourierdlem12.3 φ Q P M
fourierdlem12.4 φ X ran Q
Assertion fourierdlem12 φ i 0 ..^ M ¬ X Q i Q i + 1

Proof

Step Hyp Ref Expression
1 fourierdlem12.1 P = m p 0 m | p 0 = A p m = B i 0 ..^ m p i < p i + 1
2 fourierdlem12.2 φ M
3 fourierdlem12.3 φ Q P M
4 fourierdlem12.4 φ X ran Q
5 1 fourierdlem2 M Q P M Q 0 M Q 0 = A Q M = B i 0 ..^ M Q i < Q i + 1
6 2 5 syl φ Q P M Q 0 M Q 0 = A Q M = B i 0 ..^ M Q i < Q i + 1
7 3 6 mpbid φ Q 0 M Q 0 = A Q M = B i 0 ..^ M Q i < Q i + 1
8 7 simpld φ Q 0 M
9 elmapi Q 0 M Q : 0 M
10 ffn Q : 0 M Q Fn 0 M
11 fvelrnb Q Fn 0 M X ran Q j 0 M Q j = X
12 8 9 10 11 4syl φ X ran Q j 0 M Q j = X
13 4 12 mpbid φ j 0 M Q j = X
14 13 adantr φ i 0 ..^ M j 0 M Q j = X
15 8 9 syl φ Q : 0 M
16 15 adantr φ i 0 ..^ M Q : 0 M
17 fzofzp1 i 0 ..^ M i + 1 0 M
18 17 adantl φ i 0 ..^ M i + 1 0 M
19 16 18 ffvelcdmd φ i 0 ..^ M Q i + 1
20 19 adantr φ i 0 ..^ M i < j Q i + 1
21 20 3ad2antl1 φ i 0 ..^ M j 0 M Q j = X i < j Q i + 1
22 frn Q : 0 M ran Q
23 15 22 syl φ ran Q
24 23 4 sseldd φ X
25 24 ad2antrr φ i 0 ..^ M i < j X
26 25 3ad2antl1 φ i 0 ..^ M j 0 M Q j = X i < j X
27 16 ffvelcdmda φ i 0 ..^ M j 0 M Q j
28 27 3adant3 φ i 0 ..^ M j 0 M Q j = X Q j
29 28 adantr φ i 0 ..^ M j 0 M Q j = X i < j Q j
30 simpr i 0 ..^ M j 0 M i < j i < j
31 elfzoelz i 0 ..^ M i
32 31 ad2antrr i 0 ..^ M j 0 M i < j i
33 elfzelz j 0 M j
34 33 ad2antlr i 0 ..^ M j 0 M i < j j
35 zltp1le i j i < j i + 1 j
36 32 34 35 syl2anc i 0 ..^ M j 0 M i < j i < j i + 1 j
37 30 36 mpbid i 0 ..^ M j 0 M i < j i + 1 j
38 32 peano2zd i 0 ..^ M j 0 M i < j i + 1
39 eluz i + 1 j j i + 1 i + 1 j
40 38 34 39 syl2anc i 0 ..^ M j 0 M i < j j i + 1 i + 1 j
41 37 40 mpbird i 0 ..^ M j 0 M i < j j i + 1
42 41 adantlll φ i 0 ..^ M j 0 M i < j j i + 1
43 16 ad2antrr φ i 0 ..^ M j 0 M w i + 1 j Q : 0 M
44 0zd i 0 ..^ M j 0 M w i + 1 j 0
45 elfzel2 j 0 M M
46 45 ad2antlr i 0 ..^ M j 0 M w i + 1 j M
47 elfzelz w i + 1 j w
48 47 adantl i 0 ..^ M j 0 M w i + 1 j w
49 0red i 0 ..^ M w i + 1 j 0
50 47 zred w i + 1 j w
51 50 adantl i 0 ..^ M w i + 1 j w
52 31 peano2zd i 0 ..^ M i + 1
53 52 zred i 0 ..^ M i + 1
54 53 adantr i 0 ..^ M w i + 1 j i + 1
55 31 zred i 0 ..^ M i
56 55 adantr i 0 ..^ M w i + 1 j i
57 elfzole1 i 0 ..^ M 0 i
58 57 adantr i 0 ..^ M w i + 1 j 0 i
59 56 ltp1d i 0 ..^ M w i + 1 j i < i + 1
60 49 56 54 58 59 lelttrd i 0 ..^ M w i + 1 j 0 < i + 1
61 elfzle1 w i + 1 j i + 1 w
62 61 adantl i 0 ..^ M w i + 1 j i + 1 w
63 49 54 51 60 62 ltletrd i 0 ..^ M w i + 1 j 0 < w
64 49 51 63 ltled i 0 ..^ M w i + 1 j 0 w
65 64 adantlr i 0 ..^ M j 0 M w i + 1 j 0 w
66 50 adantl j 0 M w i + 1 j w
67 33 zred j 0 M j
68 67 adantr j 0 M w i + 1 j j
69 45 zred j 0 M M
70 69 adantr j 0 M w i + 1 j M
71 elfzle2 w i + 1 j w j
72 71 adantl j 0 M w i + 1 j w j
73 elfzle2 j 0 M j M
74 73 adantr j 0 M w i + 1 j j M
75 66 68 70 72 74 letrd j 0 M w i + 1 j w M
76 75 adantll i 0 ..^ M j 0 M w i + 1 j w M
77 44 46 48 65 76 elfzd i 0 ..^ M j 0 M w i + 1 j w 0 M
78 77 adantlll φ i 0 ..^ M j 0 M w i + 1 j w 0 M
79 43 78 ffvelcdmd φ i 0 ..^ M j 0 M w i + 1 j Q w
80 79 adantlr φ i 0 ..^ M j 0 M i < j w i + 1 j Q w
81 simp-4l φ i 0 ..^ M j 0 M i < j w i + 1 j 1 φ
82 0red i 0 ..^ M j 0 M w i + 1 j 1 0
83 elfzelz w i + 1 j 1 w
84 83 zred w i + 1 j 1 w
85 84 adantl i 0 ..^ M j 0 M w i + 1 j 1 w
86 0red i 0 ..^ M w i + 1 j 1 0
87 53 adantr i 0 ..^ M w i + 1 j 1 i + 1
88 84 adantl i 0 ..^ M w i + 1 j 1 w
89 0red i 0 ..^ M 0
90 55 ltp1d i 0 ..^ M i < i + 1
91 89 55 53 57 90 lelttrd i 0 ..^ M 0 < i + 1
92 91 adantr i 0 ..^ M w i + 1 j 1 0 < i + 1
93 elfzle1 w i + 1 j 1 i + 1 w
94 93 adantl i 0 ..^ M w i + 1 j 1 i + 1 w
95 86 87 88 92 94 ltletrd i 0 ..^ M w i + 1 j 1 0 < w
96 95 adantlr i 0 ..^ M j 0 M w i + 1 j 1 0 < w
97 82 85 96 ltled i 0 ..^ M j 0 M w i + 1 j 1 0 w
98 97 adantlll φ i 0 ..^ M j 0 M w i + 1 j 1 0 w
99 98 adantlr φ i 0 ..^ M j 0 M i < j w i + 1 j 1 0 w
100 84 adantl j 0 M w i + 1 j 1 w
101 peano2rem j j 1
102 67 101 syl j 0 M j 1
103 102 adantr j 0 M w i + 1 j 1 j 1
104 69 adantr j 0 M w i + 1 j 1 M
105 elfzle2 w i + 1 j 1 w j 1
106 105 adantl j 0 M w i + 1 j 1 w j 1
107 zlem1lt j M j M j 1 < M
108 33 45 107 syl2anc j 0 M j M j 1 < M
109 73 108 mpbid j 0 M j 1 < M
110 109 adantr j 0 M w i + 1 j 1 j 1 < M
111 100 103 104 106 110 lelttrd j 0 M w i + 1 j 1 w < M
112 111 adantlr j 0 M i < j w i + 1 j 1 w < M
113 112 adantlll φ i 0 ..^ M j 0 M i < j w i + 1 j 1 w < M
114 83 adantl φ i 0 ..^ M j 0 M i < j w i + 1 j 1 w
115 0zd φ i 0 ..^ M j 0 M i < j w i + 1 j 1 0
116 45 ad3antlr φ i 0 ..^ M j 0 M i < j w i + 1 j 1 M
117 elfzo w 0 M w 0 ..^ M 0 w w < M
118 114 115 116 117 syl3anc φ i 0 ..^ M j 0 M i < j w i + 1 j 1 w 0 ..^ M 0 w w < M
119 99 113 118 mpbir2and φ i 0 ..^ M j 0 M i < j w i + 1 j 1 w 0 ..^ M
120 15 adantr φ w 0 ..^ M Q : 0 M
121 elfzofz w 0 ..^ M w 0 M
122 121 adantl φ w 0 ..^ M w 0 M
123 120 122 ffvelcdmd φ w 0 ..^ M Q w
124 fzofzp1 w 0 ..^ M w + 1 0 M
125 124 adantl φ w 0 ..^ M w + 1 0 M
126 120 125 ffvelcdmd φ w 0 ..^ M Q w + 1
127 eleq1w i = w i 0 ..^ M w 0 ..^ M
128 127 anbi2d i = w φ i 0 ..^ M φ w 0 ..^ M
129 fveq2 i = w Q i = Q w
130 oveq1 i = w i + 1 = w + 1
131 130 fveq2d i = w Q i + 1 = Q w + 1
132 129 131 breq12d i = w Q i < Q i + 1 Q w < Q w + 1
133 128 132 imbi12d i = w φ i 0 ..^ M Q i < Q i + 1 φ w 0 ..^ M Q w < Q w + 1
134 7 simprrd φ i 0 ..^ M Q i < Q i + 1
135 134 r19.21bi φ i 0 ..^ M Q i < Q i + 1
136 133 135 chvarvv φ w 0 ..^ M Q w < Q w + 1
137 123 126 136 ltled φ w 0 ..^ M Q w Q w + 1
138 81 119 137 syl2anc φ i 0 ..^ M j 0 M i < j w i + 1 j 1 Q w Q w + 1
139 42 80 138 monoord φ i 0 ..^ M j 0 M i < j Q i + 1 Q j
140 139 3adantl3 φ i 0 ..^ M j 0 M Q j = X i < j Q i + 1 Q j
141 15 ffvelcdmda φ j 0 M Q j
142 141 3adant3 φ j 0 M Q j = X Q j
143 simp3 φ j 0 M Q j = X Q j = X
144 142 143 eqled φ j 0 M Q j = X Q j X
145 144 3adant1r φ i 0 ..^ M j 0 M Q j = X Q j X
146 145 adantr φ i 0 ..^ M j 0 M Q j = X i < j Q j X
147 21 29 26 140 146 letrd φ i 0 ..^ M j 0 M Q j = X i < j Q i + 1 X
148 21 26 147 lensymd φ i 0 ..^ M j 0 M Q j = X i < j ¬ X < Q i + 1
149 148 intnand φ i 0 ..^ M j 0 M Q j = X i < j ¬ Q i < X X < Q i + 1
150 67 ad2antlr φ i 0 ..^ M j 0 M ¬ i < j j
151 55 ad3antlr φ i 0 ..^ M j 0 M ¬ i < j i
152 simpr φ i 0 ..^ M j 0 M ¬ i < j ¬ i < j
153 150 151 152 nltled φ i 0 ..^ M j 0 M ¬ i < j j i
154 153 3adantl3 φ i 0 ..^ M j 0 M Q j = X ¬ i < j j i
155 eqcom Q j = X X = Q j
156 155 biimpi Q j = X X = Q j
157 156 adantr Q j = X j i X = Q j
158 157 3ad2antl3 φ i 0 ..^ M j 0 M Q j = X j i X = Q j
159 33 ad2antlr i 0 ..^ M j 0 M j i j
160 31 ad2antrr i 0 ..^ M j 0 M j i i
161 simpr i 0 ..^ M j 0 M j i j i
162 eluz2 i j j i j i
163 159 160 161 162 syl3anbrc i 0 ..^ M j 0 M j i i j
164 163 adantlll φ i 0 ..^ M j 0 M j i i j
165 16 ad2antrr φ i 0 ..^ M j 0 M w j i Q : 0 M
166 0zd i 0 ..^ M j 0 M w j i 0
167 45 ad2antlr i 0 ..^ M j 0 M w j i M
168 elfzelz w j i w
169 168 adantl i 0 ..^ M j 0 M w j i w
170 166 167 169 3jca i 0 ..^ M j 0 M w j i 0 M w
171 0red j 0 M w j i 0
172 67 adantr j 0 M w j i j
173 168 zred w j i w
174 173 adantl j 0 M w j i w
175 elfzle1 j 0 M 0 j
176 175 adantr j 0 M w j i 0 j
177 elfzle1 w j i j w
178 177 adantl j 0 M w j i j w
179 171 172 174 176 178 letrd j 0 M w j i 0 w
180 179 adantll i 0 ..^ M j 0 M w j i 0 w
181 173 adantl i 0 ..^ M w j i w
182 elfzoel2 i 0 ..^ M M
183 182 zred i 0 ..^ M M
184 183 adantr i 0 ..^ M w j i M
185 55 adantr i 0 ..^ M w j i i
186 elfzle2 w j i w i
187 186 adantl i 0 ..^ M w j i w i
188 elfzolt2 i 0 ..^ M i < M
189 188 adantr i 0 ..^ M w j i i < M
190 181 185 184 187 189 lelttrd i 0 ..^ M w j i w < M
191 181 184 190 ltled i 0 ..^ M w j i w M
192 191 adantlr i 0 ..^ M j 0 M w j i w M
193 170 180 192 jca32 i 0 ..^ M j 0 M w j i 0 M w 0 w w M
194 193 adantlll φ i 0 ..^ M j 0 M w j i 0 M w 0 w w M
195 elfz2 w 0 M 0 M w 0 w w M
196 194 195 sylibr φ i 0 ..^ M j 0 M w j i w 0 M
197 165 196 ffvelcdmd φ i 0 ..^ M j 0 M w j i Q w
198 197 adantlr φ i 0 ..^ M j 0 M j i w j i Q w
199 simplll φ i 0 ..^ M j 0 M w j i 1 φ
200 0red i 0 ..^ M j 0 M w j i 1 0
201 67 ad2antlr i 0 ..^ M j 0 M w j i 1 j
202 elfzelz w j i 1 w
203 202 zred w j i 1 w
204 203 adantl i 0 ..^ M j 0 M w j i 1 w
205 175 ad2antlr i 0 ..^ M j 0 M w j i 1 0 j
206 elfzle1 w j i 1 j w
207 206 adantl i 0 ..^ M j 0 M w j i 1 j w
208 200 201 204 205 207 letrd i 0 ..^ M j 0 M w j i 1 0 w
209 203 adantl i 0 ..^ M w j i 1 w
210 55 adantr i 0 ..^ M w j i 1 i
211 183 adantr i 0 ..^ M w j i 1 M
212 peano2rem i i 1
213 210 212 syl i 0 ..^ M w j i 1 i 1
214 elfzle2 w j i 1 w i 1
215 214 adantl i 0 ..^ M w j i 1 w i 1
216 210 ltm1d i 0 ..^ M w j i 1 i 1 < i
217 209 213 210 215 216 lelttrd i 0 ..^ M w j i 1 w < i
218 188 adantr i 0 ..^ M w j i 1 i < M
219 209 210 211 217 218 lttrd i 0 ..^ M w j i 1 w < M
220 219 adantlr i 0 ..^ M j 0 M w j i 1 w < M
221 202 adantl i 0 ..^ M j 0 M w j i 1 w
222 0zd i 0 ..^ M j 0 M w j i 1 0
223 182 ad2antrr i 0 ..^ M j 0 M w j i 1 M
224 221 222 223 117 syl3anc i 0 ..^ M j 0 M w j i 1 w 0 ..^ M 0 w w < M
225 208 220 224 mpbir2and i 0 ..^ M j 0 M w j i 1 w 0 ..^ M
226 225 adantlll φ i 0 ..^ M j 0 M w j i 1 w 0 ..^ M
227 199 226 137 syl2anc φ i 0 ..^ M j 0 M w j i 1 Q w Q w + 1
228 227 adantlr φ i 0 ..^ M j 0 M j i w j i 1 Q w Q w + 1
229 164 198 228 monoord φ i 0 ..^ M j 0 M j i Q j Q i
230 229 3adantl3 φ i 0 ..^ M j 0 M Q j = X j i Q j Q i
231 158 230 eqbrtrd φ i 0 ..^ M j 0 M Q j = X j i X Q i
232 24 adantr φ i 0 ..^ M X
233 elfzofz i 0 ..^ M i 0 M
234 233 adantl φ i 0 ..^ M i 0 M
235 16 234 ffvelcdmd φ i 0 ..^ M Q i
236 232 235 lenltd φ i 0 ..^ M X Q i ¬ Q i < X
237 236 adantr φ i 0 ..^ M j i X Q i ¬ Q i < X
238 237 3ad2antl1 φ i 0 ..^ M j 0 M Q j = X j i X Q i ¬ Q i < X
239 231 238 mpbid φ i 0 ..^ M j 0 M Q j = X j i ¬ Q i < X
240 154 239 syldan φ i 0 ..^ M j 0 M Q j = X ¬ i < j ¬ Q i < X
241 240 intnanrd φ i 0 ..^ M j 0 M Q j = X ¬ i < j ¬ Q i < X X < Q i + 1
242 149 241 pm2.61dan φ i 0 ..^ M j 0 M Q j = X ¬ Q i < X X < Q i + 1
243 242 intnand φ i 0 ..^ M j 0 M Q j = X ¬ Q i * Q i + 1 * X * Q i < X X < Q i + 1
244 elioo3g X Q i Q i + 1 Q i * Q i + 1 * X * Q i < X X < Q i + 1
245 243 244 sylnibr φ i 0 ..^ M j 0 M Q j = X ¬ X Q i Q i + 1
246 245 rexlimdv3a φ i 0 ..^ M j 0 M Q j = X ¬ X Q i Q i + 1
247 14 246 mpd φ i 0 ..^ M ¬ X Q i Q i + 1