Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > monoords | Structured version Visualization version Unicode version |
Description: Ordering relation for a strictly monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
monoords.fk | |
monoords.flt | ..^ |
monoords.i | |
monoords.j | |
monoords.iltj |
Ref | Expression |
---|---|
monoords |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | monoords.i | . . 3 | |
2 | 1 | ancli 574 | . . 3 |
3 | eleq1 2689 | . . . . . 6 | |
4 | 3 | anbi2d 740 | . . . . 5 |
5 | fveq2 6191 | . . . . . 6 | |
6 | 5 | eleq1d 2686 | . . . . 5 |
7 | 4, 6 | imbi12d 334 | . . . 4 |
8 | monoords.fk | . . . 4 | |
9 | 7, 8 | vtoclg 3266 | . . 3 |
10 | 1, 2, 9 | sylc 65 | . 2 |
11 | elfzel1 12341 | . . . . . . 7 | |
12 | 1, 11 | syl 17 | . . . . . 6 |
13 | elfzelz 12342 | . . . . . . 7 | |
14 | 1, 13 | syl 17 | . . . . . 6 |
15 | elfzle1 12344 | . . . . . . 7 | |
16 | 1, 15 | syl 17 | . . . . . 6 |
17 | eluz2 11693 | . . . . . 6 | |
18 | 12, 14, 16, 17 | syl3anbrc 1246 | . . . . 5 |
19 | elfzuz2 12346 | . . . . . . 7 | |
20 | 1, 19 | syl 17 | . . . . . 6 |
21 | eluzelz 11697 | . . . . . 6 | |
22 | 20, 21 | syl 17 | . . . . 5 |
23 | 14 | zred 11482 | . . . . . 6 |
24 | monoords.j | . . . . . . . 8 | |
25 | elfzelz 12342 | . . . . . . . 8 | |
26 | 24, 25 | syl 17 | . . . . . . 7 |
27 | 26 | zred 11482 | . . . . . 6 |
28 | 22 | zred 11482 | . . . . . 6 |
29 | monoords.iltj | . . . . . 6 | |
30 | elfzle2 12345 | . . . . . . 7 | |
31 | 24, 30 | syl 17 | . . . . . 6 |
32 | 23, 27, 28, 29, 31 | ltletrd 10197 | . . . . 5 |
33 | elfzo2 12473 | . . . . 5 ..^ | |
34 | 18, 22, 32, 33 | syl3anbrc 1246 | . . . 4 ..^ |
35 | fzofzp1 12565 | . . . 4 ..^ | |
36 | 34, 35 | syl 17 | . . 3 |
37 | 36 | ancli 574 | . . 3 |
38 | eleq1 2689 | . . . . . 6 | |
39 | 38 | anbi2d 740 | . . . . 5 |
40 | fveq2 6191 | . . . . . 6 | |
41 | 40 | eleq1d 2686 | . . . . 5 |
42 | 39, 41 | imbi12d 334 | . . . 4 |
43 | 42, 8 | vtoclg 3266 | . . 3 |
44 | 36, 37, 43 | sylc 65 | . 2 |
45 | 24 | ancli 574 | . . 3 |
46 | eleq1 2689 | . . . . . 6 | |
47 | 46 | anbi2d 740 | . . . . 5 |
48 | fveq2 6191 | . . . . . 6 | |
49 | 48 | eleq1d 2686 | . . . . 5 |
50 | 47, 49 | imbi12d 334 | . . . 4 |
51 | 50, 8 | vtoclg 3266 | . . 3 |
52 | 24, 45, 51 | sylc 65 | . 2 |
53 | 34 | ancli 574 | . . 3 ..^ |
54 | eleq1 2689 | . . . . . 6 ..^ ..^ | |
55 | 54 | anbi2d 740 | . . . . 5 ..^ ..^ |
56 | oveq1 6657 | . . . . . . 7 | |
57 | 56 | fveq2d 6195 | . . . . . 6 |
58 | 5, 57 | breq12d 4666 | . . . . 5 |
59 | 55, 58 | imbi12d 334 | . . . 4 ..^ ..^ |
60 | monoords.flt | . . . 4 ..^ | |
61 | 59, 60 | vtoclg 3266 | . . 3 ..^ ..^ |
62 | 34, 53, 61 | sylc 65 | . 2 |
63 | 14 | peano2zd 11485 | . . . 4 |
64 | zltp1le 11427 | . . . . . 6 | |
65 | 14, 26, 64 | syl2anc 693 | . . . . 5 |
66 | 29, 65 | mpbid 222 | . . . 4 |
67 | eluz2 11693 | . . . 4 | |
68 | 63, 26, 66, 67 | syl3anbrc 1246 | . . 3 |
69 | 12 | adantr 481 | . . . . . . 7 |
70 | 22 | adantr 481 | . . . . . . 7 |
71 | elfzelz 12342 | . . . . . . . 8 | |
72 | 71 | adantl 482 | . . . . . . 7 |
73 | 69, 70, 72 | 3jca 1242 | . . . . . 6 |
74 | 69 | zred 11482 | . . . . . . 7 |
75 | 72 | zred 11482 | . . . . . . 7 |
76 | 63 | zred 11482 | . . . . . . . . 9 |
77 | 76 | adantr 481 | . . . . . . . 8 |
78 | 23 | adantr 481 | . . . . . . . . 9 |
79 | 16 | adantr 481 | . . . . . . . . 9 |
80 | 78 | ltp1d 10954 | . . . . . . . . 9 |
81 | 74, 78, 77, 79, 80 | lelttrd 10195 | . . . . . . . 8 |
82 | elfzle1 12344 | . . . . . . . . 9 | |
83 | 82 | adantl 482 | . . . . . . . 8 |
84 | 74, 77, 75, 81, 83 | ltletrd 10197 | . . . . . . 7 |
85 | 74, 75, 84 | ltled 10185 | . . . . . 6 |
86 | 27 | adantr 481 | . . . . . . 7 |
87 | 70 | zred 11482 | . . . . . . 7 |
88 | elfzle2 12345 | . . . . . . . 8 | |
89 | 88 | adantl 482 | . . . . . . 7 |
90 | 31 | adantr 481 | . . . . . . 7 |
91 | 75, 86, 87, 89, 90 | letrd 10194 | . . . . . 6 |
92 | 73, 85, 91 | jca32 558 | . . . . 5 |
93 | elfz2 12333 | . . . . 5 | |
94 | 92, 93 | sylibr 224 | . . . 4 |
95 | 94, 8 | syldan 487 | . . 3 |
96 | 12 | adantr 481 | . . . . . . . 8 |
97 | 22 | adantr 481 | . . . . . . . 8 |
98 | elfzelz 12342 | . . . . . . . . 9 | |
99 | 98 | adantl 482 | . . . . . . . 8 |
100 | 96, 97, 99 | 3jca 1242 | . . . . . . 7 |
101 | 96 | zred 11482 | . . . . . . . 8 |
102 | 99 | zred 11482 | . . . . . . . 8 |
103 | 76 | adantr 481 | . . . . . . . . 9 |
104 | 12 | zred 11482 | . . . . . . . . . . 11 |
105 | 23 | ltp1d 10954 | . . . . . . . . . . 11 |
106 | 104, 23, 76, 16, 105 | lelttrd 10195 | . . . . . . . . . 10 |
107 | 106 | adantr 481 | . . . . . . . . 9 |
108 | elfzle1 12344 | . . . . . . . . . 10 | |
109 | 108 | adantl 482 | . . . . . . . . 9 |
110 | 101, 103, 102, 107, 109 | ltletrd 10197 | . . . . . . . 8 |
111 | 101, 102, 110 | ltled 10185 | . . . . . . 7 |
112 | 97 | zred 11482 | . . . . . . . 8 |
113 | peano2rem 10348 | . . . . . . . . . . 11 | |
114 | 27, 113 | syl 17 | . . . . . . . . . 10 |
115 | 114 | adantr 481 | . . . . . . . . 9 |
116 | elfzle2 12345 | . . . . . . . . . 10 | |
117 | 116 | adantl 482 | . . . . . . . . 9 |
118 | 27 | adantr 481 | . . . . . . . . . 10 |
119 | 118 | ltm1d 10956 | . . . . . . . . . 10 |
120 | 31 | adantr 481 | . . . . . . . . . 10 |
121 | 115, 118, 112, 119, 120 | ltletrd 10197 | . . . . . . . . 9 |
122 | 102, 115, 112, 117, 121 | lelttrd 10195 | . . . . . . . 8 |
123 | 102, 112, 122 | ltled 10185 | . . . . . . 7 |
124 | 100, 111, 123 | jca32 558 | . . . . . 6 |
125 | 124, 93 | sylibr 224 | . . . . 5 |
126 | 125, 8 | syldan 487 | . . . 4 |
127 | peano2zm 11420 | . . . . . . . . 9 | |
128 | 97, 127 | syl 17 | . . . . . . . 8 |
129 | 96, 128, 99 | 3jca 1242 | . . . . . . 7 |
130 | 128 | zred 11482 | . . . . . . . 8 |
131 | 1red 10055 | . . . . . . . . . 10 | |
132 | 27, 28, 131, 31 | lesub1dd 10643 | . . . . . . . . 9 |
133 | 132 | adantr 481 | . . . . . . . 8 |
134 | 102, 115, 130, 117, 133 | letrd 10194 | . . . . . . 7 |
135 | 129, 111, 134 | jca32 558 | . . . . . 6 |
136 | elfz2 12333 | . . . . . 6 | |
137 | 135, 136 | sylibr 224 | . . . . 5 |
138 | simpr 477 | . . . . . . . 8 | |
139 | fzoval 12471 | . . . . . . . . . . 11 ..^ | |
140 | 22, 139 | syl 17 | . . . . . . . . . 10 ..^ |
141 | 140 | eqcomd 2628 | . . . . . . . . 9 ..^ |
142 | 141 | adantr 481 | . . . . . . . 8 ..^ |
143 | 138, 142 | eleqtrd 2703 | . . . . . . 7 ..^ |
144 | fzofzp1 12565 | . . . . . . 7 ..^ | |
145 | 143, 144 | syl 17 | . . . . . 6 |
146 | simpl 473 | . . . . . . 7 | |
147 | 146, 145 | jca 554 | . . . . . 6 |
148 | eleq1 2689 | . . . . . . . . 9 | |
149 | 148 | anbi2d 740 | . . . . . . . 8 |
150 | fveq2 6191 | . . . . . . . . 9 | |
151 | 150 | eleq1d 2686 | . . . . . . . 8 |
152 | 149, 151 | imbi12d 334 | . . . . . . 7 |
153 | eleq1 2689 | . . . . . . . . . 10 | |
154 | 153 | anbi2d 740 | . . . . . . . . 9 |
155 | fveq2 6191 | . . . . . . . . . 10 | |
156 | 155 | eleq1d 2686 | . . . . . . . . 9 |
157 | 154, 156 | imbi12d 334 | . . . . . . . 8 |
158 | 157, 8 | chvarv 2263 | . . . . . . 7 |
159 | 152, 158 | vtoclg 3266 | . . . . . 6 |
160 | 145, 147, 159 | sylc 65 | . . . . 5 |
161 | 137, 160 | syldan 487 | . . . 4 |
162 | 143, 60 | syldan 487 | . . . . 5 |
163 | 137, 162 | syldan 487 | . . . 4 |
164 | 126, 161, 163 | ltled 10185 | . . 3 |
165 | 68, 95, 164 | monoord 12831 | . 2 |
166 | 10, 44, 52, 62, 165 | ltletrd 10197 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 class class class wbr 4653 cfv 5888 (class class class)co 6650 cr 9935 c1 9937 caddc 9939 clt 10074 cle 10075 cmin 10266 cz 11377 cuz 11687 cfz 12326 ..^cfzo 12465 |
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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 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-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-iun 4522 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-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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 |
This theorem is referenced by: fourierdlem34 40358 |
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