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Mirrors > Home > MPE Home > Th. List > ef01bndlem | Structured version Visualization version Unicode version |
Description: Lemma for sin01bnd 14915 and cos01bnd 14916. (Contributed by Paul Chapman, 19-Jan-2008.) |
Ref | Expression |
---|---|
ef01bnd.1 |
Ref | Expression |
---|---|
ef01bndlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 9995 | . . . . 5 | |
2 | 0xr 10086 | . . . . . . . 8 | |
3 | 1re 10039 | . . . . . . . 8 | |
4 | elioc2 12236 | . . . . . . . 8 | |
5 | 2, 3, 4 | mp2an 708 | . . . . . . 7 |
6 | 5 | simp1bi 1076 | . . . . . 6 |
7 | 6 | recnd 10068 | . . . . 5 |
8 | mulcl 10020 | . . . . 5 | |
9 | 1, 7, 8 | sylancr 695 | . . . 4 |
10 | 4nn0 11311 | . . . 4 | |
11 | ef01bnd.1 | . . . . 5 | |
12 | 11 | eftlcl 14837 | . . . 4 |
13 | 9, 10, 12 | sylancl 694 | . . 3 |
14 | 13 | abscld 14175 | . 2 |
15 | reexpcl 12877 | . . . 4 | |
16 | 6, 10, 15 | sylancl 694 | . . 3 |
17 | 4re 11097 | . . . . 5 | |
18 | 17, 3 | readdcli 10053 | . . . 4 |
19 | faccl 13070 | . . . . . 6 | |
20 | 10, 19 | ax-mp 5 | . . . . 5 |
21 | 4nn 11187 | . . . . 5 | |
22 | 20, 21 | nnmulcli 11044 | . . . 4 |
23 | nndivre 11056 | . . . 4 | |
24 | 18, 22, 23 | mp2an 708 | . . 3 |
25 | remulcl 10021 | . . 3 | |
26 | 16, 24, 25 | sylancl 694 | . 2 |
27 | 6nn 11189 | . . 3 | |
28 | nndivre 11056 | . . 3 | |
29 | 16, 27, 28 | sylancl 694 | . 2 |
30 | eqid 2622 | . . . 4 | |
31 | eqid 2622 | . . . 4 | |
32 | 21 | a1i 11 | . . . 4 |
33 | absmul 14034 | . . . . . . 7 | |
34 | 1, 7, 33 | sylancr 695 | . . . . . 6 |
35 | absi 14026 | . . . . . . . 8 | |
36 | 35 | oveq1i 6660 | . . . . . . 7 |
37 | 5 | simp2bi 1077 | . . . . . . . . . 10 |
38 | 6, 37 | elrpd 11869 | . . . . . . . . 9 |
39 | rpre 11839 | . . . . . . . . . 10 | |
40 | rpge0 11845 | . . . . . . . . . 10 | |
41 | 39, 40 | absidd 14161 | . . . . . . . . 9 |
42 | 38, 41 | syl 17 | . . . . . . . 8 |
43 | 42 | oveq2d 6666 | . . . . . . 7 |
44 | 36, 43 | syl5eq 2668 | . . . . . 6 |
45 | 7 | mulid2d 10058 | . . . . . 6 |
46 | 34, 44, 45 | 3eqtrd 2660 | . . . . 5 |
47 | 5 | simp3bi 1078 | . . . . 5 |
48 | 46, 47 | eqbrtrd 4675 | . . . 4 |
49 | 11, 30, 31, 32, 9, 48 | eftlub 14839 | . . 3 |
50 | 46 | oveq1d 6665 | . . . 4 |
51 | 50 | oveq1d 6665 | . . 3 |
52 | 49, 51 | breqtrd 4679 | . 2 |
53 | 3pos 11114 | . . . . . . . . 9 | |
54 | 0re 10040 | . . . . . . . . . 10 | |
55 | 3re 11094 | . . . . . . . . . 10 | |
56 | 5re 11099 | . . . . . . . . . 10 | |
57 | 54, 55, 56 | ltadd1i 10582 | . . . . . . . . 9 |
58 | 53, 57 | mpbi 220 | . . . . . . . 8 |
59 | 5cn 11100 | . . . . . . . . 9 | |
60 | 59 | addid2i 10224 | . . . . . . . 8 |
61 | cu2 12963 | . . . . . . . . 9 | |
62 | 5p3e8 11166 | . . . . . . . . 9 | |
63 | 3nn 11186 | . . . . . . . . . . 11 | |
64 | 63 | nncni 11030 | . . . . . . . . . 10 |
65 | 59, 64 | addcomi 10227 | . . . . . . . . 9 |
66 | 61, 62, 65 | 3eqtr2ri 2651 | . . . . . . . 8 |
67 | 58, 60, 66 | 3brtr3i 4682 | . . . . . . 7 |
68 | 2re 11090 | . . . . . . . 8 | |
69 | 1le2 11241 | . . . . . . . 8 | |
70 | 4z 11411 | . . . . . . . . 9 | |
71 | 3lt4 11197 | . . . . . . . . . 10 | |
72 | 55, 17, 71 | ltleii 10160 | . . . . . . . . 9 |
73 | 63 | nnzi 11401 | . . . . . . . . . 10 |
74 | 73 | eluz1i 11695 | . . . . . . . . 9 |
75 | 70, 72, 74 | mpbir2an 955 | . . . . . . . 8 |
76 | leexp2a 12916 | . . . . . . . 8 | |
77 | 68, 69, 75, 76 | mp3an 1424 | . . . . . . 7 |
78 | 8re 11105 | . . . . . . . . 9 | |
79 | 61, 78 | eqeltri 2697 | . . . . . . . 8 |
80 | 2nn 11185 | . . . . . . . . . 10 | |
81 | nnexpcl 12873 | . . . . . . . . . 10 | |
82 | 80, 10, 81 | mp2an 708 | . . . . . . . . 9 |
83 | 82 | nnrei 11029 | . . . . . . . 8 |
84 | 56, 79, 83 | ltletri 10165 | . . . . . . 7 |
85 | 67, 77, 84 | mp2an 708 | . . . . . 6 |
86 | 6re 11101 | . . . . . . . 8 | |
87 | 86, 83 | remulcli 10054 | . . . . . . 7 |
88 | 6pos 11119 | . . . . . . . 8 | |
89 | 82 | nngt0i 11054 | . . . . . . . 8 |
90 | 86, 83, 88, 89 | mulgt0ii 10170 | . . . . . . 7 |
91 | 56, 83, 87, 90 | ltdiv1ii 10953 | . . . . . 6 |
92 | 85, 91 | mpbi 220 | . . . . 5 |
93 | df-5 11082 | . . . . . 6 | |
94 | df-4 11081 | . . . . . . . . . . 11 | |
95 | 94 | fveq2i 6194 | . . . . . . . . . 10 |
96 | 3nn0 11310 | . . . . . . . . . . 11 | |
97 | facp1 13065 | . . . . . . . . . . 11 | |
98 | 96, 97 | ax-mp 5 | . . . . . . . . . 10 |
99 | sq2 12960 | . . . . . . . . . . . 12 | |
100 | 99, 94 | eqtr2i 2645 | . . . . . . . . . . 11 |
101 | 100 | oveq2i 6661 | . . . . . . . . . 10 |
102 | 95, 98, 101 | 3eqtri 2648 | . . . . . . . . 9 |
103 | 102 | oveq1i 6660 | . . . . . . . 8 |
104 | 99 | oveq2i 6661 | . . . . . . . 8 |
105 | fac3 13067 | . . . . . . . . . 10 | |
106 | 6cn 11102 | . . . . . . . . . 10 | |
107 | 105, 106 | eqeltri 2697 | . . . . . . . . 9 |
108 | 17 | recni 10052 | . . . . . . . . . 10 |
109 | 99, 108 | eqeltri 2697 | . . . . . . . . 9 |
110 | 107, 109, 109 | mulassi 10049 | . . . . . . . 8 |
111 | 103, 104, 110 | 3eqtr3i 2652 | . . . . . . 7 |
112 | 2p2e4 11144 | . . . . . . . . . 10 | |
113 | 112 | oveq2i 6661 | . . . . . . . . 9 |
114 | 2cn 11091 | . . . . . . . . . 10 | |
115 | 2nn0 11309 | . . . . . . . . . 10 | |
116 | expadd 12902 | . . . . . . . . . 10 | |
117 | 114, 115, 115, 116 | mp3an 1424 | . . . . . . . . 9 |
118 | 113, 117 | eqtr3i 2646 | . . . . . . . 8 |
119 | 118 | oveq2i 6661 | . . . . . . 7 |
120 | 105 | oveq1i 6660 | . . . . . . 7 |
121 | 111, 119, 120 | 3eqtr2ri 2651 | . . . . . 6 |
122 | 93, 121 | oveq12i 6662 | . . . . 5 |
123 | 82 | nncni 11030 | . . . . . . . 8 |
124 | 123 | mulid2i 10043 | . . . . . . 7 |
125 | 124 | oveq1i 6660 | . . . . . 6 |
126 | 82 | nnne0i 11055 | . . . . . . . . 9 |
127 | 123, 126 | dividi 10758 | . . . . . . . 8 |
128 | 127 | oveq2i 6661 | . . . . . . 7 |
129 | ax-1cn 9994 | . . . . . . . 8 | |
130 | 86, 88 | gt0ne0ii 10564 | . . . . . . . 8 |
131 | 129, 106, 123, 123, 130, 126 | divmuldivi 10785 | . . . . . . 7 |
132 | 86, 130 | rereccli 10790 | . . . . . . . . 9 |
133 | 132 | recni 10052 | . . . . . . . 8 |
134 | 133 | mulid1i 10042 | . . . . . . 7 |
135 | 128, 131, 134 | 3eqtr3i 2652 | . . . . . 6 |
136 | 125, 135 | eqtr3i 2646 | . . . . 5 |
137 | 92, 122, 136 | 3brtr3i 4682 | . . . 4 |
138 | rpexpcl 12879 | . . . . . 6 | |
139 | 38, 70, 138 | sylancl 694 | . . . . 5 |
140 | elrp 11834 | . . . . . 6 | |
141 | ltmul2 10874 | . . . . . . 7 | |
142 | 24, 132, 141 | mp3an12 1414 | . . . . . 6 |
143 | 140, 142 | sylbi 207 | . . . . 5 |
144 | 139, 143 | syl 17 | . . . 4 |
145 | 137, 144 | mpbii 223 | . . 3 |
146 | 16 | recnd 10068 | . . . 4 |
147 | divrec 10701 | . . . . 5 | |
148 | 106, 130, 147 | mp3an23 1416 | . . . 4 |
149 | 146, 148 | syl 17 | . . 3 |
150 | 145, 149 | breqtrrd 4681 | . 2 |
151 | 14, 26, 29, 52, 150 | lelttrd 10195 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 class class class wbr 4653 cmpt 4729 cfv 5888 (class class class)co 6650 cc 9934 cr 9935 cc0 9936 c1 9937 ci 9938 caddc 9939 cmul 9941 cxr 10073 clt 10074 cle 10075 cdiv 10684 cn 11020 c2 11070 c3 11071 c4 11072 c5 11073 c6 11074 c8 11076 cn0 11292 cz 11377 cuz 11687 crp 11832 cioc 12176 cexp 12860 cfa 13060 cabs 13974 csu 14416 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 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 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-rmo 2920 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-int 4476 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-se 5074 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-isom 5897 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-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ioc 12180 df-ico 12181 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-fac 13061 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 |
This theorem is referenced by: sin01bnd 14915 cos01bnd 14916 |
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