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Mirrors > Home > ILE Home > Th. List > ef01bndlem | Unicode version |
Description: Lemma for sin01bnd 11713 and cos01bnd 11714. (Contributed by Paul Chapman, 19-Jan-2008.) |
Ref | Expression |
---|---|
ef01bnd.1 |
Ref | Expression |
---|---|
ef01bndlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 7862 | . . . . 5 | |
2 | 0xr 7959 | . . . . . . . 8 | |
3 | 1re 7912 | . . . . . . . 8 | |
4 | elioc2 9886 | . . . . . . . 8 | |
5 | 2, 3, 4 | mp2an 424 | . . . . . . 7 |
6 | 5 | simp1bi 1007 | . . . . . 6 |
7 | 6 | recnd 7941 | . . . . 5 |
8 | mulcl 7894 | . . . . 5 | |
9 | 1, 7, 8 | sylancr 412 | . . . 4 |
10 | 4nn0 9147 | . . . 4 | |
11 | ef01bnd.1 | . . . . 5 | |
12 | 11 | eftlcl 11644 | . . . 4 |
13 | 9, 10, 12 | sylancl 411 | . . 3 |
14 | 13 | abscld 11138 | . 2 |
15 | reexpcl 10486 | . . . 4 | |
16 | 6, 10, 15 | sylancl 411 | . . 3 |
17 | 4re 8948 | . . . . 5 | |
18 | 17, 3 | readdcli 7926 | . . . 4 |
19 | faccl 10662 | . . . . . 6 | |
20 | 10, 19 | ax-mp 5 | . . . . 5 |
21 | 4nn 9034 | . . . . 5 | |
22 | 20, 21 | nnmulcli 8893 | . . . 4 |
23 | nndivre 8907 | . . . 4 | |
24 | 18, 22, 23 | mp2an 424 | . . 3 |
25 | remulcl 7895 | . . 3 | |
26 | 16, 24, 25 | sylancl 411 | . 2 |
27 | 6nn 9036 | . . 3 | |
28 | nndivre 8907 | . . 3 | |
29 | 16, 27, 28 | sylancl 411 | . 2 |
30 | eqid 2170 | . . . 4 | |
31 | eqid 2170 | . . . 4 | |
32 | 21 | a1i 9 | . . . 4 |
33 | absmul 11026 | . . . . . . 7 | |
34 | 1, 7, 33 | sylancr 412 | . . . . . 6 |
35 | absi 11016 | . . . . . . . 8 | |
36 | 35 | oveq1i 5861 | . . . . . . 7 |
37 | 5 | simp2bi 1008 | . . . . . . . . . 10 |
38 | 6, 37 | elrpd 9643 | . . . . . . . . 9 |
39 | rpre 9610 | . . . . . . . . . 10 | |
40 | rpge0 9616 | . . . . . . . . . 10 | |
41 | 39, 40 | absidd 11124 | . . . . . . . . 9 |
42 | 38, 41 | syl 14 | . . . . . . . 8 |
43 | 42 | oveq2d 5867 | . . . . . . 7 |
44 | 36, 43 | eqtrid 2215 | . . . . . 6 |
45 | 7 | mulid2d 7931 | . . . . . 6 |
46 | 34, 44, 45 | 3eqtrd 2207 | . . . . 5 |
47 | 5 | simp3bi 1009 | . . . . 5 |
48 | 46, 47 | eqbrtrd 4009 | . . . 4 |
49 | 11, 30, 31, 32, 9, 48 | eftlub 11646 | . . 3 |
50 | 46 | oveq1d 5866 | . . . 4 |
51 | 50 | oveq1d 5866 | . . 3 |
52 | 49, 51 | breqtrd 4013 | . 2 |
53 | 3pos 8965 | . . . . . . . . 9 | |
54 | 0re 7913 | . . . . . . . . . 10 | |
55 | 3re 8945 | . . . . . . . . . 10 | |
56 | 5re 8950 | . . . . . . . . . 10 | |
57 | 54, 55, 56 | ltadd1i 8414 | . . . . . . . . 9 |
58 | 53, 57 | mpbi 144 | . . . . . . . 8 |
59 | 5cn 8951 | . . . . . . . . 9 | |
60 | 59 | addid2i 8055 | . . . . . . . 8 |
61 | cu2 10567 | . . . . . . . . 9 | |
62 | 5p3e8 9018 | . . . . . . . . 9 | |
63 | 3cn 8946 | . . . . . . . . . 10 | |
64 | 59, 63 | addcomi 8056 | . . . . . . . . 9 |
65 | 61, 62, 64 | 3eqtr2ri 2198 | . . . . . . . 8 |
66 | 58, 60, 65 | 3brtr3i 4016 | . . . . . . 7 |
67 | 2re 8941 | . . . . . . . 8 | |
68 | 1le2 9079 | . . . . . . . 8 | |
69 | 4z 9235 | . . . . . . . . 9 | |
70 | 3lt4 9043 | . . . . . . . . . 10 | |
71 | 55, 17, 70 | ltleii 8015 | . . . . . . . . 9 |
72 | 3z 9234 | . . . . . . . . . 10 | |
73 | 72 | eluz1i 9487 | . . . . . . . . 9 |
74 | 69, 71, 73 | mpbir2an 937 | . . . . . . . 8 |
75 | leexp2a 10522 | . . . . . . . 8 | |
76 | 67, 68, 74, 75 | mp3an 1332 | . . . . . . 7 |
77 | 8re 8956 | . . . . . . . . 9 | |
78 | 61, 77 | eqeltri 2243 | . . . . . . . 8 |
79 | 2nn 9032 | . . . . . . . . . 10 | |
80 | nnexpcl 10482 | . . . . . . . . . 10 | |
81 | 79, 10, 80 | mp2an 424 | . . . . . . . . 9 |
82 | 81 | nnrei 8880 | . . . . . . . 8 |
83 | 56, 78, 82 | ltletri 8019 | . . . . . . 7 |
84 | 66, 76, 83 | mp2an 424 | . . . . . 6 |
85 | 6re 8952 | . . . . . . . 8 | |
86 | 85, 82 | remulcli 7927 | . . . . . . 7 |
87 | 6pos 8972 | . . . . . . . 8 | |
88 | 81 | nngt0i 8901 | . . . . . . . 8 |
89 | 85, 82, 87, 88 | mulgt0ii 8023 | . . . . . . 7 |
90 | 56, 82, 86, 89 | ltdiv1ii 8838 | . . . . . 6 |
91 | 84, 90 | mpbi 144 | . . . . 5 |
92 | df-5 8933 | . . . . . 6 | |
93 | df-4 8932 | . . . . . . . . . . 11 | |
94 | 93 | fveq2i 5497 | . . . . . . . . . 10 |
95 | 3nn0 9146 | . . . . . . . . . . 11 | |
96 | facp1 10657 | . . . . . . . . . . 11 | |
97 | 95, 96 | ax-mp 5 | . . . . . . . . . 10 |
98 | sq2 10564 | . . . . . . . . . . . 12 | |
99 | 98, 93 | eqtr2i 2192 | . . . . . . . . . . 11 |
100 | 99 | oveq2i 5862 | . . . . . . . . . 10 |
101 | 94, 97, 100 | 3eqtri 2195 | . . . . . . . . 9 |
102 | 101 | oveq1i 5861 | . . . . . . . 8 |
103 | 98 | oveq2i 5862 | . . . . . . . 8 |
104 | fac3 10659 | . . . . . . . . . 10 | |
105 | 6cn 8953 | . . . . . . . . . 10 | |
106 | 104, 105 | eqeltri 2243 | . . . . . . . . 9 |
107 | 17 | recni 7925 | . . . . . . . . . 10 |
108 | 98, 107 | eqeltri 2243 | . . . . . . . . 9 |
109 | 106, 108, 108 | mulassi 7922 | . . . . . . . 8 |
110 | 102, 103, 109 | 3eqtr3i 2199 | . . . . . . 7 |
111 | 2p2e4 8998 | . . . . . . . . . 10 | |
112 | 111 | oveq2i 5862 | . . . . . . . . 9 |
113 | 2cn 8942 | . . . . . . . . . 10 | |
114 | 2nn0 9145 | . . . . . . . . . 10 | |
115 | expadd 10511 | . . . . . . . . . 10 | |
116 | 113, 114, 114, 115 | mp3an 1332 | . . . . . . . . 9 |
117 | 112, 116 | eqtr3i 2193 | . . . . . . . 8 |
118 | 117 | oveq2i 5862 | . . . . . . 7 |
119 | 104 | oveq1i 5861 | . . . . . . 7 |
120 | 110, 118, 119 | 3eqtr2ri 2198 | . . . . . 6 |
121 | 92, 120 | oveq12i 5863 | . . . . 5 |
122 | 81 | nncni 8881 | . . . . . . . 8 |
123 | 122 | mulid2i 7916 | . . . . . . 7 |
124 | 123 | oveq1i 5861 | . . . . . 6 |
125 | 82, 88 | gt0ap0ii 8540 | . . . . . . . . 9 # |
126 | 122, 125 | dividapi 8655 | . . . . . . . 8 |
127 | 126 | oveq2i 5862 | . . . . . . 7 |
128 | ax-1cn 7860 | . . . . . . . 8 | |
129 | 85, 87 | gt0ap0ii 8540 | . . . . . . . 8 # |
130 | 128, 105, 122, 122, 129, 125 | divmuldivapi 8682 | . . . . . . 7 |
131 | 85, 129 | rerecclapi 8687 | . . . . . . . . 9 |
132 | 131 | recni 7925 | . . . . . . . 8 |
133 | 132 | mulid1i 7915 | . . . . . . 7 |
134 | 127, 130, 133 | 3eqtr3i 2199 | . . . . . 6 |
135 | 124, 134 | eqtr3i 2193 | . . . . 5 |
136 | 91, 121, 135 | 3brtr3i 4016 | . . . 4 |
137 | rpexpcl 10488 | . . . . . 6 | |
138 | 38, 69, 137 | sylancl 411 | . . . . 5 |
139 | elrp 9605 | . . . . . 6 | |
140 | ltmul2 8765 | . . . . . . 7 | |
141 | 24, 131, 140 | mp3an12 1322 | . . . . . 6 |
142 | 139, 141 | sylbi 120 | . . . . 5 |
143 | 138, 142 | syl 14 | . . . 4 |
144 | 136, 143 | mpbii 147 | . . 3 |
145 | 16 | recnd 7941 | . . . 4 |
146 | divrecap 8598 | . . . . 5 # | |
147 | 105, 129, 146 | mp3an23 1324 | . . . 4 |
148 | 145, 147 | syl 14 | . . 3 |
149 | 144, 148 | breqtrrd 4015 | . 2 |
150 | 14, 26, 29, 52, 149 | lelttrd 8037 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 class class class wbr 3987 cmpt 4048 cfv 5196 (class class class)co 5851 cc 7765 cr 7766 cc0 7767 c1 7768 ci 7769 caddc 7770 cmul 7772 cxr 7946 clt 7947 cle 7948 # cap 8493 cdiv 8582 cn 8871 c2 8922 c3 8923 c4 8924 c5 8925 c6 8926 c8 8928 cn0 9128 cz 9205 cuz 9480 crp 9603 cioc 9839 cexp 10468 cfa 10652 cabs 10954 csu 11309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 ax-arch 7886 ax-caucvg 7887 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-frec 6368 df-1o 6393 df-oadd 6397 df-er 6511 df-en 6717 df-dom 6718 df-fin 6719 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-3 8931 df-4 8932 df-5 8933 df-6 8934 df-7 8935 df-8 8936 df-n0 9129 df-z 9206 df-uz 9481 df-q 9572 df-rp 9604 df-ioc 9843 df-ico 9844 df-fz 9959 df-fzo 10092 df-seqfrec 10395 df-exp 10469 df-fac 10653 df-ihash 10703 df-shft 10772 df-cj 10799 df-re 10800 df-im 10801 df-rsqrt 10955 df-abs 10956 df-clim 11235 df-sumdc 11310 |
This theorem is referenced by: sin01bnd 11713 cos01bnd 11714 |
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