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Mirrors > Home > ILE Home > Th. List > ef01bndlem | Unicode version |
Description: Lemma for sin01bnd 11464 and cos01bnd 11465. (Contributed by Paul Chapman, 19-Jan-2008.) |
Ref | Expression |
---|---|
ef01bnd.1 |
Ref | Expression |
---|---|
ef01bndlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 7715 | . . . . 5 | |
2 | 0xr 7812 | . . . . . . . 8 | |
3 | 1re 7765 | . . . . . . . 8 | |
4 | elioc2 9719 | . . . . . . . 8 | |
5 | 2, 3, 4 | mp2an 422 | . . . . . . 7 |
6 | 5 | simp1bi 996 | . . . . . 6 |
7 | 6 | recnd 7794 | . . . . 5 |
8 | mulcl 7747 | . . . . 5 | |
9 | 1, 7, 8 | sylancr 410 | . . . 4 |
10 | 4nn0 8996 | . . . 4 | |
11 | ef01bnd.1 | . . . . 5 | |
12 | 11 | eftlcl 11394 | . . . 4 |
13 | 9, 10, 12 | sylancl 409 | . . 3 |
14 | 13 | abscld 10953 | . 2 |
15 | reexpcl 10310 | . . . 4 | |
16 | 6, 10, 15 | sylancl 409 | . . 3 |
17 | 4re 8797 | . . . . 5 | |
18 | 17, 3 | readdcli 7779 | . . . 4 |
19 | faccl 10481 | . . . . . 6 | |
20 | 10, 19 | ax-mp 5 | . . . . 5 |
21 | 4nn 8883 | . . . . 5 | |
22 | 20, 21 | nnmulcli 8742 | . . . 4 |
23 | nndivre 8756 | . . . 4 | |
24 | 18, 22, 23 | mp2an 422 | . . 3 |
25 | remulcl 7748 | . . 3 | |
26 | 16, 24, 25 | sylancl 409 | . 2 |
27 | 6nn 8885 | . . 3 | |
28 | nndivre 8756 | . . 3 | |
29 | 16, 27, 28 | sylancl 409 | . 2 |
30 | eqid 2139 | . . . 4 | |
31 | eqid 2139 | . . . 4 | |
32 | 21 | a1i 9 | . . . 4 |
33 | absmul 10841 | . . . . . . 7 | |
34 | 1, 7, 33 | sylancr 410 | . . . . . 6 |
35 | absi 10831 | . . . . . . . 8 | |
36 | 35 | oveq1i 5784 | . . . . . . 7 |
37 | 5 | simp2bi 997 | . . . . . . . . . 10 |
38 | 6, 37 | elrpd 9481 | . . . . . . . . 9 |
39 | rpre 9448 | . . . . . . . . . 10 | |
40 | rpge0 9454 | . . . . . . . . . 10 | |
41 | 39, 40 | absidd 10939 | . . . . . . . . 9 |
42 | 38, 41 | syl 14 | . . . . . . . 8 |
43 | 42 | oveq2d 5790 | . . . . . . 7 |
44 | 36, 43 | syl5eq 2184 | . . . . . 6 |
45 | 7 | mulid2d 7784 | . . . . . 6 |
46 | 34, 44, 45 | 3eqtrd 2176 | . . . . 5 |
47 | 5 | simp3bi 998 | . . . . 5 |
48 | 46, 47 | eqbrtrd 3950 | . . . 4 |
49 | 11, 30, 31, 32, 9, 48 | eftlub 11396 | . . 3 |
50 | 46 | oveq1d 5789 | . . . 4 |
51 | 50 | oveq1d 5789 | . . 3 |
52 | 49, 51 | breqtrd 3954 | . 2 |
53 | 3pos 8814 | . . . . . . . . 9 | |
54 | 0re 7766 | . . . . . . . . . 10 | |
55 | 3re 8794 | . . . . . . . . . 10 | |
56 | 5re 8799 | . . . . . . . . . 10 | |
57 | 54, 55, 56 | ltadd1i 8264 | . . . . . . . . 9 |
58 | 53, 57 | mpbi 144 | . . . . . . . 8 |
59 | 5cn 8800 | . . . . . . . . 9 | |
60 | 59 | addid2i 7905 | . . . . . . . 8 |
61 | cu2 10391 | . . . . . . . . 9 | |
62 | 5p3e8 8867 | . . . . . . . . 9 | |
63 | 3cn 8795 | . . . . . . . . . 10 | |
64 | 59, 63 | addcomi 7906 | . . . . . . . . 9 |
65 | 61, 62, 64 | 3eqtr2ri 2167 | . . . . . . . 8 |
66 | 58, 60, 65 | 3brtr3i 3957 | . . . . . . 7 |
67 | 2re 8790 | . . . . . . . 8 | |
68 | 1le2 8928 | . . . . . . . 8 | |
69 | 4z 9084 | . . . . . . . . 9 | |
70 | 3lt4 8892 | . . . . . . . . . 10 | |
71 | 55, 17, 70 | ltleii 7866 | . . . . . . . . 9 |
72 | 3z 9083 | . . . . . . . . . 10 | |
73 | 72 | eluz1i 9333 | . . . . . . . . 9 |
74 | 69, 71, 73 | mpbir2an 926 | . . . . . . . 8 |
75 | leexp2a 10346 | . . . . . . . 8 | |
76 | 67, 68, 74, 75 | mp3an 1315 | . . . . . . 7 |
77 | 8re 8805 | . . . . . . . . 9 | |
78 | 61, 77 | eqeltri 2212 | . . . . . . . 8 |
79 | 2nn 8881 | . . . . . . . . . 10 | |
80 | nnexpcl 10306 | . . . . . . . . . 10 | |
81 | 79, 10, 80 | mp2an 422 | . . . . . . . . 9 |
82 | 81 | nnrei 8729 | . . . . . . . 8 |
83 | 56, 78, 82 | ltletri 7870 | . . . . . . 7 |
84 | 66, 76, 83 | mp2an 422 | . . . . . 6 |
85 | 6re 8801 | . . . . . . . 8 | |
86 | 85, 82 | remulcli 7780 | . . . . . . 7 |
87 | 6pos 8821 | . . . . . . . 8 | |
88 | 81 | nngt0i 8750 | . . . . . . . 8 |
89 | 85, 82, 87, 88 | mulgt0ii 7874 | . . . . . . 7 |
90 | 56, 82, 86, 89 | ltdiv1ii 8687 | . . . . . 6 |
91 | 84, 90 | mpbi 144 | . . . . 5 |
92 | df-5 8782 | . . . . . 6 | |
93 | df-4 8781 | . . . . . . . . . . 11 | |
94 | 93 | fveq2i 5424 | . . . . . . . . . 10 |
95 | 3nn0 8995 | . . . . . . . . . . 11 | |
96 | facp1 10476 | . . . . . . . . . . 11 | |
97 | 95, 96 | ax-mp 5 | . . . . . . . . . 10 |
98 | sq2 10388 | . . . . . . . . . . . 12 | |
99 | 98, 93 | eqtr2i 2161 | . . . . . . . . . . 11 |
100 | 99 | oveq2i 5785 | . . . . . . . . . 10 |
101 | 94, 97, 100 | 3eqtri 2164 | . . . . . . . . 9 |
102 | 101 | oveq1i 5784 | . . . . . . . 8 |
103 | 98 | oveq2i 5785 | . . . . . . . 8 |
104 | fac3 10478 | . . . . . . . . . 10 | |
105 | 6cn 8802 | . . . . . . . . . 10 | |
106 | 104, 105 | eqeltri 2212 | . . . . . . . . 9 |
107 | 17 | recni 7778 | . . . . . . . . . 10 |
108 | 98, 107 | eqeltri 2212 | . . . . . . . . 9 |
109 | 106, 108, 108 | mulassi 7775 | . . . . . . . 8 |
110 | 102, 103, 109 | 3eqtr3i 2168 | . . . . . . 7 |
111 | 2p2e4 8847 | . . . . . . . . . 10 | |
112 | 111 | oveq2i 5785 | . . . . . . . . 9 |
113 | 2cn 8791 | . . . . . . . . . 10 | |
114 | 2nn0 8994 | . . . . . . . . . 10 | |
115 | expadd 10335 | . . . . . . . . . 10 | |
116 | 113, 114, 114, 115 | mp3an 1315 | . . . . . . . . 9 |
117 | 112, 116 | eqtr3i 2162 | . . . . . . . 8 |
118 | 117 | oveq2i 5785 | . . . . . . 7 |
119 | 104 | oveq1i 5784 | . . . . . . 7 |
120 | 110, 118, 119 | 3eqtr2ri 2167 | . . . . . 6 |
121 | 92, 120 | oveq12i 5786 | . . . . 5 |
122 | 81 | nncni 8730 | . . . . . . . 8 |
123 | 122 | mulid2i 7769 | . . . . . . 7 |
124 | 123 | oveq1i 5784 | . . . . . 6 |
125 | 82, 88 | gt0ap0ii 8390 | . . . . . . . . 9 # |
126 | 122, 125 | dividapi 8505 | . . . . . . . 8 |
127 | 126 | oveq2i 5785 | . . . . . . 7 |
128 | ax-1cn 7713 | . . . . . . . 8 | |
129 | 85, 87 | gt0ap0ii 8390 | . . . . . . . 8 # |
130 | 128, 105, 122, 122, 129, 125 | divmuldivapi 8532 | . . . . . . 7 |
131 | 85, 129 | rerecclapi 8537 | . . . . . . . . 9 |
132 | 131 | recni 7778 | . . . . . . . 8 |
133 | 132 | mulid1i 7768 | . . . . . . 7 |
134 | 127, 130, 133 | 3eqtr3i 2168 | . . . . . 6 |
135 | 124, 134 | eqtr3i 2162 | . . . . 5 |
136 | 91, 121, 135 | 3brtr3i 3957 | . . . 4 |
137 | rpexpcl 10312 | . . . . . 6 | |
138 | 38, 69, 137 | sylancl 409 | . . . . 5 |
139 | elrp 9443 | . . . . . 6 | |
140 | ltmul2 8614 | . . . . . . 7 | |
141 | 24, 131, 140 | mp3an12 1305 | . . . . . 6 |
142 | 139, 141 | sylbi 120 | . . . . 5 |
143 | 138, 142 | syl 14 | . . . 4 |
144 | 136, 143 | mpbii 147 | . . 3 |
145 | 16 | recnd 7794 | . . . 4 |
146 | divrecap 8448 | . . . . 5 # | |
147 | 105, 129, 146 | mp3an23 1307 | . . . 4 |
148 | 145, 147 | syl 14 | . . 3 |
149 | 144, 148 | breqtrrd 3956 | . 2 |
150 | 14, 26, 29, 52, 149 | lelttrd 7887 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 class class class wbr 3929 cmpt 3989 cfv 5123 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 c1 7621 ci 7622 caddc 7623 cmul 7625 cxr 7799 clt 7800 cle 7801 # cap 8343 cdiv 8432 cn 8720 c2 8771 c3 8772 c4 8773 c5 8774 c6 8775 c8 8777 cn0 8977 cz 9054 cuz 9326 crp 9441 cioc 9672 cexp 10292 cfa 10471 cabs 10769 csu 11122 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-5 8782 df-6 8783 df-7 8784 df-8 8785 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-ioc 9676 df-ico 9677 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-fac 10472 df-ihash 10522 df-shft 10587 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 |
This theorem is referenced by: sin01bnd 11464 cos01bnd 11465 |
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