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Mirrors > Home > ILE Home > Th. List > ef01bndlem | Unicode version |
Description: Lemma for sin01bnd 11720 and cos01bnd 11721. (Contributed by Paul Chapman, 19-Jan-2008.) |
Ref | Expression |
---|---|
ef01bnd.1 |
Ref | Expression |
---|---|
ef01bndlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 7869 | . . . . 5 | |
2 | 0xr 7966 | . . . . . . . 8 | |
3 | 1re 7919 | . . . . . . . 8 | |
4 | elioc2 9893 | . . . . . . . 8 | |
5 | 2, 3, 4 | mp2an 424 | . . . . . . 7 |
6 | 5 | simp1bi 1007 | . . . . . 6 |
7 | 6 | recnd 7948 | . . . . 5 |
8 | mulcl 7901 | . . . . 5 | |
9 | 1, 7, 8 | sylancr 412 | . . . 4 |
10 | 4nn0 9154 | . . . 4 | |
11 | ef01bnd.1 | . . . . 5 | |
12 | 11 | eftlcl 11651 | . . . 4 |
13 | 9, 10, 12 | sylancl 411 | . . 3 |
14 | 13 | abscld 11145 | . 2 |
15 | reexpcl 10493 | . . . 4 | |
16 | 6, 10, 15 | sylancl 411 | . . 3 |
17 | 4re 8955 | . . . . 5 | |
18 | 17, 3 | readdcli 7933 | . . . 4 |
19 | faccl 10669 | . . . . . 6 | |
20 | 10, 19 | ax-mp 5 | . . . . 5 |
21 | 4nn 9041 | . . . . 5 | |
22 | 20, 21 | nnmulcli 8900 | . . . 4 |
23 | nndivre 8914 | . . . 4 | |
24 | 18, 22, 23 | mp2an 424 | . . 3 |
25 | remulcl 7902 | . . 3 | |
26 | 16, 24, 25 | sylancl 411 | . 2 |
27 | 6nn 9043 | . . 3 | |
28 | nndivre 8914 | . . 3 | |
29 | 16, 27, 28 | sylancl 411 | . 2 |
30 | eqid 2170 | . . . 4 | |
31 | eqid 2170 | . . . 4 | |
32 | 21 | a1i 9 | . . . 4 |
33 | absmul 11033 | . . . . . . 7 | |
34 | 1, 7, 33 | sylancr 412 | . . . . . 6 |
35 | absi 11023 | . . . . . . . 8 | |
36 | 35 | oveq1i 5863 | . . . . . . 7 |
37 | 5 | simp2bi 1008 | . . . . . . . . . 10 |
38 | 6, 37 | elrpd 9650 | . . . . . . . . 9 |
39 | rpre 9617 | . . . . . . . . . 10 | |
40 | rpge0 9623 | . . . . . . . . . 10 | |
41 | 39, 40 | absidd 11131 | . . . . . . . . 9 |
42 | 38, 41 | syl 14 | . . . . . . . 8 |
43 | 42 | oveq2d 5869 | . . . . . . 7 |
44 | 36, 43 | eqtrid 2215 | . . . . . 6 |
45 | 7 | mulid2d 7938 | . . . . . 6 |
46 | 34, 44, 45 | 3eqtrd 2207 | . . . . 5 |
47 | 5 | simp3bi 1009 | . . . . 5 |
48 | 46, 47 | eqbrtrd 4011 | . . . 4 |
49 | 11, 30, 31, 32, 9, 48 | eftlub 11653 | . . 3 |
50 | 46 | oveq1d 5868 | . . . 4 |
51 | 50 | oveq1d 5868 | . . 3 |
52 | 49, 51 | breqtrd 4015 | . 2 |
53 | 3pos 8972 | . . . . . . . . 9 | |
54 | 0re 7920 | . . . . . . . . . 10 | |
55 | 3re 8952 | . . . . . . . . . 10 | |
56 | 5re 8957 | . . . . . . . . . 10 | |
57 | 54, 55, 56 | ltadd1i 8421 | . . . . . . . . 9 |
58 | 53, 57 | mpbi 144 | . . . . . . . 8 |
59 | 5cn 8958 | . . . . . . . . 9 | |
60 | 59 | addid2i 8062 | . . . . . . . 8 |
61 | cu2 10574 | . . . . . . . . 9 | |
62 | 5p3e8 9025 | . . . . . . . . 9 | |
63 | 3cn 8953 | . . . . . . . . . 10 | |
64 | 59, 63 | addcomi 8063 | . . . . . . . . 9 |
65 | 61, 62, 64 | 3eqtr2ri 2198 | . . . . . . . 8 |
66 | 58, 60, 65 | 3brtr3i 4018 | . . . . . . 7 |
67 | 2re 8948 | . . . . . . . 8 | |
68 | 1le2 9086 | . . . . . . . 8 | |
69 | 4z 9242 | . . . . . . . . 9 | |
70 | 3lt4 9050 | . . . . . . . . . 10 | |
71 | 55, 17, 70 | ltleii 8022 | . . . . . . . . 9 |
72 | 3z 9241 | . . . . . . . . . 10 | |
73 | 72 | eluz1i 9494 | . . . . . . . . 9 |
74 | 69, 71, 73 | mpbir2an 937 | . . . . . . . 8 |
75 | leexp2a 10529 | . . . . . . . 8 | |
76 | 67, 68, 74, 75 | mp3an 1332 | . . . . . . 7 |
77 | 8re 8963 | . . . . . . . . 9 | |
78 | 61, 77 | eqeltri 2243 | . . . . . . . 8 |
79 | 2nn 9039 | . . . . . . . . . 10 | |
80 | nnexpcl 10489 | . . . . . . . . . 10 | |
81 | 79, 10, 80 | mp2an 424 | . . . . . . . . 9 |
82 | 81 | nnrei 8887 | . . . . . . . 8 |
83 | 56, 78, 82 | ltletri 8026 | . . . . . . 7 |
84 | 66, 76, 83 | mp2an 424 | . . . . . 6 |
85 | 6re 8959 | . . . . . . . 8 | |
86 | 85, 82 | remulcli 7934 | . . . . . . 7 |
87 | 6pos 8979 | . . . . . . . 8 | |
88 | 81 | nngt0i 8908 | . . . . . . . 8 |
89 | 85, 82, 87, 88 | mulgt0ii 8030 | . . . . . . 7 |
90 | 56, 82, 86, 89 | ltdiv1ii 8845 | . . . . . 6 |
91 | 84, 90 | mpbi 144 | . . . . 5 |
92 | df-5 8940 | . . . . . 6 | |
93 | df-4 8939 | . . . . . . . . . . 11 | |
94 | 93 | fveq2i 5499 | . . . . . . . . . 10 |
95 | 3nn0 9153 | . . . . . . . . . . 11 | |
96 | facp1 10664 | . . . . . . . . . . 11 | |
97 | 95, 96 | ax-mp 5 | . . . . . . . . . 10 |
98 | sq2 10571 | . . . . . . . . . . . 12 | |
99 | 98, 93 | eqtr2i 2192 | . . . . . . . . . . 11 |
100 | 99 | oveq2i 5864 | . . . . . . . . . 10 |
101 | 94, 97, 100 | 3eqtri 2195 | . . . . . . . . 9 |
102 | 101 | oveq1i 5863 | . . . . . . . 8 |
103 | 98 | oveq2i 5864 | . . . . . . . 8 |
104 | fac3 10666 | . . . . . . . . . 10 | |
105 | 6cn 8960 | . . . . . . . . . 10 | |
106 | 104, 105 | eqeltri 2243 | . . . . . . . . 9 |
107 | 17 | recni 7932 | . . . . . . . . . 10 |
108 | 98, 107 | eqeltri 2243 | . . . . . . . . 9 |
109 | 106, 108, 108 | mulassi 7929 | . . . . . . . 8 |
110 | 102, 103, 109 | 3eqtr3i 2199 | . . . . . . 7 |
111 | 2p2e4 9005 | . . . . . . . . . 10 | |
112 | 111 | oveq2i 5864 | . . . . . . . . 9 |
113 | 2cn 8949 | . . . . . . . . . 10 | |
114 | 2nn0 9152 | . . . . . . . . . 10 | |
115 | expadd 10518 | . . . . . . . . . 10 | |
116 | 113, 114, 114, 115 | mp3an 1332 | . . . . . . . . 9 |
117 | 112, 116 | eqtr3i 2193 | . . . . . . . 8 |
118 | 117 | oveq2i 5864 | . . . . . . 7 |
119 | 104 | oveq1i 5863 | . . . . . . 7 |
120 | 110, 118, 119 | 3eqtr2ri 2198 | . . . . . 6 |
121 | 92, 120 | oveq12i 5865 | . . . . 5 |
122 | 81 | nncni 8888 | . . . . . . . 8 |
123 | 122 | mulid2i 7923 | . . . . . . 7 |
124 | 123 | oveq1i 5863 | . . . . . 6 |
125 | 82, 88 | gt0ap0ii 8547 | . . . . . . . . 9 # |
126 | 122, 125 | dividapi 8662 | . . . . . . . 8 |
127 | 126 | oveq2i 5864 | . . . . . . 7 |
128 | ax-1cn 7867 | . . . . . . . 8 | |
129 | 85, 87 | gt0ap0ii 8547 | . . . . . . . 8 # |
130 | 128, 105, 122, 122, 129, 125 | divmuldivapi 8689 | . . . . . . 7 |
131 | 85, 129 | rerecclapi 8694 | . . . . . . . . 9 |
132 | 131 | recni 7932 | . . . . . . . 8 |
133 | 132 | mulid1i 7922 | . . . . . . 7 |
134 | 127, 130, 133 | 3eqtr3i 2199 | . . . . . 6 |
135 | 124, 134 | eqtr3i 2193 | . . . . 5 |
136 | 91, 121, 135 | 3brtr3i 4018 | . . . 4 |
137 | rpexpcl 10495 | . . . . . 6 | |
138 | 38, 69, 137 | sylancl 411 | . . . . 5 |
139 | elrp 9612 | . . . . . 6 | |
140 | ltmul2 8772 | . . . . . . 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 7948 | . . . 4 |
146 | divrecap 8605 | . . . . 5 # | |
147 | 105, 129, 146 | mp3an23 1324 | . . . 4 |
148 | 145, 147 | syl 14 | . . 3 |
149 | 144, 148 | breqtrrd 4017 | . 2 |
150 | 14, 26, 29, 52, 149 | lelttrd 8044 | 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 3989 cmpt 4050 cfv 5198 (class class class)co 5853 cc 7772 cr 7773 cc0 7774 c1 7775 ci 7776 caddc 7777 cmul 7779 cxr 7953 clt 7954 cle 7955 # cap 8500 cdiv 8589 cn 8878 c2 8929 c3 8930 c4 8931 c5 8932 c6 8933 c8 8935 cn0 9135 cz 9212 cuz 9487 crp 9610 cioc 9846 cexp 10475 cfa 10659 cabs 10961 csu 11316 |
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 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 |
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 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941 df-7 8942 df-8 8943 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-ioc 9850 df-ico 9851 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-fac 10660 df-ihash 10710 df-shft 10779 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-sumdc 11317 |
This theorem is referenced by: sin01bnd 11720 cos01bnd 11721 |
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