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| Mirrors > Home > MPE Home > Th. List > tanregt0 | Structured version Visualization version Unicode version | ||
Description: The positivity of    
extends to complex numbers with the same
real part. (Contributed by Mario Carneiro,
5-Apr-2015.) |
| Ref | Expression |
|---|---|
| tanregt0 |
   ![/\](wedge.gif "/\"){kind=small}         
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-1cn 9994 |
. . . . . . 7
 | |
| 2 | recl 13850 |
. . . . . . . . . . 11
 | |
| 3 | 2 | adantr 481 |
. . . . . . . . . 10
|
| 4 | 3 | recnd 10068 |
. . . . . . . . 9
|
| 5 | 3 | rered 13964 |
. . . . . . . . . . 11
 |
| 6 | neghalfpire 24217 |
. . . . . . . . . . . . . 14
| |
| 7 | 6 | rexri 10097 |
. . . . . . . . . . . . 13
 |
| 8 | 0re 10040 |
. . . . . . . . . . . . . 14
| |
| 9 | pirp 24213 |
. . . . . . . . . . . . . . . 16
 | |
| 10 | rphalfcl 11858 |
. . . . . . . . . . . . . . . 16
| |
| 11 | rpgt0 11844 |
. . . . . . . . . . . . . . . 16
| |
| 12 | 9, 10, 11 | mp2b 10 |
. . . . . . . . . . . . . . 15
|
| 13 | halfpire 24216 |
. . . . . . . . . . . . . . . 16
| |
| 14 | lt0neg2 10535 |
. . . . . . . . . . . . . . . 16
| |
| 15 | 13, 14 | ax-mp 5 |
. . . . . . . . . . . . . . 15
|
| 16 | 12, 15 | mpbi 220 |
. . . . . . . . . . . . . 14
|
| 17 | 6, 8, 16 | ltleii 10160 |
. . . . . . . . . . . . 13
 |
| 18 | iooss1 12210 |
. . . . . . . . . . . . 13
 | |
| 19 | 7, 17, 18 | mp2an 708 |
. . . . . . . . . . . 12
|
| 20 | simpr 477 |
. . . . . . . . . . . 12
| |
| 21 | 19, 20 | sseldi 3601 |
. . . . . . . . . . 11
|
| 22 | 5, 21 | eqeltrd 2701 |
. . . . . . . . . 10
|
| 23 | cosne0 24276 |
. . . . . . . . . 10
  | |
| 24 | 4, 22, 23 | syl2anc 693 |
. . . . . . . . 9
|
| 25 | 4, 24 | tancld 14862 |
. . . . . . . 8
|
| 26 | ax-icn 9995 |
. . . . . . . . . 10
 | |
| 27 | imcl 13851 |
. . . . . . . . . . . 12
 | |
| 28 | 27 | adantr 481 |
. . . . . . . . . . 11
|
| 29 | 28 | recnd 10068 |
. . . . . . . . . 10
|
| 30 | mulcl 10020 |
. . . . . . . . . 10
 | |
| 31 | 26, 29, 30 | sylancr 695 |
. . . . . . . . 9
|
| 32 | rpcoshcl 14887 |
. . . . . . . . . . 11
| |
| 33 | 28, 32 | syl 17 |
. . . . . . . . . 10
|
| 34 | 33 | rpne0d 11877 |
. . . . . . . . 9
|
| 35 | 31, 34 | tancld 14862 |
. . . . . . . 8
|
| 36 | 25, 35 | mulcld 10060 |
. . . . . . 7
|
| 37 | subcl 10280 |
. . . . . . 7
| |
| 38 | 1, 36, 37 | sylancr 695 |
. . . . . 6
|
| 39 | replim 13856 |
. . . . . . . . . . . 12
 | |
| 40 | 39 | adantr 481 |
. . . . . . . . . . 11
|
| 41 | 40 | fveq2d 6195 |
. . . . . . . . . 10
|
| 42 | cosne0 24276 |
. . . . . . . . . . 11
| |
| 43 | 21, 42 | syldan 487 |
. . . . . . . . . 10
|
| 44 | 41, 43 | eqnetrrd 2862 |
. . . . . . . . 9
|
| 45 | tanaddlem 14896 |
. . . . . . . . . 10
| |
| 46 | 4, 31, 24, 34, 45 | syl22anc 1327 |
. . . . . . . . 9
|
| 47 | 44, 46 | mpbid 222 |
. . . . . . . 8
|
| 48 | 47 | necomd 2849 |
. . . . . . 7
|
| 49 | subeq0 10307 |
. . . . . . . . 9
| |
| 50 | 49 | necon3bid 2838 |
. . . . . . . 8
|
| 51 | 1, 36, 50 | sylancr 695 |
. . . . . . 7
|
| 52 | 48, 51 | mpbird 247 |
. . . . . 6
|
| 53 | 38, 52 | absrpcld 14187 |
. . . . 5
|
| 54 | 2z 11409 |
. . . . 5
 | |
| 55 | rpexpcl 12879 |
. . . . 5
 | |
| 56 | 53, 54, 55 | sylancl 694 |
. . . 4
|
| 57 | 56 | rprecred 11883 |
. . 3
|
| 58 | 38 | cjcld 13936 |
. . . . 5
|
| 59 | 25, 35 | addcld 10059 |
. . . . 5
|
| 60 | 58, 59 | mulcld 10060 |
. . . 4
|
| 61 | 60 | recld 13934 |
. . 3
|
| 62 | 56 | rpreccld 11882 |
. . . 4
|
| 63 | 62 | rpgt0d 11875 |
. . 3
|
| 64 | 3, 24 | retancld 14875 |
. . . . 5
|
| 65 | 1re 10039 |
. . . . . 6
| |
| 66 | retanhcl 14889 |
. . . . . . . 8
| |
| 67 | 28, 66 | syl 17 |
. . . . . . 7
|
| 68 | 67 | resqcld 13035 |
. . . . . 6
|
| 69 | resubcl 10345 |
. . . . . 6
| |
| 70 | 65, 68, 69 | sylancr 695 |
. . . . 5
|
| 71 | tanrpcl 24256 |
. . . . . . 7
| |
| 72 | 71 | adantl 482 |
. . . . . 6
|
| 73 | 72 | rpgt0d 11875 |
. . . . 5
|
| 74 | absresq 14042 |
. . . . . . . 8
| |
| 75 | 67, 74 | syl 17 |
. . . . . . 7
|
| 76 | tanhbnd 14891 |
. . . . . . . . . . . 12
| |
| 77 | 28, 76 | syl 17 |
. . . . . . . . . . 11
|
| 78 | eliooord 12233 |
. . . . . . . . . . 11
| |
| 79 | 77, 78 | syl 17 |
. . . . . . . . . 10
|
| 80 | abslt 14054 |
. . . . . . . . . . 11
| |
| 81 | 67, 65, 80 | sylancl 694 |
. . . . . . . . . 10
|
| 82 | 79, 81 | mpbird 247 |
. . . . . . . . 9
|
| 83 | 67 | recnd 10068 |
. . . . . . . . . . 11
|
| 84 | 83 | abscld 14175 |
. . . . . . . . . 10
|
| 85 | 65 | a1i 11 |
. . . . . . . . . 10
|
| 86 | 83 | absge0d 14183 |
. . . . . . . . . 10
|
| 87 | 0le1 10551 |
. . . . . . . . . . 11
| |
| 88 | 87 | a1i 11 |
. . . . . . . . . 10
|
| 89 | 84, 85, 86, 88 | lt2sqd 13043 |
. . . . . . . . 9
|
| 90 | 82, 89 | mpbid 222 |
. . . . . . . 8
|
| 91 | sq1 12958 |
. . . . . . . 8
| |
| 92 | 90, 91 | syl6breq 4694 |
. . . . . . 7
|
| 93 | 75, 92 | eqbrtrrd 4677 |
. . . . . 6
|
| 94 | posdif 10521 |
. . . . . . 7
| |
| 95 | 68, 65, 94 | sylancl 694 |
. . . . . 6
|
| 96 | 93, 95 | mpbid 222 |
. . . . 5
|
| 97 | 64, 70, 73, 96 | mulgt0d 10192 |
. . . 4
|
| 98 | 38 | recjd 13944 |
. . . . . . . . 9
|
| 99 | resub 13867 |
. . . . . . . . . 10
| |
| 100 | 1, 36, 99 | sylancr 695 |
. . . . . . . . 9
|
| 101 | re1 13894 |
. . . . . . . . . . 11
| |
| 102 | 101 | oveq1i 6660 |
. . . . . . . . . 10
|
| 103 | 64, 35 | remul2d 13967 |
. . . . . . . . . . . . 13
|
| 104 | negicn 10282 |
. . . . . . . . . . . . . . . . . 18
| |
| 105 | 104 | a1i 11 |
. . . . . . . . . . . . . . . . 17
|
| 106 | ine0 10465 |
. . . . . . . . . . . . . . . . . . 19
| |
| 107 | 26, 106 | negne0i 10356 |
. . . . . . . . . . . . . . . . . 18
|
| 108 | 107 | a1i 11 |
. . . . . . . . . . . . . . . . 17
|
| 109 | 35, 105, 108 | divcld 10801 |
. . . . . . . . . . . . . . . 16
|
| 110 | imre 13848 |
. . . . . . . . . . . . . . . 16
| |
| 111 | 109, 110 | syl 17 |
. . . . . . . . . . . . . . 15
|
| 112 | 26 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
|
| 113 | 106 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
|
| 114 | 35, 112, 113 | divneg2d 10815 |
. . . . . . . . . . . . . . . . 17
|
| 115 | 67 | renegcld 10457 |
. . . . . . . . . . . . . . . . 17
|
| 116 | 114, 115 | eqeltrrd 2702 |
. . . . . . . . . . . . . . . 16
|
| 117 | 116 | reim0d 13965 |
. . . . . . . . . . . . . . 15
|
| 118 | 35, 105, 108 | divcan2d 10803 |
. . . . . . . . . . . . . . . 16
|
| 119 | 118 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
|
| 120 | 111, 117, 119 | 3eqtr3rd 2665 |
. . . . . . . . . . . . . 14
|
| 121 | 120 | oveq2d 6666 |
. . . . . . . . . . . . 13
|
| 122 | 25 | mul01d 10235 |
. . . . . . . . . . . . 13
|
| 123 | 103, 121, 122 | 3eqtrd 2660 |
. . . . . . . . . . . 12
|
| 124 | 123 | oveq2d 6666 |
. . . . . . . . . . 11
|
| 125 | 1m0e1 11131 |
. . . . . . . . . . 11
| |
| 126 | 124, 125 | syl6eq 2672 |
. . . . . . . . . 10
|
| 127 | 102, 126 | syl5eq 2668 |
. . . . . . . . 9
|
| 128 | 98, 100, 127 | 3eqtrd 2660 |
. . . . . . . 8
|
| 129 | 35, 112, 113 | divcan2d 10803 |
. . . . . . . . . . 11
|
| 130 | 129 | oveq2d 6666 |
. . . . . . . . . 10
|
| 131 | 130 | fveq2d 6195 |
. . . . . . . . 9
|
| 132 | 64, 67 | crred 13971 |
. . . . . . . . 9
|
| 133 | 131, 132 | eqtr3d 2658 |
. . . . . . . 8
|
| 134 | 128, 133 | oveq12d 6668 |
. . . . . . 7
|
| 135 | mulcom 10022 |
. . . . . . . 8
| |
| 136 | 1, 25, 135 | sylancr 695 |
. . . . . . 7
|
| 137 | 134, 136 | eqtrd 2656 |
. . . . . 6
|
| 138 | 25, 83, 83 | mulassd 10063 |
. . . . . . 7
|
| 139 | 38 | imcjd 13945 |
. . . . . . . . 9
|
| 140 | imsub 13875 |
. . . . . . . . . . . 12
| |
| 141 | 1, 36, 140 | sylancr 695 |
. . . . . . . . . . 11
|
| 142 | im1 13895 |
. . . . . . . . . . . . . 14
| |
| 143 | 142 | oveq1i 6660 |
. . . . . . . . . . . . 13
|
| 144 | df-neg 10269 |
. . . . . . . . . . . . 13
| |
| 145 | 143, 144 | eqtr4i 2647 |
. . . . . . . . . . . 12
|
| 146 | 64, 35 | immul2d 13968 |
. . . . . . . . . . . . . 14
|
| 147 | imval 13847 |
. . . . . . . . . . . . . . . . 17
| |
| 148 | 35, 147 | syl 17 |
. . . . . . . . . . . . . . . 16
|
| 149 | 67 | rered 13964 |
. . . . . . . . . . . . . . . 16
|
| 150 | 148, 149 | eqtrd 2656 |
. . . . . . . . . . . . . . 15
|
| 151 | 150 | oveq2d 6666 |
. . . . . . . . . . . . . 14
|
| 152 | 146, 151 | eqtrd 2656 |
. . . . . . . . . . . . 13
|
| 153 | 152 | negeqd 10275 |
. . . . . . . . . . . 12
|
| 154 | 145, 153 | syl5eq 2668 |
. . . . . . . . . . 11
|
| 155 | 141, 154 | eqtrd 2656 |
. . . . . . . . . 10
|
| 156 | 155 | negeqd 10275 |
. . . . . . . . 9
|
| 157 | 64, 67 | remulcld 10070 |
. . . . . . . . . . 11
|
| 158 | 157 | recnd 10068 |
. . . . . . . . . 10
|
| 159 | 158 | negnegd 10383 |
. . . . . . . . 9
|
| 160 | 139, 156, 159 | 3eqtrd 2660 |
. . . . . . . 8
|
| 161 | 130 | fveq2d 6195 |
. . . . . . . . 9
|
| 162 | 64, 67 | crimd 13972 |
. . . . . . . . 9
|
| 163 | 161, 162 | eqtr3d 2658 |
. . . . . . . 8
|
| 164 | 160, 163 | oveq12d 6668 |
. . . . . . 7
|
| 165 | 83 | sqvald 13005 |
. . . . . . . 8
|
| 166 | 165 | oveq2d 6666 |
. . . . . . 7
|
| 167 | 138, 164, 166 | 3eqtr4d 2666 |
. . . . . 6
|
| 168 | 137, 167 | oveq12d 6668 |
. . . . 5
|
| 169 | 58, 59 | remuld 13958 |
. . . . 5
|
| 170 | 1 | a1i 11 |
. . . . . 6
|
| 171 | 83 | sqcld 13006 |
. . . . . 6
|
| 172 | 25, 170, 171 | subdid 10486 |
. . . . 5
|
| 173 | 168, 169, 172 | 3eqtr4d 2666 |
. . . 4
|
| 174 | 97, 173 | breqtrrd 4681 |
. . 3
|
| 175 | 57, 61, 63, 174 | mulgt0d 10192 |
. 2
|
| 176 | 40 | fveq2d 6195 |
. . . . . 6
|
| 177 | tanadd 14897 |
. . . . . . 7
| |
| 178 | 4, 31, 24, 34, 44, 177 | syl23anc 1333 |
. . . . . 6
|
| 179 | recval 14062 |
. . . . . . . . 9
| |
| 180 | 38, 52, 179 | syl2anc 693 |
. . . . . . . 8
|
| 181 | 180 | oveq1d 6665 |
. . . . . . 7
|
| 182 | 59, 38, 52 | divrec2d 10805 |
. . . . . . 7
|
| 183 | 38 | abscld 14175 |
. . . . . . . . . 10
|
| 184 | 183 | resqcld 13035 |
. . . . . . . . 9
|
| 185 | 184 | recnd 10068 |
. . . . . . . 8
|
| 186 | 56 | rpne0d 11877 |
. . . . . . . 8
|
| 187 | 58, 59, 185, 186 | div23d 10838 |
. . . . . . 7
|
| 188 | 181, 182, 187 | 3eqtr4d 2666 |
. . . . . 6
|
| 189 | 176, 178, 188 | 3eqtrd 2660 |
. . . . 5
|
| 190 | 60, 185, 186 | divrec2d 10805 |
. . . . 5
|
| 191 | 189, 190 | eqtrd 2656 |
. . . 4
|
| 192 | 191 | fveq2d 6195 |
. . 3
|
| 193 | 57, 60 | remul2d 13967 |
. . 3
|
| 194 | 192, 193 | eqtrd 2656 |
. 2
|
| 195 | 175, 194 | breqtrrd 4681 |
1
|
| Colors of variables: wff setvar class |
| Syntax hints: wi 4
wb 196 wa 384 wceq 1483 wcel 1990 wne 2794 wss 3574 class class class wbr 4653
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 cmin 10266 cneg 10267 cdiv 10684 c2 11070 cz 11377 crp 11832 cioo 12175 cexp 12860 ccj 13836 cre 13837 cim 13838 cabs 13974 ccos 14795 ctan 14796 cpi 14797 |
| 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-iin 4523 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-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 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-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 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 df-ef 14798 df-sin 14800 df-cos 14801 df-tan 14802 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 |
| This theorem is referenced by: atantan 24650 |
| Copyright terms: Public domain | W3C validator |