Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > coscn | Unicode version |
Description: Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.) |
Ref | Expression |
---|---|
coscn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cos 11560 | . 2 | |
2 | eqid 2157 | . . . . . . . 8 | |
3 | 2 | addcncntop 13048 | . . . . . . . . 9 |
4 | 3 | a1i 9 | . . . . . . . 8 |
5 | efcn 13185 | . . . . . . . . . 10 | |
6 | 5 | a1i 9 | . . . . . . . . 9 |
7 | ax-icn 7830 | . . . . . . . . . 10 | |
8 | eqid 2157 | . . . . . . . . . . 11 | |
9 | 8 | mulc1cncf 13072 | . . . . . . . . . 10 |
10 | 7, 9 | mp1i 10 | . . . . . . . . 9 |
11 | 6, 10 | cncfmpt1f 13080 | . . . . . . . 8 |
12 | negicn 8081 | . . . . . . . . . 10 | |
13 | eqid 2157 | . . . . . . . . . . 11 | |
14 | 13 | mulc1cncf 13072 | . . . . . . . . . 10 |
15 | 12, 14 | mp1i 10 | . . . . . . . . 9 |
16 | 6, 15 | cncfmpt1f 13080 | . . . . . . . 8 |
17 | 2, 4, 11, 16 | cncfmpt2fcntop 13081 | . . . . . . 7 |
18 | cncff 13060 | . . . . . . 7 | |
19 | 17, 18 | syl 14 | . . . . . 6 |
20 | eqid 2157 | . . . . . . 7 | |
21 | 20 | fmpt 5620 | . . . . . 6 |
22 | 19, 21 | sylibr 133 | . . . . 5 |
23 | eqidd 2158 | . . . . 5 | |
24 | eqidd 2158 | . . . . 5 | |
25 | oveq1 5834 | . . . . 5 | |
26 | 22, 23, 24, 25 | fmptcof 5637 | . . . 4 |
27 | 2cn 8910 | . . . . . . 7 | |
28 | 2ap0 8932 | . . . . . . 7 # | |
29 | eqid 2157 | . . . . . . . 8 | |
30 | 29 | divccncfap 13073 | . . . . . . 7 # |
31 | 27, 28, 30 | mp2an 423 | . . . . . 6 |
32 | 31 | a1i 9 | . . . . 5 |
33 | 17, 32 | cncfco 13074 | . . . 4 |
34 | 26, 33 | eqeltrrd 2235 | . . 3 |
35 | 34 | mptru 1344 | . 2 |
36 | 1, 35 | eqeltri 2230 | 1 |
Colors of variables: wff set class |
Syntax hints: wtru 1336 wcel 2128 wral 2435 class class class wbr 3967 cmpt 4028 ccom 4593 wf 5169 cfv 5173 (class class class)co 5827 cc 7733 cc0 7735 ci 7737 caddc 7738 cmul 7740 cmin 8051 cneg 8052 # cap 8461 cdiv 8550 c2 8890 cabs 10909 ce 11551 ccos 11554 cmopn 12481 ccn 12681 ctx 12748 ccncf 13053 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4082 ax-sep 4085 ax-nul 4093 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-iinf 4550 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-mulrcl 7834 ax-addcom 7835 ax-mulcom 7836 ax-addass 7837 ax-mulass 7838 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-1rid 7842 ax-0id 7843 ax-rnegex 7844 ax-precex 7845 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-apti 7850 ax-pre-ltadd 7851 ax-pre-mulgt0 7852 ax-pre-mulext 7853 ax-arch 7854 ax-caucvg 7855 ax-addf 7857 ax-mulf 7858 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-disj 3945 df-br 3968 df-opab 4029 df-mpt 4030 df-tr 4066 df-id 4256 df-po 4259 df-iso 4260 df-iord 4329 df-on 4331 df-ilim 4332 df-suc 4334 df-iom 4553 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-isom 5182 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-of 6035 df-1st 6091 df-2nd 6092 df-recs 6255 df-irdg 6320 df-frec 6341 df-1o 6366 df-oadd 6370 df-er 6483 df-map 6598 df-pm 6599 df-en 6689 df-dom 6690 df-fin 6691 df-sup 6931 df-inf 6932 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-reap 8455 df-ap 8462 df-div 8551 df-inn 8840 df-2 8898 df-3 8899 df-4 8900 df-n0 9097 df-z 9174 df-uz 9446 df-q 9536 df-rp 9568 df-xneg 9686 df-xadd 9687 df-ico 9805 df-fz 9920 df-fzo 10052 df-seqfrec 10355 df-exp 10429 df-fac 10612 df-bc 10634 df-ihash 10662 df-shft 10727 df-cj 10754 df-re 10755 df-im 10756 df-rsqrt 10910 df-abs 10911 df-clim 11188 df-sumdc 11263 df-ef 11557 df-cos 11560 df-rest 12449 df-topgen 12468 df-psmet 12483 df-xmet 12484 df-met 12485 df-bl 12486 df-mopn 12487 df-top 12492 df-topon 12505 df-bases 12537 df-ntr 12592 df-cn 12684 df-cnp 12685 df-tx 12749 df-cncf 13054 df-limced 13121 df-dvap 13122 |
This theorem is referenced by: cosz12 13197 ioocosf1o 13271 |
Copyright terms: Public domain | W3C validator |