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Mirrors > Home > MPE Home > Th. List > dvcn | Structured version Visualization version Unicode version |
Description: A differentiable function is continuous. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-Sep-2015.) |
Ref | Expression |
---|---|
dvcn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl2 1018 |
. . 3
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2 | eqid 2461 |
. . . . . 6
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3 | eqid 2461 |
. . . . . 6
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4 | 2, 3 | dvcnp2 22922 |
. . . . 5
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5 | 4 | ralrimiva 2813 |
. . . 4
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6 | raleq 2998 |
. . . . 5
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7 | 6 | biimpd 212 |
. . . 4
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8 | 5, 7 | mpan9 476 |
. . 3
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9 | 3 | cnfldtopon 21851 |
. . . . 5
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10 | simpl3 1019 |
. . . . . 6
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11 | simpl1 1017 |
. . . . . 6
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12 | 10, 11 | sstrd 3453 |
. . . . 5
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13 | resttopon 20225 |
. . . . 5
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14 | 9, 12, 13 | sylancr 674 |
. . . 4
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15 | cncnp 20344 |
. . . 4
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16 | 14, 9, 15 | sylancl 673 |
. . 3
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17 | 1, 8, 16 | mpbir2and 938 |
. 2
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18 | ssid 3462 |
. . 3
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19 | 9 | toponunii 19995 |
. . . . . . 7
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20 | 19 | restid 15380 |
. . . . . 6
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21 | 9, 20 | ax-mp 5 |
. . . . 5
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22 | 21 | eqcomi 2470 |
. . . 4
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23 | 3, 2, 22 | cncfcn 21989 |
. . 3
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24 | 12, 18, 23 | sylancl 673 |
. 2
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25 | 17, 24 | eleqtrrd 2542 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-rep 4528 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 ax-un 6609 ax-inf2 8171 ax-cnex 9620 ax-resscn 9621 ax-1cn 9622 ax-icn 9623 ax-addcl 9624 ax-addrcl 9625 ax-mulcl 9626 ax-mulrcl 9627 ax-mulcom 9628 ax-addass 9629 ax-mulass 9630 ax-distr 9631 ax-i2m1 9632 ax-1ne0 9633 ax-1rid 9634 ax-rnegex 9635 ax-rrecex 9636 ax-cnre 9637 ax-pre-lttri 9638 ax-pre-lttrn 9639 ax-pre-ltadd 9640 ax-pre-mulgt0 9641 ax-pre-sup 9642 ax-addf 9643 ax-mulf 9644 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-nel 2635 df-ral 2753 df-rex 2754 df-reu 2755 df-rmo 2756 df-rab 2757 df-v 3058 df-sbc 3279 df-csb 3375 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-pss 3431 df-nul 3743 df-if 3893 df-pw 3964 df-sn 3980 df-pr 3982 df-tp 3984 df-op 3986 df-uni 4212 df-int 4248 df-iun 4293 df-iin 4294 df-br 4416 df-opab 4475 df-mpt 4476 df-tr 4511 df-eprel 4763 df-id 4767 df-po 4773 df-so 4774 df-fr 4811 df-se 4812 df-we 4813 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-res 4864 df-ima 4865 df-pred 5398 df-ord 5444 df-on 5445 df-lim 5446 df-suc 5447 df-iota 5564 df-fun 5602 df-fn 5603 df-f 5604 df-f1 5605 df-fo 5606 df-f1o 5607 df-fv 5608 df-isom 5609 df-riota 6276 df-ov 6317 df-oprab 6318 df-mpt2 6319 df-of 6557 df-om 6719 df-1st 6819 df-2nd 6820 df-supp 6941 df-wrecs 7053 df-recs 7115 df-rdg 7153 df-1o 7207 df-2o 7208 df-oadd 7211 df-er 7388 df-map 7499 df-pm 7500 df-ixp 7548 df-en 7595 df-dom 7596 df-sdom 7597 df-fin 7598 df-fsupp 7909 df-fi 7950 df-sup 7981 df-inf 7982 df-oi 8050 df-card 8398 df-cda 8623 df-pnf 9702 df-mnf 9703 df-xr 9704 df-ltxr 9705 df-le 9706 df-sub 9887 df-neg 9888 df-div 10297 df-nn 10637 df-2 10695 df-3 10696 df-4 10697 df-5 10698 df-6 10699 df-7 10700 df-8 10701 df-9 10702 df-10 10703 df-n0 10898 df-z 10966 df-dec 11080 df-uz 11188 df-q 11293 df-rp 11331 df-xneg 11437 df-xadd 11438 df-xmul 11439 df-icc 11670 df-fz 11813 df-fzo 11946 df-seq 12245 df-exp 12304 df-hash 12547 df-cj 13210 df-re 13211 df-im 13212 df-sqrt 13346 df-abs 13347 df-struct 15171 df-ndx 15172 df-slot 15173 df-base 15174 df-sets 15175 df-ress 15176 df-plusg 15251 df-mulr 15252 df-starv 15253 df-sca 15254 df-vsca 15255 df-ip 15256 df-tset 15257 df-ple 15258 df-ds 15260 df-unif 15261 df-hom 15262 df-cco 15263 df-rest 15369 df-topn 15370 df-0g 15388 df-gsum 15389 df-topgen 15390 df-pt 15391 df-prds 15394 df-xrs 15448 df-qtop 15454 df-imas 15455 df-xps 15458 df-mre 15540 df-mrc 15541 df-acs 15543 df-mgm 16536 df-sgrp 16575 df-mnd 16585 df-submnd 16631 df-mulg 16724 df-cntz 17019 df-cmn 17480 df-psmet 19010 df-xmet 19011 df-met 19012 df-bl 19013 df-mopn 19014 df-cnfld 19019 df-top 19969 df-bases 19970 df-topon 19971 df-topsp 19972 df-ntr 20083 df-cn 20291 df-cnp 20292 df-tx 20625 df-hmeo 20818 df-xms 21383 df-ms 21384 df-tms 21385 df-cncf 21958 df-limc 22869 df-dv 22870 |
This theorem is referenced by: cpnord 22937 dvlipcn 22994 dvlip2 22995 dvivthlem1 23008 lhop1lem 23013 dvfsumlem2 23027 itgsubstlem 23048 taylthlem2 23377 efcn 23446 pige3 23520 relogcn 23631 atancn 23910 lhe4.4ex1a 36721 dvmulcncf 37834 dvdivcncf 37836 dvbdfbdioolem1 37837 ioodvbdlimc1lem2 37841 ioodvbdlimc1lem2OLD 37843 ioodvbdlimc2lem 37845 ioodvbdlimc2lemOLD 37846 fourierdlem94 38101 fourierdlem113 38120 fouriercn 38133 |
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