![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dvge0 | Structured version Visualization version GIF version |
Description: A function on a closed interval with nonnegative derivative is weakly increasing. (Contributed by Mario Carneiro, 30-Apr-2016.) |
Ref | Expression |
---|---|
dvgt0.a | β’ (π β π΄ β β) |
dvgt0.b | β’ (π β π΅ β β) |
dvgt0.f | β’ (π β πΉ β ((π΄[,]π΅)βcnββ)) |
dvge0.d | β’ (π β (β D πΉ):(π΄(,)π΅)βΆ(0[,)+β)) |
dvge0.x | β’ (π β π β (π΄[,]π΅)) |
dvge0.y | β’ (π β π β (π΄[,]π΅)) |
dvge0.l | β’ (π β π β€ π) |
Ref | Expression |
---|---|
dvge0 | β’ (π β (πΉβπ) β€ (πΉβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvge0.x | . . . . . . . 8 β’ (π β π β (π΄[,]π΅)) | |
2 | dvge0.y | . . . . . . . 8 β’ (π β π β (π΄[,]π΅)) | |
3 | dvgt0.a | . . . . . . . . . 10 β’ (π β π΄ β β) | |
4 | dvgt0.b | . . . . . . . . . 10 β’ (π β π΅ β β) | |
5 | dvgt0.f | . . . . . . . . . 10 β’ (π β πΉ β ((π΄[,]π΅)βcnββ)) | |
6 | dvge0.d | . . . . . . . . . 10 β’ (π β (β D πΉ):(π΄(,)π΅)βΆ(0[,)+β)) | |
7 | 3, 4, 5, 6 | dvgt0lem1 25879 | . . . . . . . . 9 β’ (((π β§ (π β (π΄[,]π΅) β§ π β (π΄[,]π΅))) β§ π < π) β (((πΉβπ) β (πΉβπ)) / (π β π)) β (0[,)+β)) |
8 | 7 | exp31 419 | . . . . . . . 8 β’ (π β ((π β (π΄[,]π΅) β§ π β (π΄[,]π΅)) β (π < π β (((πΉβπ) β (πΉβπ)) / (π β π)) β (0[,)+β)))) |
9 | 1, 2, 8 | mp2and 696 | . . . . . . 7 β’ (π β (π < π β (((πΉβπ) β (πΉβπ)) / (π β π)) β (0[,)+β))) |
10 | 9 | imp 406 | . . . . . 6 β’ ((π β§ π < π) β (((πΉβπ) β (πΉβπ)) / (π β π)) β (0[,)+β)) |
11 | elrege0 13432 | . . . . . . 7 β’ ((((πΉβπ) β (πΉβπ)) / (π β π)) β (0[,)+β) β ((((πΉβπ) β (πΉβπ)) / (π β π)) β β β§ 0 β€ (((πΉβπ) β (πΉβπ)) / (π β π)))) | |
12 | 11 | simprbi 496 | . . . . . 6 β’ ((((πΉβπ) β (πΉβπ)) / (π β π)) β (0[,)+β) β 0 β€ (((πΉβπ) β (πΉβπ)) / (π β π))) |
13 | 10, 12 | syl 17 | . . . . 5 β’ ((π β§ π < π) β 0 β€ (((πΉβπ) β (πΉβπ)) / (π β π))) |
14 | cncff 24757 | . . . . . . . . . 10 β’ (πΉ β ((π΄[,]π΅)βcnββ) β πΉ:(π΄[,]π΅)βΆβ) | |
15 | 5, 14 | syl 17 | . . . . . . . . 9 β’ (π β πΉ:(π΄[,]π΅)βΆβ) |
16 | 15, 2 | ffvelcdmd 7078 | . . . . . . . 8 β’ (π β (πΉβπ) β β) |
17 | 15, 1 | ffvelcdmd 7078 | . . . . . . . 8 β’ (π β (πΉβπ) β β) |
18 | 16, 17 | resubcld 11641 | . . . . . . 7 β’ (π β ((πΉβπ) β (πΉβπ)) β β) |
19 | 18 | adantr 480 | . . . . . 6 β’ ((π β§ π < π) β ((πΉβπ) β (πΉβπ)) β β) |
20 | iccssre 13407 | . . . . . . . . . 10 β’ ((π΄ β β β§ π΅ β β) β (π΄[,]π΅) β β) | |
21 | 3, 4, 20 | syl2anc 583 | . . . . . . . . 9 β’ (π β (π΄[,]π΅) β β) |
22 | 21, 2 | sseldd 3976 | . . . . . . . 8 β’ (π β π β β) |
23 | 21, 1 | sseldd 3976 | . . . . . . . 8 β’ (π β π β β) |
24 | 22, 23 | resubcld 11641 | . . . . . . 7 β’ (π β (π β π) β β) |
25 | 24 | adantr 480 | . . . . . 6 β’ ((π β§ π < π) β (π β π) β β) |
26 | 23, 22 | posdifd 11800 | . . . . . . 7 β’ (π β (π < π β 0 < (π β π))) |
27 | 26 | biimpa 476 | . . . . . 6 β’ ((π β§ π < π) β 0 < (π β π)) |
28 | ge0div 12080 | . . . . . 6 β’ ((((πΉβπ) β (πΉβπ)) β β β§ (π β π) β β β§ 0 < (π β π)) β (0 β€ ((πΉβπ) β (πΉβπ)) β 0 β€ (((πΉβπ) β (πΉβπ)) / (π β π)))) | |
29 | 19, 25, 27, 28 | syl3anc 1368 | . . . . 5 β’ ((π β§ π < π) β (0 β€ ((πΉβπ) β (πΉβπ)) β 0 β€ (((πΉβπ) β (πΉβπ)) / (π β π)))) |
30 | 13, 29 | mpbird 257 | . . . 4 β’ ((π β§ π < π) β 0 β€ ((πΉβπ) β (πΉβπ))) |
31 | 30 | ex 412 | . . 3 β’ (π β (π < π β 0 β€ ((πΉβπ) β (πΉβπ)))) |
32 | 16, 17 | subge0d 11803 | . . 3 β’ (π β (0 β€ ((πΉβπ) β (πΉβπ)) β (πΉβπ) β€ (πΉβπ))) |
33 | 31, 32 | sylibd 238 | . 2 β’ (π β (π < π β (πΉβπ) β€ (πΉβπ))) |
34 | 16 | leidd 11779 | . . 3 β’ (π β (πΉβπ) β€ (πΉβπ)) |
35 | fveq2 6882 | . . . 4 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
36 | 35 | breq1d 5149 | . . 3 β’ (π = π β ((πΉβπ) β€ (πΉβπ) β (πΉβπ) β€ (πΉβπ))) |
37 | 34, 36 | syl5ibrcom 246 | . 2 β’ (π β (π = π β (πΉβπ) β€ (πΉβπ))) |
38 | dvge0.l | . . 3 β’ (π β π β€ π) | |
39 | 23, 22 | leloed 11356 | . . 3 β’ (π β (π β€ π β (π < π β¨ π = π))) |
40 | 38, 39 | mpbid 231 | . 2 β’ (π β (π < π β¨ π = π)) |
41 | 33, 37, 40 | mpjaod 857 | 1 β’ (π β (πΉβπ) β€ (πΉβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β¨ wo 844 = wceq 1533 β wcel 2098 β wss 3941 class class class wbr 5139 βΆwf 6530 βcfv 6534 (class class class)co 7402 βcr 11106 0cc0 11107 +βcpnf 11244 < clt 11247 β€ cle 11248 β cmin 11443 / cdiv 11870 (,)cioo 13325 [,)cico 13327 [,]cicc 13328 βcnβccncf 24740 D cdv 25736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-ioo 13329 df-ico 13331 df-icc 13332 df-fz 13486 df-fzo 13629 df-seq 13968 df-exp 14029 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-mulg 18992 df-cntz 19229 df-cmn 19698 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-fbas 21231 df-fg 21232 df-cnfld 21235 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-cld 22867 df-ntr 22868 df-cls 22869 df-nei 22946 df-lp 22984 df-perf 22985 df-cn 23075 df-cnp 23076 df-haus 23163 df-cmp 23235 df-tx 23410 df-hmeo 23603 df-fil 23694 df-fm 23786 df-flim 23787 df-flf 23788 df-xms 24170 df-ms 24171 df-tms 24172 df-cncf 24742 df-limc 25739 df-dv 25740 |
This theorem is referenced by: dvle 25884 |
Copyright terms: Public domain | W3C validator |