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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 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
dvgt0.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
dvgt0.f | ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
dvge0.d | ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶(0[,)+∞)) |
dvge0.x | ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵)) |
dvge0.y | ⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵)) |
dvge0.l | ⊢ (𝜑 → 𝑋 ≤ 𝑌) |
Ref | Expression |
---|---|
dvge0 | ⊢ (𝜑 → (𝐹‘𝑋) ≤ (𝐹‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvge0.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵)) | |
2 | dvge0.y | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵)) | |
3 | dvgt0.a | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | dvgt0.b | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | dvgt0.f | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) | |
6 | dvge0.d | . . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶(0[,)+∞)) | |
7 | 3, 4, 5, 6 | dvgt0lem1 24601 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞)) |
8 | 7 | exp31 422 | . . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵)) → (𝑋 < 𝑌 → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞)))) |
9 | 1, 2, 8 | mp2and 697 | . . . . . . 7 ⊢ (𝜑 → (𝑋 < 𝑌 → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞))) |
10 | 9 | imp 409 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞)) |
11 | elrege0 12845 | . . . . . . 7 ⊢ ((((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞) ↔ ((((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ ℝ ∧ 0 ≤ (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)))) | |
12 | 11 | simprbi 499 | . . . . . 6 ⊢ ((((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞) → 0 ≤ (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋))) |
13 | 10, 12 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 ≤ (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋))) |
14 | cncff 23503 | . . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ) | |
15 | 5, 14 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
16 | 15, 2 | ffvelrnd 6854 | . . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑌) ∈ ℝ) |
17 | 15, 1 | ffvelrnd 6854 | . . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℝ) |
18 | 16, 17 | resubcld 11070 | . . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑌) − (𝐹‘𝑋)) ∈ ℝ) |
19 | 18 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝐹‘𝑌) − (𝐹‘𝑋)) ∈ ℝ) |
20 | iccssre 12821 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
21 | 3, 4, 20 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
22 | 21, 2 | sseldd 3970 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ) |
23 | 21, 1 | sseldd 3970 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ) |
24 | 22, 23 | resubcld 11070 | . . . . . . 7 ⊢ (𝜑 → (𝑌 − 𝑋) ∈ ℝ) |
25 | 24 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝑌 − 𝑋) ∈ ℝ) |
26 | 23, 22 | posdifd 11229 | . . . . . . 7 ⊢ (𝜑 → (𝑋 < 𝑌 ↔ 0 < (𝑌 − 𝑋))) |
27 | 26 | biimpa 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 < (𝑌 − 𝑋)) |
28 | ge0div 11509 | . . . . . 6 ⊢ ((((𝐹‘𝑌) − (𝐹‘𝑋)) ∈ ℝ ∧ (𝑌 − 𝑋) ∈ ℝ ∧ 0 < (𝑌 − 𝑋)) → (0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)) ↔ 0 ≤ (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)))) | |
29 | 19, 25, 27, 28 | syl3anc 1367 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)) ↔ 0 ≤ (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)))) |
30 | 13, 29 | mpbird 259 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋))) |
31 | 30 | ex 415 | . . 3 ⊢ (𝜑 → (𝑋 < 𝑌 → 0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)))) |
32 | 16, 17 | subge0d 11232 | . . 3 ⊢ (𝜑 → (0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)) ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌))) |
33 | 31, 32 | sylibd 241 | . 2 ⊢ (𝜑 → (𝑋 < 𝑌 → (𝐹‘𝑋) ≤ (𝐹‘𝑌))) |
34 | 16 | leidd 11208 | . . 3 ⊢ (𝜑 → (𝐹‘𝑌) ≤ (𝐹‘𝑌)) |
35 | fveq2 6672 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝐹‘𝑋) = (𝐹‘𝑌)) | |
36 | 35 | breq1d 5078 | . . 3 ⊢ (𝑋 = 𝑌 → ((𝐹‘𝑋) ≤ (𝐹‘𝑌) ↔ (𝐹‘𝑌) ≤ (𝐹‘𝑌))) |
37 | 34, 36 | syl5ibrcom 249 | . 2 ⊢ (𝜑 → (𝑋 = 𝑌 → (𝐹‘𝑋) ≤ (𝐹‘𝑌))) |
38 | dvge0.l | . . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑌) | |
39 | 23, 22 | leloed 10785 | . . 3 ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))) |
40 | 38, 39 | mpbid 234 | . 2 ⊢ (𝜑 → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌)) |
41 | 33, 37, 40 | mpjaod 856 | 1 ⊢ (𝜑 → (𝐹‘𝑋) ≤ (𝐹‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 class class class wbr 5068 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 +∞cpnf 10674 < clt 10677 ≤ cle 10678 − cmin 10872 / cdiv 11299 (,)cioo 12741 [,)cico 12743 [,]cicc 12744 –cn→ccncf 23486 D cdv 24463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-cmp 21997 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467 |
This theorem is referenced by: dvle 24606 |
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