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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 25164 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞)) |
8 | 7 | exp31 420 | . . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵)) → (𝑋 < 𝑌 → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞)))) |
9 | 1, 2, 8 | mp2and 696 | . . . . . . 7 ⊢ (𝜑 → (𝑋 < 𝑌 → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞))) |
10 | 9 | imp 407 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞)) |
11 | elrege0 13185 | . . . . . . 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 497 | . . . . . 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 24054 | . . . . . . . . . 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 6959 | . . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑌) ∈ ℝ) |
17 | 15, 1 | ffvelrnd 6959 | . . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℝ) |
18 | 16, 17 | resubcld 11403 | . . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑌) − (𝐹‘𝑋)) ∈ ℝ) |
19 | 18 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝐹‘𝑌) − (𝐹‘𝑋)) ∈ ℝ) |
20 | iccssre 13160 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
21 | 3, 4, 20 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
22 | 21, 2 | sseldd 3927 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ) |
23 | 21, 1 | sseldd 3927 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ) |
24 | 22, 23 | resubcld 11403 | . . . . . . 7 ⊢ (𝜑 → (𝑌 − 𝑋) ∈ ℝ) |
25 | 24 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝑌 − 𝑋) ∈ ℝ) |
26 | 23, 22 | posdifd 11562 | . . . . . . 7 ⊢ (𝜑 → (𝑋 < 𝑌 ↔ 0 < (𝑌 − 𝑋))) |
27 | 26 | biimpa 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 < (𝑌 − 𝑋)) |
28 | ge0div 11842 | . . . . . 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 1370 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)) ↔ 0 ≤ (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)))) |
30 | 13, 29 | mpbird 256 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋))) |
31 | 30 | ex 413 | . . 3 ⊢ (𝜑 → (𝑋 < 𝑌 → 0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)))) |
32 | 16, 17 | subge0d 11565 | . . 3 ⊢ (𝜑 → (0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)) ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌))) |
33 | 31, 32 | sylibd 238 | . 2 ⊢ (𝜑 → (𝑋 < 𝑌 → (𝐹‘𝑋) ≤ (𝐹‘𝑌))) |
34 | 16 | leidd 11541 | . . 3 ⊢ (𝜑 → (𝐹‘𝑌) ≤ (𝐹‘𝑌)) |
35 | fveq2 6771 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝐹‘𝑋) = (𝐹‘𝑌)) | |
36 | 35 | breq1d 5089 | . . 3 ⊢ (𝑋 = 𝑌 → ((𝐹‘𝑋) ≤ (𝐹‘𝑌) ↔ (𝐹‘𝑌) ≤ (𝐹‘𝑌))) |
37 | 34, 36 | syl5ibrcom 246 | . 2 ⊢ (𝜑 → (𝑋 = 𝑌 → (𝐹‘𝑋) ≤ (𝐹‘𝑌))) |
38 | dvge0.l | . . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑌) | |
39 | 23, 22 | leloed 11118 | . . 3 ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))) |
40 | 38, 39 | mpbid 231 | . 2 ⊢ (𝜑 → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌)) |
41 | 33, 37, 40 | mpjaod 857 | 1 ⊢ (𝜑 → (𝐹‘𝑋) ≤ (𝐹‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 class class class wbr 5079 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 ℝcr 10871 0cc0 10872 +∞cpnf 11007 < clt 11010 ≤ cle 11011 − cmin 11205 / cdiv 11632 (,)cioo 13078 [,)cico 13080 [,]cicc 13081 –cn→ccncf 24037 D cdv 25025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-fi 9148 df-sup 9179 df-inf 9180 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-ioo 13082 df-ico 13084 df-icc 13085 df-fz 13239 df-fzo 13382 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-mulg 18699 df-cntz 18921 df-cmn 19386 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-fbas 20592 df-fg 20593 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-ntr 22169 df-cls 22170 df-nei 22247 df-lp 22285 df-perf 22286 df-cn 22376 df-cnp 22377 df-haus 22464 df-cmp 22536 df-tx 22711 df-hmeo 22904 df-fil 22995 df-fm 23087 df-flim 23088 df-flf 23089 df-xms 23471 df-ms 23472 df-tms 23473 df-cncf 24039 df-limc 25028 df-dv 25029 |
This theorem is referenced by: dvle 25169 |
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