dvge0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvge0		Structured version Visualization version GIF version

Theorem dvge0 25939

Description: A function on a closed interval with nonnegative derivative is weakly increasing. (Contributed by Mario Carneiro, 30-Apr-2016.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
dvge0	⊢ (𝜑 → (𝐹‘𝑋) ≤ (𝐹‘𝑌))

**Proof of Theorem dvge0**
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 25935	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞))
8	7	exp31 419	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵)) → (𝑋 < 𝑌 → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞))))
9	1, 2, 8	mp2and 699	. . . . . . 7 ⊢ (𝜑 → (𝑋 < 𝑌 → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞)))
10	9	imp 406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞))
11		elrege0 13356	. . . . . . 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 496	. . . . . 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 24814	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
15	5, 14	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
16	15, 2	ffvelcdmd 7024	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑌) ∈ ℝ)
17	15, 1	ffvelcdmd 7024	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℝ)
18	16, 17	resubcld 11552	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑌) − (𝐹‘𝑋)) ∈ ℝ)
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝐹‘𝑌) − (𝐹‘𝑋)) ∈ ℝ)
20		iccssre 13331	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
21	3, 4, 20	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
22	21, 2	sseldd 3931	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ)
23	21, 1	sseldd 3931	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
24	22, 23	resubcld 11552	. . . . . . 7 ⊢ (𝜑 → (𝑌 − 𝑋) ∈ ℝ)
25	24	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝑌 − 𝑋) ∈ ℝ)
26	23, 22	posdifd 11711	. . . . . . 7 ⊢ (𝜑 → (𝑋 < 𝑌 ↔ 0 < (𝑌 − 𝑋)))
27	26	biimpa 476	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 < (𝑌 − 𝑋))
28		ge0div 11996	. . . . . 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 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)) ↔ 0 ≤ (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋))))
30	13, 29	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)))
31	30	ex 412	. . 3 ⊢ (𝜑 → (𝑋 < 𝑌 → 0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋))))
32	16, 17	subge0d 11714	. . 3 ⊢ (𝜑 → (0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)) ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌)))
33	31, 32	sylibd 239	. 2 ⊢ (𝜑 → (𝑋 < 𝑌 → (𝐹‘𝑋) ≤ (𝐹‘𝑌)))
34	16	leidd 11690	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ≤ (𝐹‘𝑌))
35		fveq2 6828	. . . 4 ⊢ (𝑋 = 𝑌 → (𝐹‘𝑋) = (𝐹‘𝑌))
36	35	breq1d 5103	. . 3 ⊢ (𝑋 = 𝑌 → ((𝐹‘𝑋) ≤ (𝐹‘𝑌) ↔ (𝐹‘𝑌) ≤ (𝐹‘𝑌)))
37	34, 36	syl5ibrcom 247	. 2 ⊢ (𝜑 → (𝑋 = 𝑌 → (𝐹‘𝑋) ≤ (𝐹‘𝑌)))
38		dvge0.l	. . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
39	23, 22	leloed 11263	. . 3 ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋 < 𝑌 ∨ 𝑋 = 𝑌)))
40	38, 39	mpbid 232	. 2 ⊢ (𝜑 → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))
41	33, 37, 40	mpjaod 860	1 ⊢ (𝜑 → (𝐹‘𝑋) ≤ (𝐹‘𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ⊆ wss 3898 class class class wbr 5093 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 ℝcr 11012 0cc0 11013 +∞cpnf 11150 < clt 11153 ≤ cle 11154 − cmin 11351 / cdiv 11781 (,)cioo 13247 [,)cico 13249 [,]cicc 13250 –cn→ccncf 24797 D cdv 25792

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091 ax-addf 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-fi 9302 df-sup 9333 df-inf 9334 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-q 12849 df-rp 12893 df-xneg 13013 df-xadd 13014 df-xmul 13015 df-ioo 13251 df-ico 13253 df-icc 13254 df-fz 13410 df-fzo 13557 df-seq 13911 df-exp 13971 df-hash 14240 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-starv 17178 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-unif 17186 df-hom 17187 df-cco 17188 df-rest 17328 df-topn 17329 df-0g 17347 df-gsum 17348 df-topgen 17349 df-pt 17350 df-prds 17353 df-xrs 17408 df-qtop 17413 df-imas 17414 df-xps 17416 df-mre 17490 df-mrc 17491 df-acs 17493 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-mulg 18983 df-cntz 19231 df-cmn 19696 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-fg 21291 df-cnfld 21294 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-cld 22935 df-ntr 22936 df-cls 22937 df-nei 23014 df-lp 23052 df-perf 23053 df-cn 23143 df-cnp 23144 df-haus 23231 df-cmp 23303 df-tx 23478 df-hmeo 23671 df-fil 23762 df-fm 23854 df-flim 23855 df-flf 23856 df-xms 24236 df-ms 24237 df-tms 24238 df-cncf 24799 df-limc 25795 df-dv 25796

This theorem is referenced by: dvle 25940

W3C validator