dvge0 - Metamath Proof Explorer

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Theorem dvge0 23573

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 23569	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞))
8	7	exp31 628	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵)) → (𝑋 < 𝑌 → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞))))
9	1, 2, 8	mp2and 711	. . . . . . 7 ⊢ (𝜑 → (𝑋 < 𝑌 → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞)))
10	9	imp 444	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞))
11		elrege0 12149	. . . . . . 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 479	. . . . . 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 22504	. . . . . . . . . 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 6268	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑌) ∈ ℝ)
17	15, 1	ffvelrnd 6268	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℝ)
18	16, 17	resubcld 10337	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑌) − (𝐹‘𝑋)) ∈ ℝ)
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝐹‘𝑌) − (𝐹‘𝑋)) ∈ ℝ)
20		iccssre 12126	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
21	3, 4, 20	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
22	21, 2	sseldd 3569	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ)
23	21, 1	sseldd 3569	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
24	22, 23	resubcld 10337	. . . . . . 7 ⊢ (𝜑 → (𝑌 − 𝑋) ∈ ℝ)
25	24	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝑌 − 𝑋) ∈ ℝ)
26	23, 22	posdifd 10493	. . . . . . 7 ⊢ (𝜑 → (𝑋 < 𝑌 ↔ 0 < (𝑌 − 𝑋)))
27	26	biimpa 500	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 < (𝑌 − 𝑋))
28		ge0div 10769	. . . . . 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 1318	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)) ↔ 0 ≤ (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋))))
30	13, 29	mpbird 246	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)))
31	30	ex 449	. . 3 ⊢ (𝜑 → (𝑋 < 𝑌 → 0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋))))
32	16, 17	subge0d 10496	. . 3 ⊢ (𝜑 → (0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)) ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌)))
33	31, 32	sylibd 228	. 2 ⊢ (𝜑 → (𝑋 < 𝑌 → (𝐹‘𝑋) ≤ (𝐹‘𝑌)))
34	16	leidd 10473	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ≤ (𝐹‘𝑌))
35		fveq2 6103	. . . 4 ⊢ (𝑋 = 𝑌 → (𝐹‘𝑋) = (𝐹‘𝑌))
36	35	breq1d 4593	. . 3 ⊢ (𝑋 = 𝑌 → ((𝐹‘𝑋) ≤ (𝐹‘𝑌) ↔ (𝐹‘𝑌) ≤ (𝐹‘𝑌)))
37	34, 36	syl5ibrcom 236	. 2 ⊢ (𝜑 → (𝑋 = 𝑌 → (𝐹‘𝑋) ≤ (𝐹‘𝑌)))
38		dvge0.l	. . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
39	23, 22	leloed 10059	. . 3 ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋 < 𝑌 ∨ 𝑋 = 𝑌)))
40	38, 39	mpbid 221	. 2 ⊢ (𝜑 → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))
41	33, 37, 40	mpjaod 395	1 ⊢ (𝜑 → (𝐹‘𝑋) ≤ (𝐹‘𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 +∞cpnf 9950 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 (,)cioo 12046 [,)cico 12048 [,]cicc 12049 –cn→ccncf 22487 D cdv 23433

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437

This theorem is referenced by: dvle 23574

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