dvge0 - Metamath Proof Explorer

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Theorem dvge0 26332

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 26328	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) ∧ 𝑋 < 𝑌) → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞))
8	7	exp31 425	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵)) → (𝑋 < 𝑌 → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞))))
9	1, 2, 8	mp2and 712	. . . . . . 7 ⊢ (𝜑 → (𝑋 < 𝑌 → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞)))
10	9	imp 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞))
11		elrege0 13589	. . . . . . 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 503	. . . . . 6 ⊢ ((((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)) ∈ (0[,)+∞) → 0 ≤ (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)))
13	10, 12	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 ≤ (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋)))
14		cncff 25220	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
15	5, 14	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
16	15, 2	ffvelcdmd 7086	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑌) ∈ ℝ)
17	15, 1	ffvelcdmd 7086	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℝ)
18	16, 17	resubcld 11748	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑌) − (𝐹‘𝑋)) ∈ ℝ)
19	18	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → ((𝐹‘𝑌) − (𝐹‘𝑋)) ∈ ℝ)
20		iccssre 13564	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
21	3, 4, 20	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
22	21, 2	sseldd 3909	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℝ)
23	21, 1	sseldd 3909	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
24	22, 23	resubcld 11748	. . . . . . 7 ⊢ (𝜑 → (𝑌 − 𝑋) ∈ ℝ)
25	24	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (𝑌 − 𝑋) ∈ ℝ)
26	23, 22	posdifd 11907	. . . . . . 7 ⊢ (𝜑 → (𝑋 < 𝑌 ↔ 0 < (𝑌 − 𝑋)))
27	26	biimpa 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 < (𝑌 − 𝑋))
28		ge0div 12188	. . . . . 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 1398	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)) ↔ 0 ≤ (((𝐹‘𝑌) − (𝐹‘𝑋)) / (𝑌 − 𝑋))))
30	13, 29	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → 0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)))
31	30	ex 418	. . 3 ⊢ (𝜑 → (𝑋 < 𝑌 → 0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋))))
32	16, 17	subge0d 11910	. . 3 ⊢ (𝜑 → (0 ≤ ((𝐹‘𝑌) − (𝐹‘𝑋)) ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌)))
33	31, 32	sylibd 242	. 2 ⊢ (𝜑 → (𝑋 < 𝑌 → (𝐹‘𝑋) ≤ (𝐹‘𝑌)))
34	16	leidd 11886	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ≤ (𝐹‘𝑌))
35		fveq2 6886	. . . 4 ⊢ (𝑋 = 𝑌 → (𝐹‘𝑋) = (𝐹‘𝑌))
36	35	breq1d 5112	. . 3 ⊢ (𝑋 = 𝑌 → ((𝐹‘𝑋) ≤ (𝐹‘𝑌) ↔ (𝐹‘𝑌) ≤ (𝐹‘𝑌)))
37	34, 36	syl5ibrcom 250	. 2 ⊢ (𝜑 → (𝑋 = 𝑌 → (𝐹‘𝑋) ≤ (𝐹‘𝑌)))
38		dvge0.l	. . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
39	23, 22	leloed 11457	. . 3 ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋 < 𝑌 ∨ 𝑋 = 𝑌)))
40	38, 39	mpbid 235	. 2 ⊢ (𝜑 → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))
41	33, 37, 40	mpjaod 874	1 ⊢ (𝜑 → (𝐹‘𝑋) ≤ (𝐹‘𝑌))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ⊆ wss 3895 class class class wbr 5102 ⟶wf 6535 ‘cfv 6539 (class class class)co 7421 ℝcr 11203 0cc0 11204 +∞cpnf 11344 < clt 11347 ≤ cle 11348 − cmin 11545 / cdiv 11977 (,)cioo 13480 [,)cico 13482 [,]cicc 13483 –cn→ccncf 25203 D cdv 26189

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282 ax-addf 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-map 8853 df-pm 8854 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-fi 9407 df-sup 9438 df-inf 9439 df-oi 9508 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-q 13080 df-rp 13125 df-xneg 13245 df-xadd 13246 df-xmul 13247 df-ioo 13484 df-ico 13486 df-icc 13487 df-fz 13644 df-fzo 13793 df-seq 14149 df-exp 14209 df-hash 14479 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-starv 17449 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-unif 17457 df-hom 17458 df-cco 17459 df-rest 17599 df-topn 17600 df-0g 17618 df-gsum 17619 df-topgen 17620 df-pt 17621 df-prds 17624 df-xrs 17680 df-qtop 17685 df-imas 17686 df-xps 17688 df-mre 17762 df-mrc 17763 df-acs 17765 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-mulg 19284 df-cntz 19537 df-cmn 20002 df-psmet 21676 df-xmet 21677 df-met 21678 df-bl 21679 df-mopn 21680 df-fbas 21681 df-fg 21682 df-cnfld 21685 df-top 23218 df-topon 23235 df-topsp 23257 df-bases 23270 df-cld 23343 df-ntr 23344 df-cls 23345 df-nei 23422 df-lp 23460 df-perf 23461 df-cn 23551 df-cnp 23552 df-haus 23639 df-cmp 23711 df-tx 23887 df-hmeo 24080 df-fil 24171 df-fm 24263 df-flim 24264 df-flf 24265 df-xms 24645 df-ms 24646 df-tms 24647 df-cncf 25205 df-limc 26192 df-dv 26193

This theorem is used by: dvle 26333

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