dvgt0 - Metamath Proof Explorer

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Theorem dvgt0 25990

Description: A function on a closed interval with positive derivative is increasing. (Contributed by Mario Carneiro, 19-Feb-2015.)

**Hypotheses**
Ref	Expression
dvgt0.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
dvgt0.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
dvgt0.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
dvgt0.d	⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ⁺)

**Assertion**
Ref	Expression
dvgt0	⊢ (𝜑 → 𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹))

Proof of Theorem dvgt0 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvgt0.a	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		dvgt0.b	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		dvgt0.f	. 2 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
4		dvgt0.d	. 2 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ⁺)
5		ltso 11218	. 2 ⊢ < Or ℝ
6	1, 2, 3, 4	dvgt0lem1 25988	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ ℝ⁺)
7	6	rpgt0d 12981	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 0 < (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)))
8		cncff 24879	. . . . . . . . 9 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
9	3, 8	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
10	9	ad2antrr 732	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
11		simplrr 783	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (𝐴[,]𝐵))
12	10, 11	ffvelcdmd 7027	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑦) ∈ ℝ)
13		simplrl 782	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ (𝐴[,]𝐵))
14	10, 13	ffvelcdmd 7027	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑥) ∈ ℝ)
15	12, 14	resubcld 11570	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹‘𝑦) − (𝐹‘𝑥)) ∈ ℝ)
16		iccssre 13374	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
17	1, 2, 16	syl2anc 590	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
18	17	ad2antrr 732	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐴[,]𝐵) ⊆ ℝ)
19	18, 11	sseldd 3916	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ)
20	18, 13	sseldd 3916	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ)
21	19, 20	resubcld 11570	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℝ)
22		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
23	20, 19	posdifd 11729	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥 < 𝑦 ↔ 0 < (𝑦 − 𝑥)))
24	22, 23	mpbid 233	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 0 < (𝑦 − 𝑥))
25		gt0div 12014	. . . . 5 ⊢ ((((𝐹‘𝑦) − (𝐹‘𝑥)) ∈ ℝ ∧ (𝑦 − 𝑥) ∈ ℝ ∧ 0 < (𝑦 − 𝑥)) → (0 < ((𝐹‘𝑦) − (𝐹‘𝑥)) ↔ 0 < (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))))
26	15, 21, 24, 25	syl3anc 1379	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (0 < ((𝐹‘𝑦) − (𝐹‘𝑥)) ↔ 0 < (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥))))
27	7, 26	mpbird 258	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 0 < ((𝐹‘𝑦) − (𝐹‘𝑥)))
28	14, 12	posdifd 11729	. . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹‘𝑥) < (𝐹‘𝑦) ↔ 0 < ((𝐹‘𝑦) − (𝐹‘𝑥))))
29	27, 28	mpbird 258	. 2 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑥) < (𝐹‘𝑦))
30	1, 2, 3, 4, 5, 29	dvgt0lem2 25989	1 ⊢ (𝜑 → 𝐹 Isom < , < ((𝐴[,]𝐵), ran 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2119 ⊆ wss 3883 class class class wbr 5073 ran crn 5620 ⟶wf 6482 ‘cfv 6486 Isom wiso 6487 (class class class)co 7357 ℝcr 11029 0cc0 11030 < clt 11171 − cmin 11369 / cdiv 11799 ℝ⁺crp 12934 (,)cioo 13290 [,]cicc 13293 –cn→ccncf 24862 D cdv 25849

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 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 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 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-om 7808 df-1st 7932 df-2nd 7933 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9855 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-z 12517 df-dec 12637 df-uz 12781 df-q 12891 df-rp 12935 df-xneg 13055 df-xadd 13056 df-xmul 13057 df-ioo 13294 df-ico 13296 df-icc 13297 df-fz 13454 df-fzo 13601 df-seq 13956 df-exp 14016 df-hash 14285 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-starv 17227 df-sca 17228 df-vsca 17229 df-ip 17230 df-tset 17231 df-ple 17232 df-ds 17234 df-unif 17235 df-hom 17236 df-cco 17237 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17458 df-qtop 17463 df-imas 17464 df-xps 17466 df-mre 17540 df-mrc 17541 df-acs 17543 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18744 df-mulg 19036 df-cntz 19284 df-cmn 19749 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-fbas 21345 df-fg 21346 df-cnfld 21349 df-top 22878 df-topon 22895 df-topsp 22917 df-bases 22930 df-cld 23003 df-ntr 23004 df-cls 23005 df-nei 23082 df-lp 23120 df-perf 23121 df-cn 23211 df-cnp 23212 df-haus 23299 df-cmp 23371 df-tx 23546 df-hmeo 23739 df-fil 23830 df-fm 23922 df-flim 23923 df-flf 23924 df-xms 24304 df-ms 24305 df-tms 24306 df-cncf 24864 df-limc 25852 df-dv 25853

This theorem is referenced by: dvne0 25997

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