dveq0 - Metamath Proof Explorer

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Theorem dveq0 25959

Description: If a continuous function has zero derivative at all points on the interior of a closed interval, then it must be a constant function. (Contributed by Mario Carneiro, 2-Sep-2014.) (Proof shortened by Mario Carneiro, 3-Mar-2015.)

**Hypotheses**
Ref	Expression
dveq0.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
dveq0.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
dveq0.c	⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
dveq0.d	⊢ (𝜑 → (ℝ D 𝐹) = ((𝐴(,)𝐵) × {0}))

**Assertion**
Ref	Expression
dveq0	⊢ (𝜑 → 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}))

Proof of Theorem dveq0 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dveq0.c	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
2		cncff 24840	. . . 4 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
4	3	ffnd 6661	. 2 ⊢ (𝜑 → 𝐹 Fn (𝐴[,]𝐵))
5		fvex 6845	. . 3 ⊢ (𝐹‘𝐴) ∈ V
6		fnconstg 6720	. . 3 ⊢ ((𝐹‘𝐴) ∈ V → ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) Fn (𝐴[,]𝐵))
7	5, 6	mp1i 13	. 2 ⊢ (𝜑 → ((𝐴[,]𝐵) × {(𝐹‘𝐴)}) Fn (𝐴[,]𝐵))
8	5	fvconst2 7148	. . . 4 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑥) = (𝐹‘𝐴))
9	8	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑥) = (𝐹‘𝐴))
10	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
11		dveq0.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
12	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
13	12	rexrd 11180	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ^*)
14		dveq0.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
15	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
16	15	rexrd 11180	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ^*)
17		elicc2 13325	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
18	11, 14, 17	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
19	18	biimpa 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
20	19	simp1d 1142	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
21	19	simp2d 1143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥)
22	19	simp3d 1144	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
23	12, 20, 15, 21, 22	letrd 11288	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐵)
24		lbicc2 13378	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
25	13, 16, 23, 24	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ (𝐴[,]𝐵))
26	10, 25	ffvelcdmd 7028	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝐴) ∈ ℂ)
27	3	ffvelcdmda 7027	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
28	26, 27	subcld 11490	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝐴) − (𝐹‘𝑥)) ∈ ℂ)
29		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
30	25, 29	jca 511	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐴[,]𝐵)))
31		dveq0.d	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D 𝐹) = ((𝐴(,)𝐵) × {0}))
32	31	dmeqd 5852	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D 𝐹) = dom ((𝐴(,)𝐵) × {0}))
33		c0ex 11124	. . . . . . . . . . . 12 ⊢ 0 ∈ V
34	33	snnz 4731	. . . . . . . . . . 11 ⊢ {0} ≠ ∅
35		dmxp 5876	. . . . . . . . . . 11 ⊢ ({0} ≠ ∅ → dom ((𝐴(,)𝐵) × {0}) = (𝐴(,)𝐵))
36	34, 35	ax-mp 5	. . . . . . . . . 10 ⊢ dom ((𝐴(,)𝐵) × {0}) = (𝐴(,)𝐵)
37	32, 36	eqtrdi 2785	. . . . . . . . 9 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
38		0red 11133	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
39	31	fveq1d 6834	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑦) = (((𝐴(,)𝐵) × {0})‘𝑦))
40	33	fvconst2 7148	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → (((𝐴(,)𝐵) × {0})‘𝑦) = 0)
41	39, 40	sylan9eq 2789	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑦) = 0)
42	41	abs00bd 15212	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑦)) = 0)
43		0le0 12244	. . . . . . . . . 10 ⊢ 0 ≤ 0
44	42, 43	eqbrtrdi 5135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑦)) ≤ 0)
45	11, 14, 1, 37, 38, 44	dvlip 25952	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ≤ (0 · (abs‘(𝐴 − 𝑥))))
46	30, 45	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ≤ (0 · (abs‘(𝐴 − 𝑥))))
47	12	recnd 11158	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℂ)
48	20	recnd 11158	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℂ)
49	47, 48	subcld 11490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴 − 𝑥) ∈ ℂ)
50	49	abscld 15360	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘(𝐴 − 𝑥)) ∈ ℝ)
51	50	recnd 11158	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘(𝐴 − 𝑥)) ∈ ℂ)
52	51	mul02d 11329	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (0 · (abs‘(𝐴 − 𝑥))) = 0)
53	46, 52	breqtrd 5122	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ≤ 0)
54	28	absge0d 15368	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 0 ≤ (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))))
55	28	abscld 15360	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ∈ ℝ)
56		0re 11132	. . . . . . 7 ⊢ 0 ∈ ℝ
57		letri3 11216	. . . . . . 7 ⊢ (((abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ∈ ℝ ∧ 0 ∈ ℝ) → ((abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) = 0 ↔ ((abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ≤ 0 ∧ 0 ≤ (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))))))
58	55, 56, 57	sylancl 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) = 0 ↔ ((abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ≤ 0 ∧ 0 ≤ (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))))))
59	53, 54, 58	mpbir2and 713	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) = 0)
60	28, 59	abs00d 15370	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝐴) − (𝐹‘𝑥)) = 0)
61	26, 27, 60	subeq0d 11498	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝐴) = (𝐹‘𝑥))
62	9, 61	eqtr2d 2770	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑥))
63	4, 7, 62	eqfnfvd 6977	1 ⊢ (𝜑 → 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 Vcvv 3438 ∅c0 4283 {csn 4578 class class class wbr 5096 × cxp 5620 dom cdm 5622 Fn wfn 6485 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 ℝcr 11023 0cc0 11024 · cmul 11029 ℝ^*cxr 11163 ≤ cle 11165 − cmin 11362 (,)cioo 13259 [,]cicc 13262 abscabs 15155 –cn→ccncf 24823 D cdv 25818

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 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103

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 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-mulg 18996 df-cntz 19244 df-cmn 19709 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cn 23169 df-cnp 23170 df-haus 23257 df-cmp 23329 df-tx 23504 df-hmeo 23697 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-xms 24262 df-ms 24263 df-tms 24264 df-cncf 24825 df-limc 25821 df-dv 25822

This theorem is referenced by: ftc2 26005 ftc2nc 37842

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