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Theorem dveq0 26039

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 24919	. . . 4 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
4	3	ffnd 6737	. 2 ⊢ (𝜑 → 𝐹 Fn (𝐴[,]𝐵))
5		fvex 6919	. . 3 ⊢ (𝐹‘𝐴) ∈ V
6		fnconstg 6796	. . 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 7224	. . . 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 11311	. . . . . 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 11311	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ^*)
17		elicc2 13452	. . . . . . . . . 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 1143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
21	19	simp2d 1144	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥)
22	19	simp3d 1145	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
23	12, 20, 15, 21, 22	letrd 11418	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐵)
24		lbicc2 13504	. . . . . 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 7105	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝐴) ∈ ℂ)
27	3	ffvelcdmda 7104	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
28	26, 27	subcld 11620	. . . . 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 5916	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D 𝐹) = dom ((𝐴(,)𝐵) × {0}))
33		c0ex 11255	. . . . . . . . . . . 12 ⊢ 0 ∈ V
34	33	snnz 4776	. . . . . . . . . . 11 ⊢ {0} ≠ ∅
35		dmxp 5939	. . . . . . . . . . 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 2793	. . . . . . . . 9 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
38		0red 11264	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
39	31	fveq1d 6908	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑦) = (((𝐴(,)𝐵) × {0})‘𝑦))
40	33	fvconst2 7224	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → (((𝐴(,)𝐵) × {0})‘𝑦) = 0)
41	39, 40	sylan9eq 2797	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑦) = 0)
42	41	abs00bd 15330	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑦)) = 0)
43		0le0 12367	. . . . . . . . . 10 ⊢ 0 ≤ 0
44	42, 43	eqbrtrdi 5182	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑦)) ≤ 0)
45	11, 14, 1, 37, 38, 44	dvlip 26032	. . . . . . . 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 11289	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℂ)
48	20	recnd 11289	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℂ)
49	47, 48	subcld 11620	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴 − 𝑥) ∈ ℂ)
50	49	abscld 15475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘(𝐴 − 𝑥)) ∈ ℝ)
51	50	recnd 11289	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘(𝐴 − 𝑥)) ∈ ℂ)
52	51	mul02d 11459	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (0 · (abs‘(𝐴 − 𝑥))) = 0)
53	46, 52	breqtrd 5169	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ≤ 0)
54	28	absge0d 15483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 0 ≤ (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))))
55	28	abscld 15475	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ∈ ℝ)
56		0re 11263	. . . . . . 7 ⊢ 0 ∈ ℝ
57		letri3 11346	. . . . . . 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 15485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝐴) − (𝐹‘𝑥)) = 0)
61	26, 27, 60	subeq0d 11628	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝐴) = (𝐹‘𝑥))
62	9, 61	eqtr2d 2778	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑥))
63	4, 7, 62	eqfnfvd 7054	1 ⊢ (𝜑 → 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 {csn 4626 class class class wbr 5143 × cxp 5683 dom cdm 5685 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 · cmul 11160 ℝ^*cxr 11294 ≤ cle 11296 − cmin 11492 (,)cioo 13387 [,]cicc 13390 abscabs 15273 –cn→ccncf 24902 D cdv 25898

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902

This theorem is referenced by: ftc2 26085 ftc2nc 37709

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