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Theorem dveq0 25973

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 24854	. . . 4 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
4	3	ffnd 6671	. 2 ⊢ (𝜑 → 𝐹 Fn (𝐴[,]𝐵))
5		fvex 6855	. . 3 ⊢ (𝐹‘𝐴) ∈ V
6		fnconstg 6730	. . 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 7160	. . . 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 11194	. . . . . 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 11194	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ^*)
17		elicc2 13339	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
18	11, 14, 17	syl2anc 585	. . . . . . . . 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 11302	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐵)
24		lbicc2 13392	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
25	13, 16, 23, 24	syl3anc 1374	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ (𝐴[,]𝐵))
26	10, 25	ffvelcdmd 7039	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝐴) ∈ ℂ)
27	3	ffvelcdmda 7038	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
28	26, 27	subcld 11504	. . . . 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 5862	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D 𝐹) = dom ((𝐴(,)𝐵) × {0}))
33		c0ex 11138	. . . . . . . . . . . 12 ⊢ 0 ∈ V
34	33	snnz 4735	. . . . . . . . . . 11 ⊢ {0} ≠ ∅
35		dmxp 5886	. . . . . . . . . . 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 2788	. . . . . . . . 9 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
38		0red 11147	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
39	31	fveq1d 6844	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑦) = (((𝐴(,)𝐵) × {0})‘𝑦))
40	33	fvconst2 7160	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → (((𝐴(,)𝐵) × {0})‘𝑦) = 0)
41	39, 40	sylan9eq 2792	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑦) = 0)
42	41	abs00bd 15226	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑦)) = 0)
43		0le0 12258	. . . . . . . . . 10 ⊢ 0 ≤ 0
44	42, 43	eqbrtrdi 5139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑦)) ≤ 0)
45	11, 14, 1, 37, 38, 44	dvlip 25966	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ≤ (0 · (abs‘(𝐴 − 𝑥))))
46	30, 45	syldan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ≤ (0 · (abs‘(𝐴 − 𝑥))))
47	12	recnd 11172	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℂ)
48	20	recnd 11172	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℂ)
49	47, 48	subcld 11504	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴 − 𝑥) ∈ ℂ)
50	49	abscld 15374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘(𝐴 − 𝑥)) ∈ ℝ)
51	50	recnd 11172	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘(𝐴 − 𝑥)) ∈ ℂ)
52	51	mul02d 11343	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (0 · (abs‘(𝐴 − 𝑥))) = 0)
53	46, 52	breqtrd 5126	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ≤ 0)
54	28	absge0d 15382	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 0 ≤ (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))))
55	28	abscld 15374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ∈ ℝ)
56		0re 11146	. . . . . . 7 ⊢ 0 ∈ ℝ
57		letri3 11230	. . . . . . 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 587	. . . . . 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 714	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) = 0)
60	28, 59	abs00d 15384	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝐴) − (𝐹‘𝑥)) = 0)
61	26, 27, 60	subeq0d 11512	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝐴) = (𝐹‘𝑥))
62	9, 61	eqtr2d 2773	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑥))
63	4, 7, 62	eqfnfvd 6988	1 ⊢ (𝜑 → 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3442 ∅c0 4287 {csn 4582 class class class wbr 5100 × cxp 5630 dom cdm 5632 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 · cmul 11043 ℝ^*cxr 11177 ≤ cle 11179 − cmin 11376 (,)cioo 13273 [,]cicc 13276 abscabs 15169 –cn→ccncf 24837 D cdv 25832

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-cmp 23343 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836

This theorem is referenced by: ftc2 26019 ftc2nc 37953

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