dveq0 - Metamath Proof Explorer

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Theorem dveq0 26059

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 24952	. . . 4 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
4	3	ffnd 6692	. 2 ⊢ (𝜑 → 𝐹 Fn (𝐴[,]𝐵))
5		fvex 6880	. . 3 ⊢ (𝐹‘𝐴) ∈ V
6		fnconstg 6752	. . 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 7188	. . . 4 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑥) = (𝐹‘𝐴))
9	8	adantl 485	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑥) = (𝐹‘𝐴))
10	3	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
11		dveq0.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
12	11	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
13	12	rexrd 11232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ^*)
14		dveq0.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
15	14	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
16	15	rexrd 11232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ^*)
17		elicc2 13415	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
18	11, 14, 17	syl2anc 593	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
19	18	biimpa 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
20	19	simp1d 1155	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ)
21	19	simp2d 1156	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥)
22	19	simp3d 1157	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
23	12, 20, 15, 21, 22	letrd 11340	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐵)
24		lbicc2 13468	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
25	13, 16, 23, 24	syl3anc 1390	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ (𝐴[,]𝐵))
26	10, 25	ffvelcdmd 7066	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝐴) ∈ ℂ)
27	3	ffvelcdmda 7065	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
28	26, 27	subcld 11542	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝐴) − (𝐹‘𝑥)) ∈ ℂ)
29		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
30	25, 29	jca 519	. . . . . . . 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 5881	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D 𝐹) = dom ((𝐴(,)𝐵) × {0}))
33		c0ex 11173	. . . . . . . . . . . 12 ⊢ 0 ∈ V
34	33	snnz 4735	. . . . . . . . . . 11 ⊢ {0} ≠ ∅
35		dmxp 5905	. . . . . . . . . . 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 2813	. . . . . . . . 9 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
38		0red 11184	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
39	31	fveq1d 6869	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑦) = (((𝐴(,)𝐵) × {0})‘𝑦))
40	33	fvconst2 7188	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → (((𝐴(,)𝐵) × {0})‘𝑦) = 0)
41	39, 40	sylan9eq 2817	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑦) = 0)
42	41	abs00bd 15318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑦)) = 0)
43		0le0 12319	. . . . . . . . . 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 26052	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ≤ (0 · (abs‘(𝐴 − 𝑥))))
46	30, 45	syldan 600	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ≤ (0 · (abs‘(𝐴 − 𝑥))))
47	12	recnd 11210	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℂ)
48	20	recnd 11210	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℂ)
49	47, 48	subcld 11542	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴 − 𝑥) ∈ ℂ)
50	49	abscld 15466	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘(𝐴 − 𝑥)) ∈ ℝ)
51	50	recnd 11210	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘(𝐴 − 𝑥)) ∈ ℂ)
52	51	mul02d 11381	. . . . . . 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 15474	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 0 ≤ (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))))
55	28	abscld 15466	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) ∈ ℝ)
56		0re 11183	. . . . . . 7 ⊢ 0 ∈ ℝ
57		letri3 11268	. . . . . . 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 595	. . . . . 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 723	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (abs‘((𝐹‘𝐴) − (𝐹‘𝑥))) = 0)
60	28, 59	abs00d 15476	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((𝐹‘𝐴) − (𝐹‘𝑥)) = 0)
61	26, 27, 60	subeq0d 11550	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝐴) = (𝐹‘𝑥))
62	9, 61	eqtr2d 2798	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) = (((𝐴[,]𝐵) × {(𝐹‘𝐴)})‘𝑥))
63	4, 7, 62	eqfnfvd 7014	1 ⊢ (𝜑 → 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 Vcvv 3454 ∅c0 4285 {csn 4582 class class class wbr 5100 × cxp 5645 dom cdm 5647 Fn wfn 6516 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ℂcc 11071 ℝcr 11072 0cc0 11073 · cmul 11078 ℝ^*cxr 11215 ≤ cle 11217 − cmin 11414 (,)cioo 13349 [,]cicc 13352 abscabs 15261 –cn→ccncf 24935 D cdv 25922

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-ico 13355 df-icc 13356 df-fz 13513 df-fzo 13660 df-seq 14015 df-exp 14075 df-hash 14344 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-pt 17473 df-prds 17476 df-xrs 17532 df-qtop 17537 df-imas 17538 df-xps 17540 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-mulg 19110 df-cntz 19357 df-cmn 19822 df-psmet 21413 df-xmet 21414 df-met 21415 df-bl 21416 df-mopn 21417 df-fbas 21418 df-fg 21419 df-cnfld 21422 df-top 22951 df-topon 22968 df-topsp 22990 df-bases 23003 df-cld 23076 df-ntr 23077 df-cls 23078 df-nei 23155 df-lp 23193 df-perf 23194 df-cn 23284 df-cnp 23285 df-haus 23372 df-cmp 23444 df-tx 23619 df-hmeo 23812 df-fil 23903 df-fm 23995 df-flim 23996 df-flf 23997 df-xms 24377 df-ms 24378 df-tms 24379 df-cncf 24937 df-limc 25925 df-dv 25926

This theorem is referenced by: ftc2 26103 ftc2nc 38198

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