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Theorem rollelem 25974

Description: Lemma for rolle 25975. (Contributed by Mario Carneiro, 1-Sep-2014.)

**Hypotheses**
Ref	Expression
rolle.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
rolle.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
rolle.lt	⊢ (𝜑 → 𝐴 < 𝐵)
rolle.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
rolle.d	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
rolle.r	⊢ (𝜑 → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))
rolle.u	⊢ (𝜑 → 𝑈 ∈ (𝐴[,]𝐵))
rolle.n	⊢ (𝜑 → ¬ 𝑈 ∈ {𝐴, 𝐵})

**Assertion**
Ref	Expression
rollelem	⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)

Distinct variable groups: 𝑥,𝑦,𝐴 𝜑,𝑥,𝑦 𝑥,𝐵,𝑦 𝑥,𝐹,𝑦 𝑥,𝑈,𝑦

**Proof of Theorem rollelem**
Step	Hyp	Ref	Expression
1		rolle.n	. . 3 ⊢ (𝜑 → ¬ 𝑈 ∈ {𝐴, 𝐵})
2		rolle.u	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ (𝐴[,]𝐵))
3		rolle.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4	3	rexrd 11186	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
5		rolle.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
6	5	rexrd 11186	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
7		rolle.lt	. . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐵)
8	3, 5, 7	ltled 11285	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
9		prunioo 13425	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
10	4, 6, 8, 9	syl3anc 1379	. . . . . 6 ⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
11	2, 10	eleqtrrd 2842	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}))
12		elun 4083	. . . . 5 ⊢ (𝑈 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝑈 ∈ (𝐴(,)𝐵) ∨ 𝑈 ∈ {𝐴, 𝐵}))
13	11, 12	sylib 219	. . . 4 ⊢ (𝜑 → (𝑈 ∈ (𝐴(,)𝐵) ∨ 𝑈 ∈ {𝐴, 𝐵}))
14	13	ord 870	. . 3 ⊢ (𝜑 → (¬ 𝑈 ∈ (𝐴(,)𝐵) → 𝑈 ∈ {𝐴, 𝐵}))
15	1, 14	mt3d 148	. 2 ⊢ (𝜑 → 𝑈 ∈ (𝐴(,)𝐵))
16		rolle.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
17		cncff 24878	. . . 4 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
18	16, 17	syl 17	. . 3 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
19		iccssre 13373	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
20	3, 5, 19	syl2anc 590	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
21		ioossicc 13377	. . . 4 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
22	21	a1i 11	. . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
23		rolle.d	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
24	15, 23	eleqtrrd 2842	. . 3 ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹))
25		rolle.r	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))
26		ssralv 3983	. . . 4 ⊢ ((𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈) → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)))
27	22, 25, 26	sylc 65	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))
28	18, 20, 15, 22, 24, 27	dvferm 25973	. 2 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0)
29		fveqeq2 6836	. . 3 ⊢ (𝑥 = 𝑈 → (((ℝ D 𝐹)‘𝑥) = 0 ↔ ((ℝ D 𝐹)‘𝑈) = 0))
30	29	rspcev 3560	. 2 ⊢ ((𝑈 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑈) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
31	15, 28, 30	syl2anc 590	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 853 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ∃wrex 3063 ∪ cun 3881 ⊆ wss 3883 {cpr 4557 class class class wbr 5072 dom cdm 5618 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ℝcr 11028 0cc0 11029 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 (,)cioo 13289 [,]cicc 13292 –cn→ccncf 24861 D cdv 25848

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 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

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 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9314 df-sup 9345 df-inf 9346 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17108 df-slot 17143 df-ndx 17155 df-base 17171 df-plusg 17224 df-mulr 17225 df-starv 17226 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-rest 17376 df-topn 17377 df-topgen 17397 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lp 23119 df-perf 23120 df-cn 23210 df-cnp 23211 df-haus 23298 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-cncf 24863 df-limc 25851 df-dv 25852

This theorem is referenced by: rolle 25975

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