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Theorem rollelem 24588

Description: Lemma for rolle 24589. (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 10693	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
5		rolle.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
6	5	rexrd 10693	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
7		rolle.lt	. . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐵)
8	3, 5, 7	ltled 10790	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
9		prunioo 12870	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
10	4, 6, 8, 9	syl3anc 1367	. . . . . 6 ⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
11	2, 10	eleqtrrd 2918	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}))
12		elun 4127	. . . . 5 ⊢ (𝑈 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝑈 ∈ (𝐴(,)𝐵) ∨ 𝑈 ∈ {𝐴, 𝐵}))
13	11, 12	sylib 220	. . . 4 ⊢ (𝜑 → (𝑈 ∈ (𝐴(,)𝐵) ∨ 𝑈 ∈ {𝐴, 𝐵}))
14	13	ord 860	. . 3 ⊢ (𝜑 → (¬ 𝑈 ∈ (𝐴(,)𝐵) → 𝑈 ∈ {𝐴, 𝐵}))
15	1, 14	mt3d 150	. 2 ⊢ (𝜑 → 𝑈 ∈ (𝐴(,)𝐵))
16		rolle.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
17		cncff 23503	. . . 4 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
18	16, 17	syl 17	. . 3 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
19		iccssre 12821	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
20	3, 5, 19	syl2anc 586	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
21		ioossicc 12825	. . . 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 2918	. . 3 ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹))
25		rolle.r	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))
26		ssralv 4035	. . . 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 24587	. 2 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0)
29		fveqeq2 6681	. . 3 ⊢ (𝑥 = 𝑈 → (((ℝ D 𝐹)‘𝑥) = 0 ↔ ((ℝ D 𝐹)‘𝑈) = 0))
30	29	rspcev 3625	. 2 ⊢ ((𝑈 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑈) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
31	15, 28, 30	syl2anc 586	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ∪ cun 3936 ⊆ wss 3938 {cpr 4571 class class class wbr 5068 dom cdm 5557 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 ℝ^*cxr 10676 < clt 10677 ≤ cle 10678 (,)cioo 12741 [,]cicc 12744 –cn→ccncf 23486 D cdv 24463

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-rest 16698 df-topn 16699 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-cncf 23488 df-limc 24466 df-dv 24467

This theorem is referenced by: rolle 24589

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