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Theorem rollelem 25900

Description: Lemma for rolle 25901. (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 11231	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
5		rolle.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
6	5	rexrd 11231	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
7		rolle.lt	. . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐵)
8	3, 5, 7	ltled 11329	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
9		prunioo 13449	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
10	4, 6, 8, 9	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))
11	2, 10	eleqtrrd 2832	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}))
12		elun 4119	. . . . 5 ⊢ (𝑈 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝑈 ∈ (𝐴(,)𝐵) ∨ 𝑈 ∈ {𝐴, 𝐵}))
13	11, 12	sylib 218	. . . 4 ⊢ (𝜑 → (𝑈 ∈ (𝐴(,)𝐵) ∨ 𝑈 ∈ {𝐴, 𝐵}))
14	13	ord 864	. . 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 24793	. . . 4 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
18	16, 17	syl 17	. . 3 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
19		iccssre 13397	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
20	3, 5, 19	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
21		ioossicc 13401	. . . 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 2832	. . 3 ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹))
25		rolle.r	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))
26		ssralv 4018	. . . 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 25899	. 2 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0)
29		fveqeq2 6870	. . 3 ⊢ (𝑥 = 𝑈 → (((ℝ D 𝐹)‘𝑥) = 0 ↔ ((ℝ D 𝐹)‘𝑈) = 0))
30	29	rspcev 3591	. 2 ⊢ ((𝑈 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑈) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
31	15, 28, 30	syl2anc 584	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 ∪ cun 3915 ⊆ wss 3917 {cpr 4594 class class class wbr 5110 dom cdm 5641 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ℝcr 11074 0cc0 11075 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 (,)cioo 13313 [,]cicc 13316 –cn→ccncf 24776 D cdv 25771

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9369 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ico 13319 df-icc 13320 df-fz 13476 df-seq 13974 df-exp 14034 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17124 df-slot 17159 df-ndx 17171 df-base 17187 df-plusg 17240 df-mulr 17241 df-starv 17242 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-rest 17392 df-topn 17393 df-topgen 17413 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-haus 23209 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-cncf 24778 df-limc 25774 df-dv 25775

This theorem is referenced by: rolle 25901

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