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Mirrors > Home > MPE Home > Th. List > rollelem | Structured version Visualization version GIF version |
Description: Lemma for rolle 24734. (Contributed by Mario Carneiro, 1-Sep-2014.) |
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}) |
Ref | Expression |
---|---|
rollelem | ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) |
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 10762 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
5 | rolle.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | 5 | rexrd 10762 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
7 | rolle.lt | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐵) | |
8 | 3, 5, 7 | ltled 10859 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
9 | prunioo 12948 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) | |
10 | 4, 6, 8, 9 | syl3anc 1372 | . . . . . 6 ⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) |
11 | 2, 10 | eleqtrrd 2836 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵})) |
12 | elun 4037 | . . . . 5 ⊢ (𝑈 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝑈 ∈ (𝐴(,)𝐵) ∨ 𝑈 ∈ {𝐴, 𝐵})) | |
13 | 11, 12 | sylib 221 | . . . 4 ⊢ (𝜑 → (𝑈 ∈ (𝐴(,)𝐵) ∨ 𝑈 ∈ {𝐴, 𝐵})) |
14 | 13 | ord 863 | . . 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 23638 | . . . 4 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
19 | iccssre 12896 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
20 | 3, 5, 19 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
21 | ioossicc 12900 | . . . 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 2836 | . . 3 ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹)) |
25 | rolle.r | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) | |
26 | ssralv 3941 | . . . 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 24732 | . 2 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0) |
29 | fveqeq2 6677 | . . 3 ⊢ (𝑥 = 𝑈 → (((ℝ D 𝐹)‘𝑥) = 0 ↔ ((ℝ D 𝐹)‘𝑈) = 0)) | |
30 | 29 | rspcev 3524 | . 2 ⊢ ((𝑈 ∈ (𝐴(,)𝐵) ∧ ((ℝ D 𝐹)‘𝑈) = 0) → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) |
31 | 15, 28, 30 | syl2anc 587 | 1 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 846 = wceq 1542 ∈ wcel 2113 ∀wral 3053 ∃wrex 3054 ∪ cun 3839 ⊆ wss 3841 {cpr 4515 class class class wbr 5027 dom cdm 5519 ⟶wf 6329 ‘cfv 6333 (class class class)co 7164 ℝcr 10607 0cc0 10608 ℝ*cxr 10745 < clt 10746 ≤ cle 10747 (,)cioo 12814 [,]cicc 12817 –cn→ccncf 23621 D cdv 24607 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-pm 8433 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fi 8941 df-sup 8972 df-inf 8973 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-q 12424 df-rp 12466 df-xneg 12583 df-xadd 12584 df-xmul 12585 df-ioo 12818 df-ico 12820 df-icc 12821 df-fz 12975 df-seq 13454 df-exp 13515 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-plusg 16674 df-mulr 16675 df-starv 16676 df-tset 16680 df-ple 16681 df-ds 16683 df-unif 16684 df-rest 16792 df-topn 16793 df-topgen 16813 df-psmet 20202 df-xmet 20203 df-met 20204 df-bl 20205 df-mopn 20206 df-fbas 20207 df-fg 20208 df-cnfld 20211 df-top 21638 df-topon 21655 df-topsp 21677 df-bases 21690 df-cld 21763 df-ntr 21764 df-cls 21765 df-nei 21842 df-lp 21880 df-perf 21881 df-cn 21971 df-cnp 21972 df-haus 22059 df-fil 22590 df-fm 22682 df-flim 22683 df-flf 22684 df-xms 23066 df-ms 23067 df-cncf 23623 df-limc 24610 df-dv 24611 |
This theorem is referenced by: rolle 24734 |
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