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Mirrors > Home > MPE Home > Th. List > rollelem | Structured version Visualization version GIF version |
Description: Lemma for rolle 25154. (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 11025 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
5 | rolle.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | 5 | rexrd 11025 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
7 | rolle.lt | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐵) | |
8 | 3, 5, 7 | ltled 11123 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
9 | prunioo 13213 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) | |
10 | 4, 6, 8, 9 | syl3anc 1370 | . . . . . 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 217 | . . . 4 ⊢ (𝜑 → (𝑈 ∈ (𝐴(,)𝐵) ∨ 𝑈 ∈ {𝐴, 𝐵})) |
14 | 13 | ord 861 | . . 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 24056 | . . . 4 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
19 | iccssre 13161 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
20 | 3, 5, 19 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
21 | ioossicc 13165 | . . . 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 3987 | . . . 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 25152 | . 2 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0) |
29 | fveqeq2 6783 | . . 3 ⊢ (𝑥 = 𝑈 → (((ℝ D 𝐹)‘𝑥) = 0 ↔ ((ℝ D 𝐹)‘𝑈) = 0)) | |
30 | 29 | rspcev 3561 | . 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 844 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ∪ cun 3885 ⊆ wss 3887 {cpr 4563 class class class wbr 5074 dom cdm 5589 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 ℝ*cxr 11008 < clt 11009 ≤ cle 11010 (,)cioo 13079 [,]cicc 13082 –cn→ccncf 24039 D cdv 25027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-mulr 16976 df-starv 16977 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-rest 17133 df-topn 17134 df-topgen 17154 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-cncf 24041 df-limc 25030 df-dv 25031 |
This theorem is referenced by: rolle 25154 |
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