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Mirrors > Home > MPE Home > Th. List > rollelem | Structured version Visualization version GIF version |
Description: Lemma for rolle 26043. (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 11309 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
5 | rolle.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | 5 | rexrd 11309 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
7 | rolle.lt | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐵) | |
8 | 3, 5, 7 | ltled 11407 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
9 | prunioo 13518 | . . . . . . 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 4163 | . . . . 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 24933 | . . . 4 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
19 | iccssre 13466 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
20 | 3, 5, 19 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
21 | ioossicc 13470 | . . . 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 4064 | . . . 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 26041 | . 2 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0) |
29 | fveqeq2 6916 | . . 3 ⊢ (𝑥 = 𝑈 → (((ℝ D 𝐹)‘𝑥) = 0 ↔ ((ℝ D 𝐹)‘𝑈) = 0)) | |
30 | 29 | rspcev 3622 | . 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 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ∪ cun 3961 ⊆ wss 3963 {cpr 4633 class class class wbr 5148 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 (,)cioo 13384 [,]cicc 13387 –cn→ccncf 24916 D cdv 25913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-rest 17469 df-topn 17470 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-cncf 24918 df-limc 25916 df-dv 25917 |
This theorem is referenced by: rolle 26043 |
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