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Mirrors > Home > MPE Home > Th. List > evthicc | Structured version Visualization version GIF version |
Description: Specialization of the Extreme Value Theorem to a closed interval of ℝ. (Contributed by Mario Carneiro, 12-Aug-2014.) |
Ref | Expression |
---|---|
evthicc.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
evthicc.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
evthicc.3 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
evthicc.4 | ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
Ref | Expression |
---|---|
evthicc | ⊢ (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . . 4 ⊢ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) | |
2 | eqid 2734 | . . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
3 | evthicc.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | evthicc.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | eqid 2734 | . . . . . 6 ⊢ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) | |
6 | 2, 5 | icccmp 24860 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Comp) |
7 | 3, 4, 6 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Comp) |
8 | evthicc.4 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) | |
9 | iccssre 13465 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
10 | 3, 4, 9 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
11 | ax-resscn 11209 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
12 | 10, 11 | sstrdi 4007 | . . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ) |
13 | eqid 2734 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) = ((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) | |
14 | eqid 2734 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
15 | eqid 2734 | . . . . . . . 8 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) | |
16 | eqid 2734 | . . . . . . . . 9 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) | |
17 | 14, 16 | tgioo 24831 | . . . . . . . 8 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) |
18 | 13, 14, 15, 17 | cncfmet 24948 | . . . . . . 7 ⊢ (((𝐴[,]𝐵) ⊆ ℂ ∧ ℝ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,)))) |
19 | 12, 11, 18 | sylancl 586 | . . . . . 6 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℝ) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,)))) |
20 | 2, 15 | resubmet 24837 | . . . . . . . 8 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) = ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))) |
21 | 10, 20 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) = ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))) |
22 | 21 | oveq1d 7445 | . . . . . 6 ⊢ (𝜑 → ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,))) = (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (topGen‘ran (,)))) |
23 | 19, 22 | eqtrd 2774 | . . . . 5 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℝ) = (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (topGen‘ran (,)))) |
24 | 8, 23 | eleqtrd 2840 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (topGen‘ran (,)))) |
25 | retop 24797 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
26 | uniretop 24798 | . . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
27 | 26 | restuni 23185 | . . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ) → (𝐴[,]𝐵) = ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))) |
28 | 25, 10, 27 | sylancr 587 | . . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) = ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))) |
29 | 3 | rexrd 11308 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
30 | 4 | rexrd 11308 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
31 | evthicc.3 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
32 | lbicc2 13500 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
33 | 29, 30, 31, 32 | syl3anc 1370 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
34 | 33 | ne0d 4347 | . . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ≠ ∅) |
35 | 28, 34 | eqnetrrd 3006 | . . . 4 ⊢ (𝜑 → ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ≠ ∅) |
36 | 1, 2, 7, 24, 35 | evth 25004 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))∀𝑦 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))(𝐹‘𝑦) ≤ (𝐹‘𝑥)) |
37 | 28 | raleqdv 3323 | . . . 4 ⊢ (𝜑 → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥) ↔ ∀𝑦 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))(𝐹‘𝑦) ≤ (𝐹‘𝑥))) |
38 | 28, 37 | rexeqbidv 3344 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥) ↔ ∃𝑥 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))∀𝑦 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))(𝐹‘𝑦) ≤ (𝐹‘𝑥))) |
39 | 36, 38 | mpbird 257 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥)) |
40 | 1, 2, 7, 24, 35 | evth2 25005 | . . 3 ⊢ (𝜑 → ∃𝑧 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))∀𝑤 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))(𝐹‘𝑧) ≤ (𝐹‘𝑤)) |
41 | 28 | raleqdv 3323 | . . . 4 ⊢ (𝜑 → (∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤) ↔ ∀𝑤 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))(𝐹‘𝑧) ≤ (𝐹‘𝑤))) |
42 | 28, 41 | rexeqbidv 3344 | . . 3 ⊢ (𝜑 → (∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤) ↔ ∃𝑧 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))∀𝑤 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))(𝐹‘𝑧) ≤ (𝐹‘𝑤))) |
43 | 40, 42 | mpbird 257 | . 2 ⊢ (𝜑 → ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤)) |
44 | 39, 43 | jca 511 | 1 ⊢ (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 ⊆ wss 3962 ∅c0 4338 ∪ cuni 4911 class class class wbr 5147 × cxp 5686 ran crn 5689 ↾ cres 5690 ∘ ccom 5692 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 ℝ*cxr 11291 ≤ cle 11293 − cmin 11489 (,)cioo 13383 [,]cicc 13386 abscabs 15269 ↾t crest 17466 topGenctg 17483 MetOpencmopn 21371 Topctop 22914 Cn ccn 23247 Compccmp 23409 –cn→ccncf 24915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cn 23250 df-cnp 23251 df-cmp 23410 df-tx 23585 df-hmeo 23778 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 |
This theorem is referenced by: evthicc2 25508 cniccbdd 25509 rolle 26042 dvivthlem1 26061 itgsubst 26104 evthiccabs 45448 cncficcgt0 45843 |
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