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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 2731 | . . . 4 ⊢ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) | |
2 | eqid 2731 | . . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
3 | evthicc.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | evthicc.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | eqid 2731 | . . . . . 6 ⊢ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) | |
6 | 2, 5 | icccmp 24662 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Comp) |
7 | 3, 4, 6 | syl2anc 583 | . . . 4 ⊢ (𝜑 → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Comp) |
8 | evthicc.4 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) | |
9 | iccssre 13413 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
10 | 3, 4, 9 | syl2anc 583 | . . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
11 | ax-resscn 11173 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
12 | 10, 11 | sstrdi 3994 | . . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ) |
13 | eqid 2731 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) = ((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) | |
14 | eqid 2731 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
15 | eqid 2731 | . . . . . . . 8 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) | |
16 | eqid 2731 | . . . . . . . . 9 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) | |
17 | 14, 16 | tgioo 24633 | . . . . . . . 8 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) |
18 | 13, 14, 15, 17 | cncfmet 24750 | . . . . . . 7 ⊢ (((𝐴[,]𝐵) ⊆ ℂ ∧ ℝ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,)))) |
19 | 12, 11, 18 | sylancl 585 | . . . . . 6 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℝ) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,)))) |
20 | 2, 15 | resubmet 24639 | . . . . . . . 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 7427 | . . . . . 6 ⊢ (𝜑 → ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,))) = (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (topGen‘ran (,)))) |
23 | 19, 22 | eqtrd 2771 | . . . . 5 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℝ) = (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (topGen‘ran (,)))) |
24 | 8, 23 | eleqtrd 2834 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (topGen‘ran (,)))) |
25 | retop 24599 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
26 | uniretop 24600 | . . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
27 | 26 | restuni 22987 | . . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ) → (𝐴[,]𝐵) = ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))) |
28 | 25, 10, 27 | sylancr 586 | . . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) = ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))) |
29 | 3 | rexrd 11271 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
30 | 4 | rexrd 11271 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
31 | evthicc.3 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
32 | lbicc2 13448 | . . . . . . 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 4335 | . . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ≠ ∅) |
35 | 28, 34 | eqnetrrd 3008 | . . . 4 ⊢ (𝜑 → ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ≠ ∅) |
36 | 1, 2, 7, 24, 35 | evth 24806 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))∀𝑦 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))(𝐹‘𝑦) ≤ (𝐹‘𝑥)) |
37 | 28 | raleqdv 3324 | . . . 4 ⊢ (𝜑 → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥) ↔ ∀𝑦 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))(𝐹‘𝑦) ≤ (𝐹‘𝑥))) |
38 | 28, 37 | rexeqbidv 3342 | . . 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 24807 | . . 3 ⊢ (𝜑 → ∃𝑧 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))∀𝑤 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))(𝐹‘𝑧) ≤ (𝐹‘𝑤)) |
41 | 28 | raleqdv 3324 | . . . 4 ⊢ (𝜑 → (∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤) ↔ ∀𝑤 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))(𝐹‘𝑧) ≤ (𝐹‘𝑤))) |
42 | 28, 41 | rexeqbidv 3342 | . . 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 1540 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 ⊆ wss 3948 ∅c0 4322 ∪ cuni 4908 class class class wbr 5148 × cxp 5674 ran crn 5677 ↾ cres 5678 ∘ ccom 5680 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ℝcr 11115 ℝ*cxr 11254 ≤ cle 11256 − cmin 11451 (,)cioo 13331 [,]cicc 13334 abscabs 15188 ↾t crest 17373 topGenctg 17390 MetOpencmopn 21224 Topctop 22716 Cn ccn 23049 Compccmp 23211 –cn→ccncf 24717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-icc 13338 df-fz 13492 df-fzo 13635 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19229 df-cmn 19698 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-cnfld 21235 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cn 23052 df-cnp 23053 df-cmp 23212 df-tx 23387 df-hmeo 23580 df-xms 24147 df-ms 24148 df-tms 24149 df-cncf 24719 |
This theorem is referenced by: evthicc2 25310 cniccbdd 25311 rolle 25843 dvivthlem1 25862 itgsubst 25905 evthiccabs 44671 cncficcgt0 45066 |
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