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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 2821 | . . . 4 ⊢ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) | |
2 | eqid 2821 | . . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
3 | evthicc.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | evthicc.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | eqid 2821 | . . . . . 6 ⊢ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) | |
6 | 2, 5 | icccmp 23433 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Comp) |
7 | 3, 4, 6 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Comp) |
8 | evthicc.4 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) | |
9 | iccssre 12819 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
10 | 3, 4, 9 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
11 | ax-resscn 10594 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
12 | 10, 11 | sstrdi 3979 | . . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ) |
13 | eqid 2821 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) = ((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) | |
14 | eqid 2821 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
15 | eqid 2821 | . . . . . . . 8 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) | |
16 | eqid 2821 | . . . . . . . . 9 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) | |
17 | 14, 16 | tgioo 23404 | . . . . . . . 8 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) |
18 | 13, 14, 15, 17 | cncfmet 23516 | . . . . . . 7 ⊢ (((𝐴[,]𝐵) ⊆ ℂ ∧ ℝ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,)))) |
19 | 12, 11, 18 | sylancl 588 | . . . . . 6 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℝ) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,)))) |
20 | 2, 15 | resubmet 23410 | . . . . . . . 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 7171 | . . . . . 6 ⊢ (𝜑 → ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,))) = (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (topGen‘ran (,)))) |
23 | 19, 22 | eqtrd 2856 | . . . . 5 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℝ) = (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (topGen‘ran (,)))) |
24 | 8, 23 | eleqtrd 2915 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (topGen‘ran (,)))) |
25 | retop 23370 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
26 | uniretop 23371 | . . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
27 | 26 | restuni 21770 | . . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ) → (𝐴[,]𝐵) = ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))) |
28 | 25, 10, 27 | sylancr 589 | . . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) = ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))) |
29 | 3 | rexrd 10691 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
30 | 4 | rexrd 10691 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
31 | evthicc.3 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
32 | lbicc2 12853 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
33 | 29, 30, 31, 32 | syl3anc 1367 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
34 | 33 | ne0d 4301 | . . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ≠ ∅) |
35 | 28, 34 | eqnetrrd 3084 | . . . 4 ⊢ (𝜑 → ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ≠ ∅) |
36 | 1, 2, 7, 24, 35 | evth 23563 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))∀𝑦 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))(𝐹‘𝑦) ≤ (𝐹‘𝑥)) |
37 | 28 | raleqdv 3415 | . . . 4 ⊢ (𝜑 → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥) ↔ ∀𝑦 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))(𝐹‘𝑦) ≤ (𝐹‘𝑥))) |
38 | 28, 37 | rexeqbidv 3402 | . . 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 259 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥)) |
40 | 1, 2, 7, 24, 35 | evth2 23564 | . . 3 ⊢ (𝜑 → ∃𝑧 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))∀𝑤 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))(𝐹‘𝑧) ≤ (𝐹‘𝑤)) |
41 | 28 | raleqdv 3415 | . . . 4 ⊢ (𝜑 → (∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤) ↔ ∀𝑤 ∈ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))(𝐹‘𝑧) ≤ (𝐹‘𝑤))) |
42 | 28, 41 | rexeqbidv 3402 | . . 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 259 | . 2 ⊢ (𝜑 → ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤)) |
44 | 39, 43 | jca 514 | 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 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 ∅c0 4291 ∪ cuni 4838 class class class wbr 5066 × cxp 5553 ran crn 5556 ↾ cres 5557 ∘ ccom 5559 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 ℝ*cxr 10674 ≤ cle 10676 − cmin 10870 (,)cioo 12739 [,]cicc 12742 abscabs 14593 ↾t crest 16694 topGenctg 16711 MetOpencmopn 20535 Topctop 21501 Cn ccn 21832 Compccmp 21994 –cn→ccncf 23484 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cn 21835 df-cnp 21836 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 |
This theorem is referenced by: evthicc2 24061 cniccbdd 24062 rolle 24587 dvivthlem1 24605 itgsubst 24646 evthiccabs 41820 cncficcgt0 42220 |
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