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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 2765 | . . . 4 ⊢ ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) | |
| 2 | eqid 2765 | . . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
| 3 | evthicc.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 4 | evthicc.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 5 | eqid 2765 | . . . . . 6 ⊢ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) | |
| 6 | 2, 5 | icccmp 24940 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Comp) |
| 7 | 3, 4, 6 | syl2anc 595 | . . . 4 ⊢ (𝜑 → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Comp) |
| 8 | evthicc.4 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) | |
| 9 | iccssre 13444 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
| 10 | 3, 4, 9 | syl2anc 595 | . . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
| 11 | ax-resscn 11145 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
| 12 | 10, 11 | sstrdi 3951 | . . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ) |
| 13 | eqid 2765 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) = ((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) | |
| 14 | eqid 2765 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
| 15 | eqid 2765 | . . . . . . . 8 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) | |
| 16 | eqid 2765 | . . . . . . . . 9 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) | |
| 17 | 14, 16 | tgioo 24910 | . . . . . . . 8 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) |
| 18 | 13, 14, 15, 17 | cncfmet 25025 | . . . . . . 7 ⊢ (((𝐴[,]𝐵) ⊆ ℂ ∧ ℝ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,)))) |
| 19 | 12, 11, 18 | sylancl 597 | . . . . . 6 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℝ) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,)))) |
| 20 | 2, 15 | resubmet 24916 | . . . . . . . 8 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) = ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))) |
| 21 | 10, 20 | syl 18 | . . . . . . 7 ⊢ (𝜑 → (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) = ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))) |
| 22 | 21 | oveq1d 7415 | . . . . . 6 ⊢ (𝜑 → ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,))) = (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (topGen‘ran (,)))) |
| 23 | 19, 22 | eqtrd 2800 | . . . . 5 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℝ) = (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (topGen‘ran (,)))) |
| 24 | 8, 23 | eleqtrd 2867 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) Cn (topGen‘ran (,)))) |
| 25 | retop 24875 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
| 26 | uniretop 24876 | . . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
| 27 | 26 | restuni 23276 | . . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ) → (𝐴[,]𝐵) = ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))) |
| 28 | 25, 10, 27 | sylancr 598 | . . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) = ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵))) |
| 29 | 3 | rexrd 11247 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
| 30 | 4 | rexrd 11247 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
| 31 | evthicc.3 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
| 32 | lbicc2 13479 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
| 33 | 29, 30, 31, 32 | syl3anc 1394 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
| 34 | 33 | ne0d 4297 | . . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ≠ ∅) |
| 35 | 28, 34 | eqnetrrd 3028 | . . . 4 ⊢ (𝜑 → ∪ ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ≠ ∅) |
| 36 | 1, 2, 7, 24, 35 | evth 25075 | . . 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 3340 | . . 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 260 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥)) |
| 40 | 1, 2, 7, 24, 35 | evth2 25076 | . . 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 3340 | . . 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 260 | . 2 ⊢ (𝜑 → ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤)) |
| 44 | 39, 43 | jca 520 | 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 400 = wceq 1563 ∈ wcel 2145 ∀wral 3079 ∃wrex 3089 ⊆ wss 3907 ∅c0 4288 ∪ cuni 4867 class class class wbr 5104 × cxp 5649 ran crn 5652 ↾ cres 5653 ∘ ccom 5655 ‘cfv 6525 (class class class)co 7400 ℂcc 11086 ℝcr 11087 ℝ*cxr 11230 ≤ cle 11232 − cmin 11429 (,)cioo 13360 [,]cicc 13363 abscabs 15273 ↾t crest 17461 topGenctg 17478 MetOpencmopn 21469 Topctop 23007 Cn ccn 23338 Compccmp 23500 –cn→ccncf 24992 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5231 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12222 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12493 df-z 12580 df-dec 12700 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13364 df-icc 13367 df-fz 13524 df-fzo 13671 df-seq 14026 df-exp 14086 df-hash 14355 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17195 df-sets 17212 df-slot 17230 df-ndx 17242 df-base 17258 df-ress 17279 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17463 df-topn 17464 df-0g 17482 df-gsum 17483 df-topgen 17484 df-pt 17485 df-prds 17488 df-xrs 17544 df-qtop 17549 df-imas 17550 df-xps 17552 df-mre 17626 df-mrc 17627 df-acs 17629 df-mgm 18686 df-sgrp 18765 df-mnd 18781 df-submnd 18830 df-mulg 19122 df-cntz 19375 df-cmn 19840 df-psmet 21471 df-xmet 21472 df-met 21473 df-bl 21474 df-mopn 21475 df-cnfld 21480 df-top 23008 df-topon 23025 df-topsp 23047 df-bases 23060 df-cn 23341 df-cnp 23342 df-cmp 23501 df-tx 23676 df-hmeo 23869 df-xms 24434 df-ms 24435 df-tms 24436 df-cncf 24994 |
| This theorem is referenced by: evthicc2 25576 cniccbdd 25577 rolle 26106 dvivthlem1 26124 itgsubst 26165 evthiccabs 46071 cncficcgt0 46461 |
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