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Theorem evthicc 24059

Description: Specialization of the Extreme Value Theorem to a closed interval of ℝ. (Contributed by Mario Carneiro, 12-Aug-2014.)

**Hypotheses**
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))

**Assertion**
Ref	Expression
evthicc	⊢ (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤)))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑧,𝑤,𝐴 𝑥,𝐵,𝑦 𝑤,𝐵,𝑧 𝑥,𝐹,𝑦 𝜑,𝑥,𝑦 𝜑,𝑤,𝑧 𝑤,𝐹,𝑧

**Proof of Theorem evthicc**
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 23432	. . . . 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 12817	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
10	3, 4, 9	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
11		ax-resscn 10593	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
12	10, 11	sstrdi 3978	. . . . . . 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 23403	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
18	13, 14, 15, 17	cncfmet 23515	. . . . . . 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 23409	. . . . . . . 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 7170	. . . . . 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 23369	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
26		uniretop 23370	. . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,))
27	26	restuni 21769	. . . . . 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 10690	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
30	4	rexrd 10690	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
31		evthicc.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
32		lbicc2 12851	. . . . . . 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 4300	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ≠ ∅)
35	28, 34	eqnetrrd 3084	. . . 4 ⊢ (𝜑 → ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ≠ ∅)
36	1, 2, 7, 24, 35	evth 23562	. . 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 23563	. . 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 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ⊆ wss 3935 ∅c0 4290 ∪ cuni 4837 class class class wbr 5065 × cxp 5552 ran crn 5555 ↾ cres 5556 ∘ ccom 5558 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ℝcr 10535 ℝ^*cxr 10673 ≤ cle 10675 − cmin 10869 (,)cioo 12737 [,]cicc 12740 abscabs 14592 ↾_t crest 16693 topGenctg 16710 MetOpencmopn 20534 Topctop 21500 Cn ccn 21831 Compccmp 21993 –cn→ccncf 23483

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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-mulf 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-icc 12744 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cn 21834 df-cnp 21835 df-cmp 21994 df-tx 22169 df-hmeo 22362 df-xms 22929 df-ms 22930 df-tms 22931 df-cncf 23485

This theorem is referenced by: evthicc2 24060 cniccbdd 24061 rolle 24586 dvivthlem1 24604 itgsubst 24645 evthiccabs 41769 cncficcgt0 42169

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