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Theorem evthicc 24063

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 2798	. . . 4 ⊢ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))
2		eqid 2798	. . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
3		evthicc.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		evthicc.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5		eqid 2798	. . . . . 6 ⊢ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))
6	2, 5	icccmp 23430	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp)
7	3, 4, 6	syl2anc 587	. . . 4 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp)
8		evthicc.4	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
9		iccssre 12807	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
10	3, 4, 9	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
11		ax-resscn 10583	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
12	10, 11	sstrdi 3927	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
13		eqid 2798	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) = ((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))
14		eqid 2798	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
15		eqid 2798	. . . . . . . 8 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
16		eqid 2798	. . . . . . . . 9 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
17	14, 16	tgioo 23401	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
18	13, 14, 15, 17	cncfmet 23514	. . . . . . 7 ⊢ (((𝐴[,]𝐵) ⊆ ℂ ∧ ℝ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,))))
19	12, 11, 18	sylancl 589	. . . . . 6 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℝ) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,))))
20	2, 15	resubmet 23407	. . . . . . . 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 7150	. . . . . 6 ⊢ (𝜑 → ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,))) = (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn (topGen‘ran (,))))
23	19, 22	eqtrd 2833	. . . . 5 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℝ) = (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn (topGen‘ran (,))))
24	8, 23	eleqtrd 2892	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn (topGen‘ran (,))))
25		retop 23367	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
26		uniretop 23368	. . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,))
27	26	restuni 21767	. . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ) → (𝐴[,]𝐵) = ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
28	25, 10, 27	sylancr 590	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) = ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
29	3	rexrd 10680	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
30	4	rexrd 10680	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
31		evthicc.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
32		lbicc2 12842	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
33	29, 30, 31, 32	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
34	33	ne0d 4251	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ≠ ∅)
35	28, 34	eqnetrrd 3055	. . . 4 ⊢ (𝜑 → ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ≠ ∅)
36	1, 2, 7, 24, 35	evth 23564	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))∀𝑦 ∈ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))(𝐹‘𝑦) ≤ (𝐹‘𝑥))
37	28	raleqdv 3364	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥) ↔ ∀𝑦 ∈ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))(𝐹‘𝑦) ≤ (𝐹‘𝑥)))
38	28, 37	rexeqbidv 3355	. . 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 23565	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))∀𝑤 ∈ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))(𝐹‘𝑧) ≤ (𝐹‘𝑤))
41	28	raleqdv 3364	. . . 4 ⊢ (𝜑 → (∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤) ↔ ∀𝑤 ∈ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))(𝐹‘𝑧) ≤ (𝐹‘𝑤)))
42	28, 41	rexeqbidv 3355	. . 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 515	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 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 ∅c0 4243 ∪ cuni 4800 class class class wbr 5030 × cxp 5517 ran crn 5520 ↾ cres 5521 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 ℝ^*cxr 10663 ≤ cle 10665 − cmin 10859 (,)cioo 12726 [,]cicc 12729 abscabs 14585 ↾_t crest 16686 topGenctg 16703 MetOpencmopn 20081 Topctop 21498 Cn ccn 21829 Compccmp 21991 –cn→ccncf 23481

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cn 21832 df-cnp 21833 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483

This theorem is referenced by: evthicc2 24064 cniccbdd 24065 rolle 24593 dvivthlem1 24611 itgsubst 24652 evthiccabs 42133 cncficcgt0 42530

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