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Theorem evthicc 25494

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 2737	. . . 4 ⊢ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))
2		eqid 2737	. . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
3		evthicc.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		evthicc.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5		eqid 2737	. . . . . 6 ⊢ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))
6	2, 5	icccmp 24847	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp)
7	3, 4, 6	syl2anc 584	. . . 4 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp)
8		evthicc.4	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
9		iccssre 13469	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
10	3, 4, 9	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
11		ax-resscn 11212	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
12	10, 11	sstrdi 3996	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
13		eqid 2737	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) = ((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))
14		eqid 2737	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
15		eqid 2737	. . . . . . . 8 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
16		eqid 2737	. . . . . . . . 9 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
17	14, 16	tgioo 24817	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
18	13, 14, 15, 17	cncfmet 24935	. . . . . . 7 ⊢ (((𝐴[,]𝐵) ⊆ ℂ ∧ ℝ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,))))
19	12, 11, 18	sylancl 586	. . . . . 6 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℝ) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,))))
20	2, 15	resubmet 24823	. . . . . . . 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 7446	. . . . . 6 ⊢ (𝜑 → ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,))) = (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn (topGen‘ran (,))))
23	19, 22	eqtrd 2777	. . . . 5 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℝ) = (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn (topGen‘ran (,))))
24	8, 23	eleqtrd 2843	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn (topGen‘ran (,))))
25		retop 24782	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
26		uniretop 24783	. . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,))
27	26	restuni 23170	. . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ) → (𝐴[,]𝐵) = ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
28	25, 10, 27	sylancr 587	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) = ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
29	3	rexrd 11311	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
30	4	rexrd 11311	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
31		evthicc.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
32		lbicc2 13504	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
33	29, 30, 31, 32	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
34	33	ne0d 4342	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ≠ ∅)
35	28, 34	eqnetrrd 3009	. . . 4 ⊢ (𝜑 → ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ≠ ∅)
36	1, 2, 7, 24, 35	evth 24991	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))∀𝑦 ∈ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))(𝐹‘𝑦) ≤ (𝐹‘𝑥))
37	28	raleqdv 3326	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑥) ↔ ∀𝑦 ∈ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))(𝐹‘𝑦) ≤ (𝐹‘𝑥)))
38	28, 37	rexeqbidv 3347	. . 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 24992	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))∀𝑤 ∈ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))(𝐹‘𝑧) ≤ (𝐹‘𝑤))
41	28	raleqdv 3326	. . . 4 ⊢ (𝜑 → (∀𝑤 ∈ (𝐴[,]𝐵)(𝐹‘𝑧) ≤ (𝐹‘𝑤) ↔ ∀𝑤 ∈ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))(𝐹‘𝑧) ≤ (𝐹‘𝑤)))
42	28, 41	rexeqbidv 3347	. . 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 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 class class class wbr 5143 × cxp 5683 ran crn 5686 ↾ cres 5687 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 ℝ^*cxr 11294 ≤ cle 11296 − cmin 11492 (,)cioo 13387 [,]cicc 13390 abscabs 15273 ↾_t crest 17465 topGenctg 17482 MetOpencmopn 21354 Topctop 22899 Cn ccn 23232 Compccmp 23394 –cn→ccncf 24902

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cn 23235 df-cnp 23236 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904

This theorem is referenced by: evthicc2 25495 cniccbdd 25496 rolle 26028 dvivthlem1 26047 itgsubst 26090 evthiccabs 45509 cncficcgt0 45903

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