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Theorem evthicc 25655

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 2766	. . . 4 ⊢ ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))
2		eqid 2766	. . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
3		evthicc.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		evthicc.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5		eqid 2766	. . . . . 6 ⊢ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))
6	2, 5	icccmp 25020	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp)
7	3, 4, 6	syl2anc 596	. . . 4 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp)
8		evthicc.4	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
9		iccssre 13474	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
10	3, 4, 9	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
11		ax-resscn 11175	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
12	10, 11	sstrdi 3952	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
13		eqid 2766	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) = ((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))
14		eqid 2766	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
15		eqid 2766	. . . . . . . 8 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
16		eqid 2766	. . . . . . . . 9 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
17	14, 16	tgioo 24990	. . . . . . . 8 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
18	13, 14, 15, 17	cncfmet 25105	. . . . . . 7 ⊢ (((𝐴[,]𝐵) ⊆ ℂ ∧ ℝ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,))))
19	12, 11, 18	sylancl 598	. . . . . 6 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℝ) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,))))
20	2, 15	resubmet 24996	. . . . . . . 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 7438	. . . . . 6 ⊢ (𝜑 → ((MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) Cn (topGen‘ran (,))) = (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn (topGen‘ran (,))))
23	19, 22	eqtrd 2801	. . . . 5 ⊢ (𝜑 → ((𝐴[,]𝐵)–cn→ℝ) = (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn (topGen‘ran (,))))
24	8, 23	eleqtrd 2868	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) Cn (topGen‘ran (,))))
25		retop 24955	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
26		uniretop 24956	. . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,))
27	26	restuni 23356	. . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ) → (𝐴[,]𝐵) = ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
28	25, 10, 27	sylancr 599	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) = ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
29	3	rexrd 11277	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
30	4	rexrd 11277	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
31		evthicc.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
32		lbicc2 13509	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
33	29, 30, 31, 32	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
34	33	ne0d 4298	. . . . 5 ⊢ (𝜑 → (𝐴[,]𝐵) ≠ ∅)
35	28, 34	eqnetrrd 3029	. . . 4 ⊢ (𝜑 → ∪ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ≠ ∅)
36	1, 2, 7, 24, 35	evth 25155	. . 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 3342	. . 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 25156	. . 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 3342	. . 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 521	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∀wral 3082 ∃wrex 3092 ⊆ wss 3908 ∅c0 4289 ∪ cuni 4877 class class class wbr 5114 × cxp 5664 ran crn 5667 ↾ cres 5668 ∘ ccom 5670 ‘cfv 6543 (class class class)co 7423 ℂcc 11116 ℝcr 11117 ℝ^*cxr 11260 ≤ cle 11262 − cmin 11459 (,)cioo 13390 [,]cicc 13393 abscabs 15311 ↾_t crest 17498 topGenctg 17515 MetOpencmopn 21549 Topctop 23087 Cn ccn 23418 Compccmp 23580 –cn→ccncf 25072

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13394 df-icc 13397 df-fz 13554 df-fzo 13702 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-hom 17359 df-cco 17360 df-rest 17500 df-topn 17501 df-0g 17519 df-gsum 17520 df-topgen 17521 df-pt 17522 df-prds 17525 df-xrs 17581 df-qtop 17586 df-imas 17587 df-xps 17589 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-submnd 18873 df-mulg 19165 df-cntz 19418 df-cmn 19883 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554 df-mopn 21555 df-cnfld 21560 df-top 23088 df-topon 23105 df-topsp 23127 df-bases 23140 df-cn 23421 df-cnp 23422 df-cmp 23581 df-tx 23756 df-hmeo 23949 df-xms 24514 df-ms 24515 df-tms 24516 df-cncf 25074

This theorem is used by: evthicc2 25656 cniccbdd 25657 rolle 26186 dvivthlem1 26204 itgsubst 26245 evthiccabs 46253 cncficcgt0 46643

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