evthiccabs - Mathbox for Glauco Siliprandi

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Theorem evthiccabs 44209

Description: Extreme Value Theorem on y closed interval, for the absolute value of y continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
evthiccabs.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
evthiccabs.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
evthiccabs.aleb	⊢ (𝜑 → 𝐴 ≤ 𝐵)
evthiccabs.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))

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

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

**Proof of Theorem evthiccabs**
Step	Hyp	Ref	Expression
1		evthiccabs.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		evthiccabs.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		evthiccabs.aleb	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
4		ax-resscn 11167	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
5		ssid 4005	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
6		cncfss 24415	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
7	4, 5, 6	mp2an 691	. . . . . . 7 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
8		evthiccabs.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
9	7, 8	sselid 3981	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
10		abscncf 24417	. . . . . . 7 ⊢ abs ∈ (ℂ–cn→ℝ)
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → abs ∈ (ℂ–cn→ℝ))
12	9, 11	cncfco 24423	. . . . 5 ⊢ (𝜑 → (abs ∘ 𝐹) ∈ ((𝐴[,]𝐵)–cn→ℝ))
13	1, 2, 3, 12	evthicc 24976	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑦) ≤ ((abs ∘ 𝐹)‘𝑥) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑧) ≤ ((abs ∘ 𝐹)‘𝑤)))
14	13	simpld 496	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑦) ≤ ((abs ∘ 𝐹)‘𝑥))
15		cncff 24409	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
16		ffun 6721	. . . . . . . . . 10 ⊢ (𝐹:(𝐴[,]𝐵)⟶ℝ → Fun 𝐹)
17	8, 15, 16	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐹)
18	17	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → Fun 𝐹)
19		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ (𝐴[,]𝐵))
20		fdm 6727	. . . . . . . . . . . 12 ⊢ (𝐹:(𝐴[,]𝐵)⟶ℝ → dom 𝐹 = (𝐴[,]𝐵))
21	8, 15, 20	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = (𝐴[,]𝐵))
22	21	eqcomd 2739	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,]𝐵) = dom 𝐹)
23	22	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = dom 𝐹)
24	19, 23	eleqtrd 2836	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ dom 𝐹)
25		fvco 6990	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → ((abs ∘ 𝐹)‘𝑦) = (abs‘(𝐹‘𝑦)))
26	18, 24, 25	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑦) = (abs‘(𝐹‘𝑦)))
27	26	adantlr 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑦) = (abs‘(𝐹‘𝑦)))
28	17	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → Fun 𝐹)
29		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
30	22	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = dom 𝐹)
31	29, 30	eleqtrd 2836	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ dom 𝐹)
32		fvco 6990	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥)))
33	28, 31, 32	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥)))
34	33	adantr 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥)))
35	27, 34	breq12d 5162	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (((abs ∘ 𝐹)‘𝑦) ≤ ((abs ∘ 𝐹)‘𝑥) ↔ (abs‘(𝐹‘𝑦)) ≤ (abs‘(𝐹‘𝑥))))
36	35	ralbidva 3176	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (∀𝑦 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑦) ≤ ((abs ∘ 𝐹)‘𝑥) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑦)) ≤ (abs‘(𝐹‘𝑥))))
37	36	rexbidva 3177	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑦) ≤ ((abs ∘ 𝐹)‘𝑥) ↔ ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑦)) ≤ (abs‘(𝐹‘𝑥))))
38	14, 37	mpbid 231	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑦)) ≤ (abs‘(𝐹‘𝑥)))
39	13	simprd 497	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑧) ≤ ((abs ∘ 𝐹)‘𝑤))
40	17	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → Fun 𝐹)
41		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑧 ∈ (𝐴[,]𝐵))
42	22	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = dom 𝐹)
43	41, 42	eleqtrd 2836	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑧 ∈ dom 𝐹)
44		fvco 6990	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → ((abs ∘ 𝐹)‘𝑧) = (abs‘(𝐹‘𝑧)))
45	40, 43, 44	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑧) = (abs‘(𝐹‘𝑧)))
46	45	adantr 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑧) = (abs‘(𝐹‘𝑧)))
47	17	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴[,]𝐵)) → Fun 𝐹)
48		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴[,]𝐵)) → 𝑤 ∈ (𝐴[,]𝐵))
49	22	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = dom 𝐹)
50	48, 49	eleqtrd 2836	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴[,]𝐵)) → 𝑤 ∈ dom 𝐹)
51		fvco 6990	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑤 ∈ dom 𝐹) → ((abs ∘ 𝐹)‘𝑤) = (abs‘(𝐹‘𝑤)))
52	47, 50, 51	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑤) = (abs‘(𝐹‘𝑤)))
53	52	adantlr 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → ((abs ∘ 𝐹)‘𝑤) = (abs‘(𝐹‘𝑤)))
54	46, 53	breq12d 5162	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) ∧ 𝑤 ∈ (𝐴[,]𝐵)) → (((abs ∘ 𝐹)‘𝑧) ≤ ((abs ∘ 𝐹)‘𝑤) ↔ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑤))))
55	54	ralbidva 3176	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (∀𝑤 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑧) ≤ ((abs ∘ 𝐹)‘𝑤) ↔ ∀𝑤 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑤))))
56	55	rexbidva 3177	. . 3 ⊢ (𝜑 → (∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)((abs ∘ 𝐹)‘𝑧) ≤ ((abs ∘ 𝐹)‘𝑤) ↔ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑤))))
57	39, 56	mpbid 231	. 2 ⊢ (𝜑 → ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑤)))
58	38, 57	jca 513	1 ⊢ (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑦)) ≤ (abs‘(𝐹‘𝑥)) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑤))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ⊆ wss 3949 class class class wbr 5149 dom cdm 5677 ∘ ccom 5681 Fun wfun 6538 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 ≤ cle 11249 [,]cicc 13327 abscabs 15181 –cn→ccncf 24392

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-icc 13331 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cn 22731 df-cnp 22732 df-cmp 22891 df-tx 23066 df-hmeo 23259 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394

This theorem is referenced by: fourierdlem77 44899 fourierdlem83 44905

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