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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cncmpmax | Structured version Visualization version GIF version |
Description: When the hypothesis for the extreme value theorem hold, then the sup of the range of the function belongs to the range, it is real and it an upper bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
cncmpmax.1 | ⊢ 𝑇 = ∪ 𝐽 |
cncmpmax.2 | ⊢ 𝐾 = (topGen‘ran (,)) |
cncmpmax.3 | ⊢ (𝜑 → 𝐽 ∈ Comp) |
cncmpmax.4 | ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
cncmpmax.5 | ⊢ (𝜑 → 𝑇 ≠ ∅) |
Ref | Expression |
---|---|
cncmpmax | ⊢ (𝜑 → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncmpmax.1 | . . 3 ⊢ 𝑇 = ∪ 𝐽 | |
2 | cncmpmax.2 | . . 3 ⊢ 𝐾 = (topGen‘ran (,)) | |
3 | cncmpmax.3 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Comp) | |
4 | cncmpmax.4 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
5 | cncmpmax.5 | . . 3 ⊢ (𝜑 → 𝑇 ≠ ∅) | |
6 | 1, 2, 3, 4, 5 | evth 23166 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑇 ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) |
7 | eqid 2778 | . . . . . . . . 9 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾) | |
8 | 2, 1, 7, 4 | fcnre 40117 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
9 | 8 | frnd 6298 | . . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
10 | 9 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → ran 𝐹 ⊆ ℝ) |
11 | 8 | ffund 6295 | . . . . . . . . 9 ⊢ (𝜑 → Fun 𝐹) |
12 | 11 | adantr 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → Fun 𝐹) |
13 | simpr 479 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝑇) | |
14 | 8 | adantr 474 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐹:𝑇⟶ℝ) |
15 | 14 | fdmd 6300 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → dom 𝐹 = 𝑇) |
16 | 13, 15 | eleqtrrd 2862 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ dom 𝐹) |
17 | fvelrn 6616 | . . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) | |
18 | 12, 16, 17 | syl2anc 579 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐹‘𝑥) ∈ ran 𝐹) |
19 | 18 | adantrr 707 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → (𝐹‘𝑥) ∈ ran 𝐹) |
20 | ffn 6291 | . . . . . . . . . . . . 13 ⊢ (𝐹:𝑇⟶ℝ → 𝐹 Fn 𝑇) | |
21 | fvelrnb 6503 | . . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝑇 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑠 ∈ 𝑇 (𝐹‘𝑠) = 𝑦)) | |
22 | 8, 20, 21 | 3syl 18 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑠 ∈ 𝑇 (𝐹‘𝑠) = 𝑦)) |
23 | 22 | biimpa 470 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑠 ∈ 𝑇 (𝐹‘𝑠) = 𝑦) |
24 | df-rex 3096 | . . . . . . . . . . 11 ⊢ (∃𝑠 ∈ 𝑇 (𝐹‘𝑠) = 𝑦 ↔ ∃𝑠(𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) | |
25 | 23, 24 | sylib 210 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑠(𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) |
26 | 25 | adantlr 705 | . . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑠(𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) |
27 | simprr 763 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → (𝐹‘𝑠) = 𝑦) | |
28 | simpllr 766 | . . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) | |
29 | simprl 761 | . . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → 𝑠 ∈ 𝑇) | |
30 | fveq2 6446 | . . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑠 → (𝐹‘𝑡) = (𝐹‘𝑠)) | |
31 | 30 | breq1d 4896 | . . . . . . . . . . . 12 ⊢ (𝑡 = 𝑠 → ((𝐹‘𝑡) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑠) ≤ (𝐹‘𝑥))) |
32 | 31 | rspccva 3510 | . . . . . . . . . . 11 ⊢ ((∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥) ∧ 𝑠 ∈ 𝑇) → (𝐹‘𝑠) ≤ (𝐹‘𝑥)) |
33 | 28, 29, 32 | syl2anc 579 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → (𝐹‘𝑠) ≤ (𝐹‘𝑥)) |
34 | 27, 33 | eqbrtrrd 4910 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → 𝑦 ≤ (𝐹‘𝑥)) |
35 | 26, 34 | exlimddv 1978 | . . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ≤ (𝐹‘𝑥)) |
36 | 35 | ralrimiva 3148 | . . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) → ∀𝑦 ∈ ran 𝐹 𝑦 ≤ (𝐹‘𝑥)) |
37 | 36 | adantrl 706 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → ∀𝑦 ∈ ran 𝐹 𝑦 ≤ (𝐹‘𝑥)) |
38 | ubelsupr 40112 | . . . . . 6 ⊢ ((ran 𝐹 ⊆ ℝ ∧ (𝐹‘𝑥) ∈ ran 𝐹 ∧ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ (𝐹‘𝑥)) → (𝐹‘𝑥) = sup(ran 𝐹, ℝ, < )) | |
39 | 10, 19, 37, 38 | syl3anc 1439 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → (𝐹‘𝑥) = sup(ran 𝐹, ℝ, < )) |
40 | 39 | eqcomd 2784 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → sup(ran 𝐹, ℝ, < ) = (𝐹‘𝑥)) |
41 | 40, 19 | eqeltrd 2859 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹) |
42 | 10, 41 | sseldd 3822 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → sup(ran 𝐹, ℝ, < ) ∈ ℝ) |
43 | simplrr 768 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) ∧ 𝑠 ∈ 𝑇) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) | |
44 | 43, 32 | sylancom 582 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) ∧ 𝑠 ∈ 𝑇) → (𝐹‘𝑠) ≤ (𝐹‘𝑥)) |
45 | 40 | adantr 474 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) ∧ 𝑠 ∈ 𝑇) → sup(ran 𝐹, ℝ, < ) = (𝐹‘𝑥)) |
46 | 44, 45 | breqtrrd 4914 | . . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) ∧ 𝑠 ∈ 𝑇) → (𝐹‘𝑠) ≤ sup(ran 𝐹, ℝ, < )) |
47 | 46 | ralrimiva 3148 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → ∀𝑠 ∈ 𝑇 (𝐹‘𝑠) ≤ sup(ran 𝐹, ℝ, < )) |
48 | 30 | breq1d 4896 | . . . . 5 ⊢ (𝑡 = 𝑠 → ((𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < ) ↔ (𝐹‘𝑠) ≤ sup(ran 𝐹, ℝ, < ))) |
49 | 48 | cbvralv 3367 | . . . 4 ⊢ (∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < ) ↔ ∀𝑠 ∈ 𝑇 (𝐹‘𝑠) ≤ sup(ran 𝐹, ℝ, < )) |
50 | 47, 49 | sylibr 226 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < )) |
51 | 41, 42, 50 | 3jca 1119 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < ))) |
52 | 6, 51 | rexlimddv 3218 | 1 ⊢ (𝜑 → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∃wex 1823 ∈ wcel 2107 ≠ wne 2969 ∀wral 3090 ∃wrex 3091 ⊆ wss 3792 ∅c0 4141 ∪ cuni 4671 class class class wbr 4886 dom cdm 5355 ran crn 5356 Fun wfun 6129 Fn wfn 6130 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 supcsup 8634 ℝcr 10271 < clt 10411 ≤ cle 10412 (,)cioo 12487 topGenctg 16484 Cn ccn 21436 Compccmp 21598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-icc 12494 df-fz 12644 df-fzo 12785 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cn 21439 df-cnp 21440 df-cmp 21599 df-tx 21774 df-hmeo 21967 df-xms 22533 df-ms 22534 df-tms 22535 |
This theorem is referenced by: stoweidlem36 41180 |
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