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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 24929 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑇 ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) |
7 | eqid 2725 | . . . . . . . . 9 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾) | |
8 | 2, 1, 7, 4 | fcnre 44526 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
9 | 8 | frnd 6731 | . . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
10 | 9 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → ran 𝐹 ⊆ ℝ) |
11 | 8 | ffund 6727 | . . . . . . . . 9 ⊢ (𝜑 → Fun 𝐹) |
12 | 11 | adantr 479 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → Fun 𝐹) |
13 | simpr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝑇) | |
14 | 8 | adantr 479 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐹:𝑇⟶ℝ) |
15 | 14 | fdmd 6733 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → dom 𝐹 = 𝑇) |
16 | 13, 15 | eleqtrrd 2828 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ dom 𝐹) |
17 | fvelrn 7085 | . . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) | |
18 | 12, 16, 17 | syl2anc 582 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐹‘𝑥) ∈ ran 𝐹) |
19 | 18 | adantrr 715 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → (𝐹‘𝑥) ∈ ran 𝐹) |
20 | ffn 6723 | . . . . . . . . . . . . 13 ⊢ (𝐹:𝑇⟶ℝ → 𝐹 Fn 𝑇) | |
21 | fvelrnb 6958 | . . . . . . . . . . . . 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 475 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑠 ∈ 𝑇 (𝐹‘𝑠) = 𝑦) |
24 | df-rex 3060 | . . . . . . . . . . 11 ⊢ (∃𝑠 ∈ 𝑇 (𝐹‘𝑠) = 𝑦 ↔ ∃𝑠(𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) | |
25 | 23, 24 | sylib 217 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑠(𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) |
26 | 25 | adantlr 713 | . . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) → ∃𝑠(𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) |
27 | simprr 771 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → (𝐹‘𝑠) = 𝑦) | |
28 | simpllr 774 | . . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) | |
29 | simprl 769 | . . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → 𝑠 ∈ 𝑇) | |
30 | fveq2 6896 | . . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑠 → (𝐹‘𝑡) = (𝐹‘𝑠)) | |
31 | 30 | breq1d 5159 | . . . . . . . . . . . 12 ⊢ (𝑡 = 𝑠 → ((𝐹‘𝑡) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑠) ≤ (𝐹‘𝑥))) |
32 | 31 | rspccva 3605 | . . . . . . . . . . 11 ⊢ ((∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥) ∧ 𝑠 ∈ 𝑇) → (𝐹‘𝑠) ≤ (𝐹‘𝑥)) |
33 | 28, 29, 32 | syl2anc 582 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → (𝐹‘𝑠) ≤ (𝐹‘𝑥)) |
34 | 27, 33 | eqbrtrrd 5173 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) ∧ (𝑠 ∈ 𝑇 ∧ (𝐹‘𝑠) = 𝑦)) → 𝑦 ≤ (𝐹‘𝑥)) |
35 | 26, 34 | exlimddv 1930 | . . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ≤ (𝐹‘𝑥)) |
36 | 35 | ralrimiva 3135 | . . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) → ∀𝑦 ∈ ran 𝐹 𝑦 ≤ (𝐹‘𝑥)) |
37 | 36 | adantrl 714 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → ∀𝑦 ∈ ran 𝐹 𝑦 ≤ (𝐹‘𝑥)) |
38 | ubelsupr 44521 | . . . . . 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 1368 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → (𝐹‘𝑥) = sup(ran 𝐹, ℝ, < )) |
40 | 39 | eqcomd 2731 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → sup(ran 𝐹, ℝ, < ) = (𝐹‘𝑥)) |
41 | 40, 19 | eqeltrd 2825 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹) |
42 | 10, 41 | sseldd 3977 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → sup(ran 𝐹, ℝ, < ) ∈ ℝ) |
43 | simplrr 776 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) ∧ 𝑠 ∈ 𝑇) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥)) | |
44 | 43, 32 | sylancom 586 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) ∧ 𝑠 ∈ 𝑇) → (𝐹‘𝑠) ≤ (𝐹‘𝑥)) |
45 | 40 | adantr 479 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) ∧ 𝑠 ∈ 𝑇) → sup(ran 𝐹, ℝ, < ) = (𝐹‘𝑥)) |
46 | 44, 45 | breqtrrd 5177 | . . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) ∧ 𝑠 ∈ 𝑇) → (𝐹‘𝑠) ≤ sup(ran 𝐹, ℝ, < )) |
47 | 46 | ralrimiva 3135 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → ∀𝑠 ∈ 𝑇 (𝐹‘𝑠) ≤ sup(ran 𝐹, ℝ, < )) |
48 | 30 | breq1d 5159 | . . . . 5 ⊢ (𝑡 = 𝑠 → ((𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < ) ↔ (𝐹‘𝑠) ≤ sup(ran 𝐹, ℝ, < ))) |
49 | 48 | cbvralvw 3224 | . . . 4 ⊢ (∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < ) ↔ ∀𝑠 ∈ 𝑇 (𝐹‘𝑠) ≤ sup(ran 𝐹, ℝ, < )) |
50 | 47, 49 | sylibr 233 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ (𝐹‘𝑥))) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < )) |
51 | 41, 42, 50 | 3jca 1125 | . 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 3150 | 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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 ∃wrex 3059 ⊆ wss 3944 ∅c0 4322 ∪ cuni 4909 class class class wbr 5149 dom cdm 5678 ran crn 5679 Fun wfun 6543 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 supcsup 9465 ℝcr 11139 < clt 11280 ≤ cle 11281 (,)cioo 13359 topGenctg 17422 Cn ccn 23172 Compccmp 23334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-icc 13366 df-fz 13520 df-fzo 13663 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-pt 17429 df-prds 17432 df-xrs 17487 df-qtop 17492 df-imas 17493 df-xps 17495 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-mulg 19032 df-cntz 19280 df-cmn 19749 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cn 23175 df-cnp 23176 df-cmp 23335 df-tx 23510 df-hmeo 23703 df-xms 24270 df-ms 24271 df-tms 24272 |
This theorem is referenced by: stoweidlem36 45559 |
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