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Mirrors > Home > MPE Home > Th. List > fsequb | Structured version Visualization version GIF version |
Description: The values of a finite real sequence have an upper bound. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fsequb | ⊢ (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) < 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfi 13886 | . . 3 ⊢ (𝑀...𝑁) ∈ Fin | |
2 | fimaxre3 12109 | . . 3 ⊢ (((𝑀...𝑁) ∈ Fin ∧ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ≤ 𝑦) | |
3 | 1, 2 | mpan 689 | . 2 ⊢ (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → ∃𝑦 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ≤ 𝑦) |
4 | r19.26 3111 | . . . . . 6 ⊢ (∀𝑘 ∈ (𝑀...𝑁)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) ≤ 𝑦) ↔ (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ ∧ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ≤ 𝑦)) | |
5 | peano2re 11336 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ) | |
6 | ltp1 12003 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → 𝑦 < (𝑦 + 1)) | |
7 | 6 | adantr 482 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑘) ∈ ℝ) → 𝑦 < (𝑦 + 1)) |
8 | simpr 486 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑘) ∈ ℝ) → (𝐹‘𝑘) ∈ ℝ) | |
9 | simpl 484 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑘) ∈ ℝ) → 𝑦 ∈ ℝ) | |
10 | 5 | adantr 482 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑘) ∈ ℝ) → (𝑦 + 1) ∈ ℝ) |
11 | lelttr 11253 | . . . . . . . . . . 11 ⊢ (((𝐹‘𝑘) ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ) → (((𝐹‘𝑘) ≤ 𝑦 ∧ 𝑦 < (𝑦 + 1)) → (𝐹‘𝑘) < (𝑦 + 1))) | |
12 | 8, 9, 10, 11 | syl3anc 1372 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑘) ∈ ℝ) → (((𝐹‘𝑘) ≤ 𝑦 ∧ 𝑦 < (𝑦 + 1)) → (𝐹‘𝑘) < (𝑦 + 1))) |
13 | 7, 12 | mpan2d 693 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑘) ∈ ℝ) → ((𝐹‘𝑘) ≤ 𝑦 → (𝐹‘𝑘) < (𝑦 + 1))) |
14 | 13 | expimpd 455 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) ≤ 𝑦) → (𝐹‘𝑘) < (𝑦 + 1))) |
15 | 14 | ralimdv 3163 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑁)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) ≤ 𝑦) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) < (𝑦 + 1))) |
16 | brralrspcev 5169 | . . . . . . 7 ⊢ (((𝑦 + 1) ∈ ℝ ∧ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) < (𝑦 + 1)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) < 𝑥) | |
17 | 5, 15, 16 | syl6an 683 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑁)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) < 𝑥)) |
18 | 4, 17 | biimtrrid 242 | . . . . 5 ⊢ (𝑦 ∈ ℝ → ((∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ ∧ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) < 𝑥)) |
19 | 18 | expd 417 | . . . 4 ⊢ (𝑦 ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ≤ 𝑦 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) < 𝑥))) |
20 | 19 | impcom 409 | . . 3 ⊢ ((∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ≤ 𝑦 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) < 𝑥)) |
21 | 20 | rexlimdva 3149 | . 2 ⊢ (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → (∃𝑦 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ≤ 𝑦 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) < 𝑥)) |
22 | 3, 21 | mpd 15 | 1 ⊢ (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) < 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ∀wral 3061 ∃wrex 3070 class class class wbr 5109 ‘cfv 6500 (class class class)co 7361 Fincfn 8889 ℝcr 11058 1c1 11060 + caddc 11062 < clt 11197 ≤ cle 11198 ...cfz 13433 |
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-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 |
This theorem is referenced by: (None) |
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