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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 13970 | . . 3 ⊢ (𝑀...𝑁) ∈ Fin | |
2 | fimaxre3 12191 | . . 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 3108 | . . . . . 6 ⊢ (∀𝑘 ∈ (𝑀...𝑁)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) ≤ 𝑦) ↔ (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ ∧ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ≤ 𝑦)) | |
5 | peano2re 11418 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ) | |
6 | ltp1 12085 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → 𝑦 < (𝑦 + 1)) | |
7 | 6 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑘) ∈ ℝ) → 𝑦 < (𝑦 + 1)) |
8 | simpr 484 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑘) ∈ ℝ) → (𝐹‘𝑘) ∈ ℝ) | |
9 | simpl 482 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑘) ∈ ℝ) → 𝑦 ∈ ℝ) | |
10 | 5 | adantr 480 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑘) ∈ ℝ) → (𝑦 + 1) ∈ ℝ) |
11 | lelttr 11335 | . . . . . . . . . . 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 1369 | . . . . . . . . . 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 453 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) ≤ 𝑦) → (𝐹‘𝑘) < (𝑦 + 1))) |
15 | 14 | ralimdv 3166 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (∀𝑘 ∈ (𝑀...𝑁)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) ≤ 𝑦) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) < (𝑦 + 1))) |
16 | brralrspcev 5208 | . . . . . . 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 415 | . . . 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 407 | . . 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 3152 | . 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 395 ∈ wcel 2099 ∀wral 3058 ∃wrex 3067 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 Fincfn 8964 ℝcr 11138 1c1 11140 + caddc 11142 < clt 11279 ≤ cle 11280 ...cfz 13517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 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 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 |
This theorem is referenced by: (None) |
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