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Mirrors > Home > MPE Home > Th. List > fsequb2 | Structured version Visualization version GIF version |
Description: The values of a finite real sequence have an upper bound. (Contributed by NM, 20-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fsequb2 | ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfi 13334 | . . 3 ⊢ (𝑀...𝑁) ∈ Fin | |
2 | ffvelrn 6844 | . . . 4 ⊢ ((𝐹:(𝑀...𝑁)⟶ℝ ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) | |
3 | 2 | ralrimiva 3182 | . . 3 ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ) |
4 | fimaxre3 11581 | . . 3 ⊢ (((𝑀...𝑁) ∈ Fin ∧ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ≤ 𝑥) | |
5 | 1, 3, 4 | sylancr 589 | . 2 ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ≤ 𝑥) |
6 | ffn 6509 | . . . 4 ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → 𝐹 Fn (𝑀...𝑁)) | |
7 | breq1 5062 | . . . . 5 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑦 ≤ 𝑥 ↔ (𝐹‘𝑘) ≤ 𝑥)) | |
8 | 7 | ralrn 6849 | . . . 4 ⊢ (𝐹 Fn (𝑀...𝑁) → (∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥 ↔ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ≤ 𝑥)) |
9 | 6, 8 | syl 17 | . . 3 ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → (∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥 ↔ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ≤ 𝑥)) |
10 | 9 | rexbidv 3297 | . 2 ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ≤ 𝑥)) |
11 | 5, 10 | mpbird 259 | 1 ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 class class class wbr 5059 ran crn 5551 Fn wfn 6345 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 Fincfn 8503 ℝcr 10530 ≤ cle 10670 ...cfz 12886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 |
This theorem is referenced by: fseqsupubi 13340 |
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