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Mirrors > Home > MPE Home > Th. List > fseqsupubi | Structured version Visualization version GIF version |
Description: The values of a finite real sequence are bounded by their supremum. (Contributed by NM, 20-Sep-2005.) |
Ref | Expression |
---|---|
fseqsupubi | ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → (𝐹‘𝐾) ≤ sup(ran 𝐹, ℝ, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frn 5966 | . . 3 ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → ran 𝐹 ⊆ ℝ) | |
2 | 1 | adantl 481 | . 2 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → ran 𝐹 ⊆ ℝ) |
3 | fdm 5964 | . . 3 ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → dom 𝐹 = (𝑀...𝑁)) | |
4 | ne0i 3880 | . . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) ≠ ∅) | |
5 | dm0rn0 5263 | . . . . . 6 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅) | |
6 | eqeq1 2614 | . . . . . . 7 ⊢ (dom 𝐹 = (𝑀...𝑁) → (dom 𝐹 = ∅ ↔ (𝑀...𝑁) = ∅)) | |
7 | 6 | biimpd 218 | . . . . . 6 ⊢ (dom 𝐹 = (𝑀...𝑁) → (dom 𝐹 = ∅ → (𝑀...𝑁) = ∅)) |
8 | 5, 7 | syl5bir 232 | . . . . 5 ⊢ (dom 𝐹 = (𝑀...𝑁) → (ran 𝐹 = ∅ → (𝑀...𝑁) = ∅)) |
9 | 8 | necon3d 2803 | . . . 4 ⊢ (dom 𝐹 = (𝑀...𝑁) → ((𝑀...𝑁) ≠ ∅ → ran 𝐹 ≠ ∅)) |
10 | 4, 9 | mpan9 485 | . . 3 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ dom 𝐹 = (𝑀...𝑁)) → ran 𝐹 ≠ ∅) |
11 | 3, 10 | sylan2 490 | . 2 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → ran 𝐹 ≠ ∅) |
12 | fsequb2 12637 | . . 3 ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) | |
13 | 12 | adantl 481 | . 2 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) |
14 | ffn 5958 | . . 3 ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → 𝐹 Fn (𝑀...𝑁)) | |
15 | fnfvelrn 6264 | . . . 4 ⊢ ((𝐹 Fn (𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐹‘𝐾) ∈ ran 𝐹) | |
16 | 15 | ancoms 468 | . . 3 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐹 Fn (𝑀...𝑁)) → (𝐹‘𝐾) ∈ ran 𝐹) |
17 | 14, 16 | sylan2 490 | . 2 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → (𝐹‘𝐾) ∈ ran 𝐹) |
18 | suprub 10863 | . 2 ⊢ (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝐹‘𝐾) ∈ ran 𝐹) → (𝐹‘𝐾) ≤ sup(ran 𝐹, ℝ, < )) | |
19 | 2, 11, 13, 17, 18 | syl31anc 1321 | 1 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → (𝐹‘𝐾) ≤ sup(ran 𝐹, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 dom cdm 5038 ran crn 5039 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 supcsup 8229 ℝcr 9814 < clt 9953 ≤ cle 9954 ...cfz 12197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 |
This theorem is referenced by: (None) |
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