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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 6609 | . . 3 ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → ran 𝐹 ⊆ ℝ) | |
2 | 1 | adantl 482 | . 2 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → ran 𝐹 ⊆ ℝ) |
3 | fdm 6611 | . . 3 ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → dom 𝐹 = (𝑀...𝑁)) | |
4 | ne0i 4270 | . . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) ≠ ∅) | |
5 | dm0rn0 5836 | . . . . . 6 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅) | |
6 | eqeq1 2742 | . . . . . . 7 ⊢ (dom 𝐹 = (𝑀...𝑁) → (dom 𝐹 = ∅ ↔ (𝑀...𝑁) = ∅)) | |
7 | 6 | biimpd 228 | . . . . . 6 ⊢ (dom 𝐹 = (𝑀...𝑁) → (dom 𝐹 = ∅ → (𝑀...𝑁) = ∅)) |
8 | 5, 7 | syl5bir 242 | . . . . 5 ⊢ (dom 𝐹 = (𝑀...𝑁) → (ran 𝐹 = ∅ → (𝑀...𝑁) = ∅)) |
9 | 8 | necon3d 2964 | . . . 4 ⊢ (dom 𝐹 = (𝑀...𝑁) → ((𝑀...𝑁) ≠ ∅ → ran 𝐹 ≠ ∅)) |
10 | 4, 9 | mpan9 507 | . . 3 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ dom 𝐹 = (𝑀...𝑁)) → ran 𝐹 ≠ ∅) |
11 | 3, 10 | sylan2 593 | . 2 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → ran 𝐹 ≠ ∅) |
12 | fsequb2 13694 | . . 3 ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) | |
13 | 12 | adantl 482 | . 2 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) |
14 | ffn 6602 | . . 3 ⊢ (𝐹:(𝑀...𝑁)⟶ℝ → 𝐹 Fn (𝑀...𝑁)) | |
15 | fnfvelrn 6960 | . . . 4 ⊢ ((𝐹 Fn (𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐹‘𝐾) ∈ ran 𝐹) | |
16 | 15 | ancoms 459 | . . 3 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐹 Fn (𝑀...𝑁)) → (𝐹‘𝐾) ∈ ran 𝐹) |
17 | 14, 16 | sylan2 593 | . 2 ⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → (𝐹‘𝐾) ∈ ran 𝐹) |
18 | 2, 11, 13, 17 | suprubd 11935 | 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 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 ⊆ wss 3888 ∅c0 4258 class class class wbr 5076 dom cdm 5591 ran crn 5592 Fn wfn 6430 ⟶wf 6431 ‘cfv 6435 (class class class)co 7277 supcsup 9197 ℝcr 10868 < clt 11007 ≤ cle 11008 ...cfz 13237 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-pre-sup 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-sup 9199 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-n0 12232 df-z 12318 df-uz 12581 df-fz 13238 |
This theorem is referenced by: (None) |
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