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Mirrors > Home > MPE Home > Th. List > Mathboxes > supfz | Structured version Visualization version GIF version |
Description: The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) |
Ref | Expression |
---|---|
supfz | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zssre 11989 | . . . 4 ⊢ ℤ ⊆ ℝ | |
2 | ltso 10721 | . . . 4 ⊢ < Or ℝ | |
3 | soss 5493 | . . . 4 ⊢ (ℤ ⊆ ℝ → ( < Or ℝ → < Or ℤ)) | |
4 | 1, 2, 3 | mp2 9 | . . 3 ⊢ < Or ℤ |
5 | 4 | a1i 11 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → < Or ℤ) |
6 | eluzelz 12254 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
7 | eluzfz2 12916 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) | |
8 | elfzle2 12912 | . . . 4 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ≤ 𝑁) | |
9 | 8 | adantl 484 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ≤ 𝑁) |
10 | elfzelz 12909 | . . . . 5 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) | |
11 | 10 | zred 12088 | . . . 4 ⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℝ) |
12 | eluzelre 12255 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ) | |
13 | lenlt 10719 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑥 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑥)) | |
14 | 11, 12, 13 | syl2anr 598 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑥 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑥)) |
15 | 9, 14 | mpbid 234 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝑥 ∈ (𝑀...𝑁)) → ¬ 𝑁 < 𝑥) |
16 | 5, 6, 7, 15 | supmax 8931 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 class class class wbr 5066 Or wor 5473 ‘cfv 6355 (class class class)co 7156 supcsup 8904 ℝcr 10536 < clt 10675 ≤ cle 10676 ℤcz 11982 ℤ≥cuz 12244 ...cfz 12893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-neg 10873 df-z 11983 df-uz 12245 df-fz 12894 |
This theorem is referenced by: (None) |
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