Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > suprltrp | Structured version Visualization version GIF version |
Description: The supremum of a nonempty bounded set of reals can be approximated from below by elements of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
suprltrp.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
suprltrp.n0 | ⊢ (𝜑 → 𝐴 ≠ ∅) |
suprltrp.bnd | ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
suprltrp.x | ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
Ref | Expression |
---|---|
suprltrp | ⊢ (𝜑 → ∃𝑧 ∈ 𝐴 (sup(𝐴, ℝ, < ) − 𝑋) < 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suprltrp.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
2 | suprltrp.n0 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
3 | suprltrp.bnd | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) | |
4 | suprcl 11757 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ) | |
5 | 1, 2, 3, 4 | syl3anc 1373 | . . 3 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ) |
6 | suprltrp.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ+) | |
7 | 5, 6 | ltsubrpd 12625 | . 2 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) − 𝑋) < sup(𝐴, ℝ, < )) |
8 | 6 | rpred 12593 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) |
9 | 5, 8 | resubcld 11225 | . . 3 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) − 𝑋) ∈ ℝ) |
10 | suprlub 11761 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ∧ (sup(𝐴, ℝ, < ) − 𝑋) ∈ ℝ) → ((sup(𝐴, ℝ, < ) − 𝑋) < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 (sup(𝐴, ℝ, < ) − 𝑋) < 𝑧)) | |
11 | 1, 2, 3, 9, 10 | syl31anc 1375 | . 2 ⊢ (𝜑 → ((sup(𝐴, ℝ, < ) − 𝑋) < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 (sup(𝐴, ℝ, < ) − 𝑋) < 𝑧)) |
12 | 7, 11 | mpbid 235 | 1 ⊢ (𝜑 → ∃𝑧 ∈ 𝐴 (sup(𝐴, ℝ, < ) − 𝑋) < 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 ∃wrex 3052 ⊆ wss 3853 ∅c0 4223 class class class wbr 5039 (class class class)co 7191 supcsup 9034 ℝcr 10693 < clt 10832 ≤ cle 10833 − cmin 11027 ℝ+crp 12551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-rp 12552 |
This theorem is referenced by: sge0ltfirp 43556 |
Copyright terms: Public domain | W3C validator |