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Mathbox for Glauco Siliprandi |
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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 12139 | . . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ) | |
5 | 1, 2, 3, 4 | syl3anc 1371 | . . 3 ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ) |
6 | suprltrp.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ+) | |
7 | 5, 6 | ltsubrpd 13013 | . 2 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) − 𝑋) < sup(𝐴, ℝ, < )) |
8 | 6 | rpred 12981 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) |
9 | 5, 8 | resubcld 11607 | . . 3 ⊢ (𝜑 → (sup(𝐴, ℝ, < ) − 𝑋) ∈ ℝ) |
10 | suprlub 12143 | . . 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 1373 | . 2 ⊢ (𝜑 → ((sup(𝐴, ℝ, < ) − 𝑋) < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 (sup(𝐴, ℝ, < ) − 𝑋) < 𝑧)) |
12 | 7, 11 | mpbid 231 | 1 ⊢ (𝜑 → ∃𝑧 ∈ 𝐴 (sup(𝐴, ℝ, < ) − 𝑋) < 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ⊆ wss 3928 ∅c0 4302 class class class wbr 5125 (class class class)co 7377 supcsup 9400 ℝcr 11074 < clt 11213 ≤ cle 11214 − cmin 11409 ℝ+crp 12939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2702 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-po 5565 df-so 5566 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-er 8670 df-en 8906 df-dom 8907 df-sdom 8908 df-sup 9402 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-rp 12940 |
This theorem is referenced by: sge0ltfirp 44794 |
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