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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > infxrlesupxr | Structured version Visualization version GIF version |
Description: The supremum of a nonempty set is greater than or equal to the infimum. The second condition is needed, see supxrltinfxr 45399. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
infxrlesupxr.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
infxrlesupxr.2 | ⊢ (𝜑 → 𝐴 ≠ ∅) |
Ref | Expression |
---|---|
infxrlesupxr | ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ≤ sup(𝐴, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infxrlesupxr.2 | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
2 | n0 4359 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
3 | 2 | biimpi 216 | . . 3 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) |
5 | infxrlesupxr.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) | |
6 | 5 | infxrcld 45339 | . . . . . 6 ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
7 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
8 | 5 | sselda 3995 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
9 | 5 | supxrcld 45047 | . . . . . 6 ⊢ (𝜑 → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
11 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ*) |
12 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
13 | infxrlb 13373 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝑥) | |
14 | 11, 12, 13 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝑥) |
15 | eqid 2735 | . . . . . 6 ⊢ sup(𝐴, ℝ*, < ) = sup(𝐴, ℝ*, < ) | |
16 | 11, 12, 15 | supxrubd 45053 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐴, ℝ*, < )) |
17 | 7, 8, 10, 14, 16 | xrletrd 13201 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ sup(𝐴, ℝ*, < )) |
18 | 17 | ex 412 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → inf(𝐴, ℝ*, < ) ≤ sup(𝐴, ℝ*, < ))) |
19 | 18 | exlimdv 1931 | . 2 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → inf(𝐴, ℝ*, < ) ≤ sup(𝐴, ℝ*, < ))) |
20 | 4, 19 | mpd 15 | 1 ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ≤ sup(𝐴, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 supcsup 9478 infcinf 9479 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 |
This theorem is referenced by: liminflelimsuplem 45731 |
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