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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 44890. (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 4343 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
3 | 2 | biimpi 215 | . . 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 44830 | . . . . . 6 ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
7 | 6 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
8 | 5 | sselda 3973 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
9 | 5 | supxrcld 44534 | . . . . . 6 ⊢ (𝜑 → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
10 | 9 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
11 | 5 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ*) |
12 | simpr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
13 | infxrlb 13340 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝑥) | |
14 | 11, 12, 13 | syl2anc 582 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝑥) |
15 | eqid 2725 | . . . . . 6 ⊢ sup(𝐴, ℝ*, < ) = sup(𝐴, ℝ*, < ) | |
16 | 11, 12, 15 | supxrubd 44540 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐴, ℝ*, < )) |
17 | 7, 8, 10, 14, 16 | xrletrd 13168 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ sup(𝐴, ℝ*, < )) |
18 | 17 | ex 411 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → inf(𝐴, ℝ*, < ) ≤ sup(𝐴, ℝ*, < ))) |
19 | 18 | exlimdv 1928 | . 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 394 ∃wex 1773 ∈ wcel 2098 ≠ wne 2930 ⊆ wss 3941 ∅c0 4319 class class class wbr 5144 supcsup 9458 infcinf 9459 ℝ*cxr 11272 < clt 11273 ≤ cle 11274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 |
This theorem is referenced by: liminflelimsuplem 45222 |
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