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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > infxrge0gelb | Structured version Visualization version GIF version |
Description: The infimum of a set of nonnegative extended reals is greater than or equal to a lower bound. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.) |
Ref | Expression |
---|---|
infxrge0glb.a | ⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞)) |
infxrge0glb.b | ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) |
Ref | Expression |
---|---|
infxrge0gelb | ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infxrge0glb.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞)) | |
2 | infxrge0glb.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) | |
3 | 1, 2 | infxrge0glb 32575 | . . 3 ⊢ (𝜑 → (inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)) |
4 | 3 | notbid 317 | . 2 ⊢ (𝜑 → (¬ inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)) |
5 | iccssxr 13434 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
6 | 5, 2 | sselid 3971 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
7 | xrltso 13147 | . . . . . . 7 ⊢ < Or ℝ* | |
8 | soss 5605 | . . . . . . 7 ⊢ ((0[,]+∞) ⊆ ℝ* → ( < Or ℝ* → < Or (0[,]+∞))) | |
9 | 5, 7, 8 | mp2 9 | . . . . . 6 ⊢ < Or (0[,]+∞) |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝜑 → < Or (0[,]+∞)) |
11 | xrge0infss 32570 | . . . . . 6 ⊢ (𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | |
12 | 1, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) |
13 | 10, 12 | infcl 9506 | . . . 4 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ∈ (0[,]+∞)) |
14 | 5, 13 | sselid 3971 | . . 3 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ∈ ℝ*) |
15 | 6, 14 | xrlenltd 11305 | . 2 ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ¬ inf(𝐴, (0[,]+∞), < ) < 𝐵)) |
16 | 6 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
17 | 1, 5 | sstrdi 3986 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
18 | 17 | sselda 3973 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
19 | 16, 18 | xrlenltd 11305 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝑥 ↔ ¬ 𝑥 < 𝐵)) |
20 | 19 | ralbidva 3166 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵)) |
21 | ralnex 3062 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵) | |
22 | 20, 21 | bitrdi 286 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)) |
23 | 4, 15, 22 | 3bitr4d 310 | 1 ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 ∀wral 3051 ∃wrex 3060 ⊆ wss 3941 class class class wbr 5144 Or wor 5584 (class class class)co 7413 infcinf 9459 0cc0 11133 +∞cpnf 11270 ℝ*cxr 11272 < clt 11273 ≤ cle 11274 [,]cicc 13354 |
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-iun 4994 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-1st 7987 df-2nd 7988 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 df-icc 13358 |
This theorem is referenced by: (None) |
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