Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 30488 | . . 3 ⊢ (𝜑 → (inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)) |
4 | 3 | notbid 320 | . 2 ⊢ (𝜑 → (¬ inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)) |
5 | iccssxr 12818 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
6 | 5, 2 | sseldi 3964 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
7 | xrltso 12533 | . . . . . . 7 ⊢ < Or ℝ* | |
8 | soss 5492 | . . . . . . 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 30483 | . . . . . 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 8951 | . . . 4 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ∈ (0[,]+∞)) |
14 | 5, 13 | sseldi 3964 | . . 3 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ∈ ℝ*) |
15 | 6, 14 | xrlenltd 10706 | . 2 ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ¬ inf(𝐴, (0[,]+∞), < ) < 𝐵)) |
16 | 6 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
17 | 1, 5 | sstrdi 3978 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
18 | 17 | sselda 3966 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
19 | 16, 18 | xrlenltd 10706 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝑥 ↔ ¬ 𝑥 < 𝐵)) |
20 | 19 | ralbidva 3196 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵)) |
21 | ralnex 3236 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵) | |
22 | 20, 21 | syl6bb 289 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)) |
23 | 4, 15, 22 | 3bitr4d 313 | 1 ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ⊆ wss 3935 class class class wbr 5065 Or wor 5472 (class class class)co 7155 infcinf 8904 0cc0 10536 +∞cpnf 10671 ℝ*cxr 10673 < clt 10674 ≤ cle 10675 [,]cicc 12740 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-icc 12744 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |