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Mirrors > Home > MPE Home > Th. List > Mathboxes > infxrge0lb | Structured version Visualization version GIF version |
Description: A member of a set of nonnegative extended reals is greater than or equal to the set's infimum. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.) |
Ref | Expression |
---|---|
infxrge0lb.a | ⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞)) |
infxrge0lb.b | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
Ref | Expression |
---|---|
infxrge0lb | ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssxr 12807 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | xrltso 12522 | . . . . . 6 ⊢ < Or ℝ* | |
3 | soss 5486 | . . . . . 6 ⊢ ((0[,]+∞) ⊆ ℝ* → ( < Or ℝ* → < Or (0[,]+∞))) | |
4 | 1, 2, 3 | mp2 9 | . . . . 5 ⊢ < Or (0[,]+∞) |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → < Or (0[,]+∞)) |
6 | infxrge0lb.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞)) | |
7 | xrge0infss 30410 | . . . . 5 ⊢ (𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) |
9 | 5, 8 | infcl 8940 | . . 3 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ∈ (0[,]+∞)) |
10 | 1, 9 | sseldi 3962 | . 2 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ∈ ℝ*) |
11 | infxrge0lb.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
12 | 6, 11 | sseldd 3965 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) |
13 | 1, 12 | sseldi 3962 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
14 | 5, 8 | inflb 8941 | . . 3 ⊢ (𝜑 → (𝐵 ∈ 𝐴 → ¬ 𝐵 < inf(𝐴, (0[,]+∞), < ))) |
15 | 11, 14 | mpd 15 | . 2 ⊢ (𝜑 → ¬ 𝐵 < inf(𝐴, (0[,]+∞), < )) |
16 | 10, 13, 15 | xrnltled 10697 | 1 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ⊆ wss 3933 class class class wbr 5057 Or wor 5466 (class class class)co 7145 infcinf 8893 0cc0 10525 +∞cpnf 10660 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 [,]cicc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-icc 12733 |
This theorem is referenced by: (None) |
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