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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > infxrge0glb | Structured version Visualization version GIF version |
Description: The infimum of a set of nonnegative extended reals is the greatest 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 |
---|---|
infxrge0glb | ⊢ (𝜑 → (inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infxrge0glb.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) | |
2 | iccssxr 12808 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
3 | xrltso 12522 | . . . . . 6 ⊢ < Or ℝ* | |
4 | soss 5457 | . . . . . 6 ⊢ ((0[,]+∞) ⊆ ℝ* → ( < Or ℝ* → < Or (0[,]+∞))) | |
5 | 2, 3, 4 | mp2 9 | . . . . 5 ⊢ < Or (0[,]+∞) |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → < Or (0[,]+∞)) |
7 | infxrge0glb.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞)) | |
8 | xrge0infss 30510 | . . . . 5 ⊢ (𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) |
10 | 6, 9, 7 | infglbb 8939 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (0[,]+∞)) → (inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ∃𝑧 ∈ 𝐴 𝑧 < 𝐵)) |
11 | 1, 10 | mpdan 686 | . 2 ⊢ (𝜑 → (inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ∃𝑧 ∈ 𝐴 𝑧 < 𝐵)) |
12 | breq1 5033 | . . 3 ⊢ (𝑥 = 𝑧 → (𝑥 < 𝐵 ↔ 𝑧 < 𝐵)) | |
13 | 12 | cbvrexvw 3397 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 < 𝐵 ↔ ∃𝑧 ∈ 𝐴 𝑧 < 𝐵) |
14 | 11, 13 | syl6bbr 292 | 1 ⊢ (𝜑 → (inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 class class class wbr 5030 Or wor 5437 (class class class)co 7135 infcinf 8889 0cc0 10526 +∞cpnf 10661 ℝ*cxr 10663 < clt 10664 [,]cicc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-icc 12733 |
This theorem is referenced by: infxrge0gelb 30516 |
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