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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 12573 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | xrltso 12289 | . . . . . 6 ⊢ < Or ℝ* | |
3 | soss 5295 | . . . . . 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 30104 | . . . . 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 8684 | . . 3 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ∈ (0[,]+∞)) |
10 | 1, 9 | sseldi 3819 | . 2 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ∈ ℝ*) |
11 | infxrge0lb.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
12 | 6, 11 | sseldd 3822 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) |
13 | 1, 12 | sseldi 3819 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
14 | 5, 8 | inflb 8685 | . . 3 ⊢ (𝜑 → (𝐵 ∈ 𝐴 → ¬ 𝐵 < inf(𝐴, (0[,]+∞), < ))) |
15 | 11, 14 | mpd 15 | . 2 ⊢ (𝜑 → ¬ 𝐵 < inf(𝐴, (0[,]+∞), < )) |
16 | 10, 13, 15 | xrnltled 10447 | 1 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∈ wcel 2107 ∀wral 3090 ∃wrex 3091 ⊆ wss 3792 class class class wbr 4888 Or wor 5275 (class class class)co 6924 infcinf 8637 0cc0 10274 +∞cpnf 10410 ℝ*cxr 10412 < clt 10413 ≤ cle 10414 [,]cicc 12495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-po 5276 df-so 5277 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-sup 8638 df-inf 8639 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-icc 12499 |
This theorem is referenced by: (None) |
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