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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 13461 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | xrltso 13174 | . . . . . 6 ⊢ < Or ℝ* | |
3 | soss 5614 | . . . . . 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 32664 | . . . . 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 9531 | . . 3 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ∈ (0[,]+∞)) |
10 | 1, 9 | sselid 3977 | . 2 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ∈ ℝ*) |
11 | infxrge0lb.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
12 | 6, 11 | sseldd 3980 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) |
13 | 1, 12 | sselid 3977 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
14 | 5, 8 | inflb 9532 | . . 3 ⊢ (𝜑 → (𝐵 ∈ 𝐴 → ¬ 𝐵 < inf(𝐴, (0[,]+∞), < ))) |
15 | 11, 14 | mpd 15 | . 2 ⊢ (𝜑 → ¬ 𝐵 < inf(𝐴, (0[,]+∞), < )) |
16 | 10, 13, 15 | xrnltled 11332 | 1 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 ⊆ wss 3947 class class class wbr 5153 Or wor 5593 (class class class)co 7424 infcinf 9484 0cc0 11158 +∞cpnf 11295 ℝ*cxr 11297 < clt 11298 ≤ cle 11299 [,]cicc 13381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-inf 9486 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-icc 13385 |
This theorem is referenced by: (None) |
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