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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 13467 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | xrltso 13180 | . . . . . 6 ⊢ < Or ℝ* | |
3 | soss 5617 | . . . . . 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 32771 | . . . . 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 9526 | . . 3 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ∈ (0[,]+∞)) |
10 | 1, 9 | sselid 3993 | . 2 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ∈ ℝ*) |
11 | infxrge0lb.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
12 | 6, 11 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) |
13 | 1, 12 | sselid 3993 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
14 | 5, 8 | inflb 9527 | . . 3 ⊢ (𝜑 → (𝐵 ∈ 𝐴 → ¬ 𝐵 < inf(𝐴, (0[,]+∞), < ))) |
15 | 11, 14 | mpd 15 | . 2 ⊢ (𝜑 → ¬ 𝐵 < inf(𝐴, (0[,]+∞), < )) |
16 | 10, 13, 15 | xrnltled 11327 | 1 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 class class class wbr 5148 Or wor 5596 (class class class)co 7431 infcinf 9479 0cc0 11153 +∞cpnf 11290 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 [,]cicc 13387 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-icc 13391 |
This theorem is referenced by: (None) |
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