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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > infxrge0gelb | Structured version Visualization version GIF version |
Description: The infimum of a set of nonnegative extended reals is greater than or equal to a 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 |
---|---|
infxrge0gelb | ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infxrge0glb.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞)) | |
2 | infxrge0glb.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) | |
3 | 1, 2 | infxrge0glb 32556 | . . 3 ⊢ (𝜑 → (inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)) |
4 | 3 | notbid 317 | . 2 ⊢ (𝜑 → (¬ inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)) |
5 | iccssxr 13447 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
6 | 5, 2 | sselid 3980 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
7 | xrltso 13160 | . . . . . . 7 ⊢ < Or ℝ* | |
8 | soss 5614 | . . . . . . 7 ⊢ ((0[,]+∞) ⊆ ℝ* → ( < Or ℝ* → < Or (0[,]+∞))) | |
9 | 5, 7, 8 | mp2 9 | . . . . . 6 ⊢ < Or (0[,]+∞) |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝜑 → < Or (0[,]+∞)) |
11 | xrge0infss 32551 | . . . . . 6 ⊢ (𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) | |
12 | 1, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))) |
13 | 10, 12 | infcl 9519 | . . . 4 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ∈ (0[,]+∞)) |
14 | 5, 13 | sselid 3980 | . . 3 ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ∈ ℝ*) |
15 | 6, 14 | xrlenltd 11318 | . 2 ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ¬ inf(𝐴, (0[,]+∞), < ) < 𝐵)) |
16 | 6 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
17 | 1, 5 | sstrdi 3994 | . . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
18 | 17 | sselda 3982 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
19 | 16, 18 | xrlenltd 11318 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝑥 ↔ ¬ 𝑥 < 𝐵)) |
20 | 19 | ralbidva 3173 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵)) |
21 | ralnex 3069 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵) | |
22 | 20, 21 | bitrdi 286 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)) |
23 | 4, 15, 22 | 3bitr4d 310 | 1 ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 ∀wral 3058 ∃wrex 3067 ⊆ wss 3949 class class class wbr 5152 Or wor 5593 (class class class)co 7426 infcinf 9472 0cc0 11146 +∞cpnf 11283 ℝ*cxr 11285 < clt 11286 ≤ cle 11287 [,]cicc 13367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 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 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-icc 13371 |
This theorem is referenced by: (None) |
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