Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > inficc | Structured version Visualization version GIF version |
Description: The infimum of a nonempty set, included in a closed interval, is a member of the interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
inficc.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
inficc.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
inficc.s | ⊢ (𝜑 → 𝑆 ⊆ (𝐴[,]𝐵)) |
inficc.n0 | ⊢ (𝜑 → 𝑆 ≠ ∅) |
Ref | Expression |
---|---|
inficc | ⊢ (𝜑 → inf(𝑆, ℝ*, < ) ∈ (𝐴[,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inficc.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | inficc.b | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | inficc.s | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝐴[,]𝐵)) | |
4 | iccssxr 12818 | . . . . 5 ⊢ (𝐴[,]𝐵) ⊆ ℝ* | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ*) |
6 | 3, 5 | sstrd 3976 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℝ*) |
7 | infxrcl 12725 | . . 3 ⊢ (𝑆 ⊆ ℝ* → inf(𝑆, ℝ*, < ) ∈ ℝ*) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → inf(𝑆, ℝ*, < ) ∈ ℝ*) |
9 | 1 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ*) |
10 | 2 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ ℝ*) |
11 | 3 | sselda 3966 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (𝐴[,]𝐵)) |
12 | iccgelb 12792 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥) | |
13 | 9, 10, 11, 12 | syl3anc 1367 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ≤ 𝑥) |
14 | 13 | ralrimiva 3182 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐴 ≤ 𝑥) |
15 | infxrgelb 12727 | . . . 4 ⊢ ((𝑆 ⊆ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ inf(𝑆, ℝ*, < ) ↔ ∀𝑥 ∈ 𝑆 𝐴 ≤ 𝑥)) | |
16 | 6, 1, 15 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐴 ≤ inf(𝑆, ℝ*, < ) ↔ ∀𝑥 ∈ 𝑆 𝐴 ≤ 𝑥)) |
17 | 14, 16 | mpbird 259 | . 2 ⊢ (𝜑 → 𝐴 ≤ inf(𝑆, ℝ*, < )) |
18 | inficc.n0 | . . . 4 ⊢ (𝜑 → 𝑆 ≠ ∅) | |
19 | n0 4309 | . . . 4 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆) | |
20 | 18, 19 | sylib 220 | . . 3 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝑆) |
21 | 8 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ∈ ℝ*) |
22 | 4, 11 | sseldi 3964 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℝ*) |
23 | 6 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ ℝ*) |
24 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆) | |
25 | infxrlb 12726 | . . . . . . 7 ⊢ ((𝑆 ⊆ ℝ* ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ≤ 𝑥) | |
26 | 23, 24, 25 | syl2anc 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ≤ 𝑥) |
27 | iccleub 12791 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵) | |
28 | 9, 10, 11, 27 | syl3anc 1367 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ≤ 𝐵) |
29 | 21, 22, 10, 26, 28 | xrletrd 12554 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ≤ 𝐵) |
30 | 29 | ex 415 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑆 → inf(𝑆, ℝ*, < ) ≤ 𝐵)) |
31 | 30 | exlimdv 1930 | . . 3 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝑆 → inf(𝑆, ℝ*, < ) ≤ 𝐵)) |
32 | 20, 31 | mpd 15 | . 2 ⊢ (𝜑 → inf(𝑆, ℝ*, < ) ≤ 𝐵) |
33 | 1, 2, 8, 17, 32 | eliccxrd 41801 | 1 ⊢ (𝜑 → inf(𝑆, ℝ*, < ) ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ⊆ wss 3935 ∅c0 4290 class class class wbr 5065 (class class class)co 7155 infcinf 8904 ℝ*cxr 10673 < clt 10674 ≤ cle 10675 [,]cicc 12740 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-icc 12744 |
This theorem is referenced by: ovnf 42844 |
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