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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 13467 | . . . . 5 ⊢ (𝐴[,]𝐵) ⊆ ℝ* | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ*) |
6 | 3, 5 | sstrd 4006 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℝ*) |
7 | infxrcl 13372 | . . 3 ⊢ (𝑆 ⊆ ℝ* → inf(𝑆, ℝ*, < ) ∈ ℝ*) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → inf(𝑆, ℝ*, < ) ∈ ℝ*) |
9 | 1 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ*) |
10 | 2 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ ℝ*) |
11 | 3 | sselda 3995 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (𝐴[,]𝐵)) |
12 | iccgelb 13440 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥) | |
13 | 9, 10, 11, 12 | syl3anc 1370 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ≤ 𝑥) |
14 | 13 | ralrimiva 3144 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐴 ≤ 𝑥) |
15 | infxrgelb 13374 | . . . 4 ⊢ ((𝑆 ⊆ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ inf(𝑆, ℝ*, < ) ↔ ∀𝑥 ∈ 𝑆 𝐴 ≤ 𝑥)) | |
16 | 6, 1, 15 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐴 ≤ inf(𝑆, ℝ*, < ) ↔ ∀𝑥 ∈ 𝑆 𝐴 ≤ 𝑥)) |
17 | 14, 16 | mpbird 257 | . 2 ⊢ (𝜑 → 𝐴 ≤ inf(𝑆, ℝ*, < )) |
18 | inficc.n0 | . . . 4 ⊢ (𝜑 → 𝑆 ≠ ∅) | |
19 | n0 4359 | . . . 4 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆) | |
20 | 18, 19 | sylib 218 | . . 3 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝑆) |
21 | 8 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ∈ ℝ*) |
22 | 4, 11 | sselid 3993 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℝ*) |
23 | 6 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ ℝ*) |
24 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆) | |
25 | infxrlb 13373 | . . . . . . 7 ⊢ ((𝑆 ⊆ ℝ* ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ≤ 𝑥) | |
26 | 23, 24, 25 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ≤ 𝑥) |
27 | iccleub 13439 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵) | |
28 | 9, 10, 11, 27 | syl3anc 1370 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ≤ 𝐵) |
29 | 21, 22, 10, 26, 28 | xrletrd 13201 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ≤ 𝐵) |
30 | 29 | ex 412 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑆 → inf(𝑆, ℝ*, < ) ≤ 𝐵)) |
31 | 30 | exlimdv 1931 | . . 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 45480 | 1 ⊢ (𝜑 → inf(𝑆, ℝ*, < ) ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 (class class class)co 7431 infcinf 9479 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: ovnf 46519 |
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