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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 13405 | . . . . 5 ⊢ (𝐴[,]𝐵) ⊆ ℝ* | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ*) |
6 | 3, 5 | sstrd 3985 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℝ*) |
7 | infxrcl 13310 | . . 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 3975 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (𝐴[,]𝐵)) |
12 | iccgelb 13378 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥) | |
13 | 9, 10, 11, 12 | syl3anc 1368 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ≤ 𝑥) |
14 | 13 | ralrimiva 3138 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐴 ≤ 𝑥) |
15 | infxrgelb 13312 | . . . 4 ⊢ ((𝑆 ⊆ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ inf(𝑆, ℝ*, < ) ↔ ∀𝑥 ∈ 𝑆 𝐴 ≤ 𝑥)) | |
16 | 6, 1, 15 | syl2anc 583 | . . 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 4339 | . . . 4 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆) | |
20 | 18, 19 | sylib 217 | . . 3 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝑆) |
21 | 8 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ∈ ℝ*) |
22 | 4, 11 | sselid 3973 | . . . . . 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 13311 | . . . . . . 7 ⊢ ((𝑆 ⊆ ℝ* ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ≤ 𝑥) | |
26 | 23, 24, 25 | syl2anc 583 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ≤ 𝑥) |
27 | iccleub 13377 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵) | |
28 | 9, 10, 11, 27 | syl3anc 1368 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ≤ 𝐵) |
29 | 21, 22, 10, 26, 28 | xrletrd 13139 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ≤ 𝐵) |
30 | 29 | ex 412 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑆 → inf(𝑆, ℝ*, < ) ≤ 𝐵)) |
31 | 30 | exlimdv 1928 | . . 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 44750 | 1 ⊢ (𝜑 → inf(𝑆, ℝ*, < ) ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∃wex 1773 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ⊆ wss 3941 ∅c0 4315 class class class wbr 5139 (class class class)co 7402 infcinf 9433 ℝ*cxr 11245 < clt 11246 ≤ cle 11247 [,]cicc 13325 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-inf 9435 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-icc 13329 |
This theorem is referenced by: ovnf 45789 |
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