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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 13348 | . . . . 5 ⊢ (𝐴[,]𝐵) ⊆ ℝ* | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ*) |
6 | 3, 5 | sstrd 3955 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℝ*) |
7 | infxrcl 13253 | . . 3 ⊢ (𝑆 ⊆ ℝ* → inf(𝑆, ℝ*, < ) ∈ ℝ*) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → inf(𝑆, ℝ*, < ) ∈ ℝ*) |
9 | 1 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ*) |
10 | 2 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ ℝ*) |
11 | 3 | sselda 3945 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (𝐴[,]𝐵)) |
12 | iccgelb 13321 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥) | |
13 | 9, 10, 11, 12 | syl3anc 1372 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ≤ 𝑥) |
14 | 13 | ralrimiva 3144 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐴 ≤ 𝑥) |
15 | infxrgelb 13255 | . . . 4 ⊢ ((𝑆 ⊆ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ inf(𝑆, ℝ*, < ) ↔ ∀𝑥 ∈ 𝑆 𝐴 ≤ 𝑥)) | |
16 | 6, 1, 15 | syl2anc 585 | . . 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 4307 | . . . 4 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆) | |
20 | 18, 19 | sylib 217 | . . 3 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝑆) |
21 | 8 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ∈ ℝ*) |
22 | 4, 11 | sselid 3943 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ℝ*) |
23 | 6 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ ℝ*) |
24 | simpr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆) | |
25 | infxrlb 13254 | . . . . . . 7 ⊢ ((𝑆 ⊆ ℝ* ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ≤ 𝑥) | |
26 | 23, 24, 25 | syl2anc 585 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ≤ 𝑥) |
27 | iccleub 13320 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵) | |
28 | 9, 10, 11, 27 | syl3anc 1372 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ≤ 𝐵) |
29 | 21, 22, 10, 26, 28 | xrletrd 13082 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → inf(𝑆, ℝ*, < ) ≤ 𝐵) |
30 | 29 | ex 414 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑆 → inf(𝑆, ℝ*, < ) ≤ 𝐵)) |
31 | 30 | exlimdv 1937 | . . 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 43772 | 1 ⊢ (𝜑 → inf(𝑆, ℝ*, < ) ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∃wex 1782 ∈ wcel 2107 ≠ wne 2944 ∀wral 3065 ⊆ wss 3911 ∅c0 4283 class class class wbr 5106 (class class class)co 7358 infcinf 9378 ℝ*cxr 11189 < clt 11190 ≤ cle 11191 [,]cicc 13268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-sup 9379 df-inf 9380 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-icc 13272 |
This theorem is referenced by: ovnf 44811 |
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