Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ressiocsup | Structured version Visualization version GIF version |
Description: If the supremum belongs to a set of reals, the set is a subset of the unbounded below, right-closed interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
ressiocsup.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
ressiocsup.s | ⊢ 𝑆 = sup(𝐴, ℝ*, < ) |
ressiocsup.e | ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
ressiocsup.5 | ⊢ 𝐼 = (-∞(,]𝑆) |
Ref | Expression |
---|---|
ressiocsup | ⊢ (𝜑 → 𝐴 ⊆ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 10697 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -∞ ∈ ℝ*) |
3 | ressiocsup.s | . . . . . 6 ⊢ 𝑆 = sup(𝐴, ℝ*, < ) | |
4 | ressiocsup.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
5 | ressxr 10684 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
6 | 5 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℝ*) |
7 | 4, 6 | sstrd 3976 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
8 | 7 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ*) |
9 | 8 | supxrcld 41371 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
10 | 3, 9 | eqeltrid 2917 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ ℝ*) |
11 | 7 | sselda 3966 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
12 | 4 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ) |
13 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
14 | 12, 13 | sseldd 3967 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
15 | 14 | mnfltd 12518 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -∞ < 𝑥) |
16 | supxrub 12716 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐴, ℝ*, < )) | |
17 | 8, 13, 16 | syl2anc 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐴, ℝ*, < )) |
18 | 3 | a1i 11 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = sup(𝐴, ℝ*, < )) |
19 | 18 | eqcomd 2827 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℝ*, < ) = 𝑆) |
20 | 17, 19 | breqtrd 5091 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ 𝑆) |
21 | 2, 10, 11, 15, 20 | eliocd 41781 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (-∞(,]𝑆)) |
22 | ressiocsup.5 | . . . 4 ⊢ 𝐼 = (-∞(,]𝑆) | |
23 | 21, 22 | eleqtrrdi 2924 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐼) |
24 | 23 | ralrimiva 3182 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐼) |
25 | dfss3 3955 | . 2 ⊢ (𝐴 ⊆ 𝐼 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐼) | |
26 | 24, 25 | sylibr 236 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 class class class wbr 5065 (class class class)co 7155 supcsup 8903 ℝcr 10535 -∞cmnf 10672 ℝ*cxr 10673 < clt 10674 ≤ cle 10675 (,]cioc 12738 |
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-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-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-ioc 12742 |
This theorem is referenced by: pimdecfgtioc 42992 pimincfltioc 42993 |
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