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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 11316 | . . . . . 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 11303 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
6 | 5 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℝ*) |
7 | 4, 6 | sstrd 4006 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
8 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ*) |
9 | 8 | supxrcld 45047 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
10 | 3, 9 | eqeltrid 2843 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ ℝ*) |
11 | 7 | sselda 3995 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
12 | 4 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ) |
13 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
14 | 12, 13 | sseldd 3996 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
15 | 14 | mnfltd 13164 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -∞ < 𝑥) |
16 | supxrub 13363 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐴, ℝ*, < )) | |
17 | 8, 13, 16 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐴, ℝ*, < )) |
18 | 3 | a1i 11 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 = sup(𝐴, ℝ*, < )) |
19 | 18 | eqcomd 2741 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(𝐴, ℝ*, < ) = 𝑆) |
20 | 17, 19 | breqtrd 5174 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ 𝑆) |
21 | 2, 10, 11, 15, 20 | eliocd 45460 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (-∞(,]𝑆)) |
22 | ressiocsup.5 | . . . 4 ⊢ 𝐼 = (-∞(,]𝑆) | |
23 | 21, 22 | eleqtrrdi 2850 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐼) |
24 | 23 | ralrimiva 3144 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐼) |
25 | dfss3 3984 | . 2 ⊢ (𝐴 ⊆ 𝐼 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐼) | |
26 | 24, 25 | sylibr 234 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 class class class wbr 5148 (class class class)co 7431 supcsup 9478 ℝcr 11152 -∞cmnf 11291 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 (,]cioc 13385 |
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-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-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-ioc 13389 |
This theorem is referenced by: pimdecfgtioc 46671 pimincfltioc 46672 |
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