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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ressiooinf | Structured version Visualization version GIF version |
Description: If the infimum does not belong to a set of reals, the set is a subset of the unbounded above, left-open interval, with lower bound equal to the infimum. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
ressiooinf.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
ressiooinf.s | ⊢ 𝑆 = inf(𝐴, ℝ*, < ) |
ressiooinf.n | ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴) |
ressiooinf.i | ⊢ 𝐼 = (𝑆(,)+∞) |
Ref | Expression |
---|---|
ressiooinf | ⊢ (𝜑 → 𝐴 ⊆ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressiooinf.s | . . . . . 6 ⊢ 𝑆 = inf(𝐴, ℝ*, < ) | |
2 | ressiooinf.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
3 | ressxr 10420 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
4 | 3 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℝ*) |
5 | 2, 4 | sstrd 3830 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
6 | 5 | adantr 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ*) |
7 | 6 | infxrcld 40500 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
8 | 1, 7 | syl5eqel 2862 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ ℝ*) |
9 | pnfxr 10430 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
10 | 9 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → +∞ ∈ ℝ*) |
11 | 2 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ) |
12 | simpr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
13 | 11, 12 | sseldd 3821 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
14 | 5 | sselda 3820 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
15 | infxrlb 12476 | . . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝑥) | |
16 | 6, 12, 15 | syl2anc 579 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝑥) |
17 | 1, 16 | syl5eqbr 4921 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ≤ 𝑥) |
18 | id 22 | . . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑆 → 𝑥 = 𝑆) | |
19 | 18 | eqcomd 2783 | . . . . . . . . . . . 12 ⊢ (𝑥 = 𝑆 → 𝑆 = 𝑥) |
20 | 19 | adantl 475 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑆) → 𝑆 = 𝑥) |
21 | simpl 476 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑆) → 𝑥 ∈ 𝐴) | |
22 | 20, 21 | eqeltrd 2858 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑆) → 𝑆 ∈ 𝐴) |
23 | 22 | adantll 704 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝑆) → 𝑆 ∈ 𝐴) |
24 | ressiooinf.n | . . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴) | |
25 | 24 | ad2antrr 716 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝑆) → ¬ 𝑆 ∈ 𝐴) |
26 | 23, 25 | pm2.65da 807 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 = 𝑆) |
27 | 26 | neqned 2975 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ 𝑆) |
28 | 27 | necomd 3023 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ≠ 𝑥) |
29 | 8, 14, 17, 28 | xrleneltd 40429 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 < 𝑥) |
30 | 13 | ltpnfd 12266 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 < +∞) |
31 | 8, 10, 13, 29, 30 | eliood 40614 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝑆(,)+∞)) |
32 | ressiooinf.i | . . . 4 ⊢ 𝐼 = (𝑆(,)+∞) | |
33 | 31, 32 | syl6eleqr 2869 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐼) |
34 | 33 | ralrimiva 3147 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐼) |
35 | dfss3 3809 | . 2 ⊢ (𝐴 ⊆ 𝐼 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐼) | |
36 | 34, 35 | sylibr 226 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∀wral 3089 ⊆ wss 3791 class class class wbr 4886 (class class class)co 6922 infcinf 8635 ℝcr 10271 +∞cpnf 10408 ℝ*cxr 10410 < clt 10411 ≤ cle 10412 (,)cioo 12487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-ioo 12491 |
This theorem is referenced by: (None) |
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