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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 11289 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
4 | 3 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℝ*) |
5 | 2, 4 | sstrd 3990 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
6 | 5 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ*) |
7 | 6 | infxrcld 44771 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
8 | 1, 7 | eqeltrid 2833 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ ℝ*) |
9 | pnfxr 11299 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
10 | 9 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → +∞ ∈ ℝ*) |
11 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ) |
12 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
13 | 11, 12 | sseldd 3981 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
14 | 5 | sselda 3980 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
15 | infxrlb 13346 | . . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝑥) | |
16 | 6, 12, 15 | syl2anc 583 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝑥) |
17 | 1, 16 | eqbrtrid 5183 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ≤ 𝑥) |
18 | id 22 | . . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑆 → 𝑥 = 𝑆) | |
19 | 18 | eqcomd 2734 | . . . . . . . . . . . 12 ⊢ (𝑥 = 𝑆 → 𝑆 = 𝑥) |
20 | 19 | adantl 481 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑆) → 𝑆 = 𝑥) |
21 | simpl 482 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑆) → 𝑥 ∈ 𝐴) | |
22 | 20, 21 | eqeltrd 2829 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑆) → 𝑆 ∈ 𝐴) |
23 | 22 | adantll 713 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝑆) → 𝑆 ∈ 𝐴) |
24 | ressiooinf.n | . . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴) | |
25 | 24 | ad2antrr 725 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝑆) → ¬ 𝑆 ∈ 𝐴) |
26 | 23, 25 | pm2.65da 816 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 = 𝑆) |
27 | 26 | neqned 2944 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ 𝑆) |
28 | 27 | necomd 2993 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ≠ 𝑥) |
29 | 8, 14, 17, 28 | xrleneltd 44705 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 < 𝑥) |
30 | 13 | ltpnfd 13134 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 < +∞) |
31 | 8, 10, 13, 29, 30 | eliood 44883 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝑆(,)+∞)) |
32 | ressiooinf.i | . . . 4 ⊢ 𝐼 = (𝑆(,)+∞) | |
33 | 31, 32 | eleqtrrdi 2840 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐼) |
34 | 33 | ralrimiva 3143 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐼) |
35 | dfss3 3968 | . 2 ⊢ (𝐴 ⊆ 𝐼 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐼) | |
36 | 34, 35 | sylibr 233 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ⊆ wss 3947 class class class wbr 5148 (class class class)co 7420 infcinf 9465 ℝcr 11138 +∞cpnf 11276 ℝ*cxr 11278 < clt 11279 ≤ cle 11280 (,)cioo 13357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-ioo 13361 |
This theorem is referenced by: (None) |
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