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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 11303 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
4 | 3 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℝ*) |
5 | 2, 4 | sstrd 4006 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ*) |
6 | 5 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ ℝ*) |
7 | 6 | infxrcld 45339 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ∈ ℝ*) |
8 | 1, 7 | eqeltrid 2843 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ ℝ*) |
9 | pnfxr 11313 | . . . . . 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 3996 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
14 | 5 | sselda 3995 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ*) |
15 | infxrlb 13373 | . . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝑥 ∈ 𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝑥) | |
16 | 6, 12, 15 | syl2anc 584 | . . . . . . 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 2741 | . . . . . . . . . . . 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 2839 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑆) → 𝑆 ∈ 𝐴) |
23 | 22 | adantll 714 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝑆) → 𝑆 ∈ 𝐴) |
24 | ressiooinf.n | . . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴) | |
25 | 24 | ad2antrr 726 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝑆) → ¬ 𝑆 ∈ 𝐴) |
26 | 23, 25 | pm2.65da 817 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 = 𝑆) |
27 | 26 | neqned 2945 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ 𝑆) |
28 | 27 | necomd 2994 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ≠ 𝑥) |
29 | 8, 14, 17, 28 | xrleneltd 45273 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 < 𝑥) |
30 | 13 | ltpnfd 13161 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 < +∞) |
31 | 8, 10, 13, 29, 30 | eliood 45451 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝑆(,)+∞)) |
32 | ressiooinf.i | . . . 4 ⊢ 𝐼 = (𝑆(,)+∞) | |
33 | 31, 32 | eleqtrrdi 2850 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐼) |
34 | 33 | ralrimiva 3144 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐼) |
35 | dfss3 3984 | . 2 ⊢ (𝐴 ⊆ 𝐼 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐼) | |
36 | 34, 35 | sylibr 234 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 class class class wbr 5148 (class class class)co 7431 infcinf 9479 ℝcr 11152 +∞cpnf 11290 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 (,)cioo 13384 |
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-iun 4998 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-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-ioo 13388 |
This theorem is referenced by: (None) |
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