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Mirrors > Home > MPE Home > Th. List > infregelb | Structured version Visualization version GIF version |
Description: Any lower bound of a nonempty set of real numbers is less than or equal to its infimum. (Contributed by Jeff Hankins, 1-Sep-2013.) (Revised by AV, 4-Sep-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
infregelb | ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltso 10156 | . . . . . 6 ⊢ < Or ℝ | |
2 | 1 | a1i 11 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → < Or ℝ) |
3 | infm3 11020 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑤 ∈ 𝐴 𝑤 < 𝑦))) | |
4 | simp1 1081 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → 𝐴 ⊆ ℝ) | |
5 | 2, 3, 4 | infglbb 8438 | . . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (inf(𝐴, ℝ, < ) < 𝐵 ↔ ∃𝑤 ∈ 𝐴 𝑤 < 𝐵)) |
6 | 5 | notbid 307 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (¬ inf(𝐴, ℝ, < ) < 𝐵 ↔ ¬ ∃𝑤 ∈ 𝐴 𝑤 < 𝐵)) |
7 | infrecl 11043 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → inf(𝐴, ℝ, < ) ∈ ℝ) | |
8 | 7 | anim1i 591 | . . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (inf(𝐴, ℝ, < ) ∈ ℝ ∧ 𝐵 ∈ ℝ)) |
9 | 8 | ancomd 466 | . . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (𝐵 ∈ ℝ ∧ inf(𝐴, ℝ, < ) ∈ ℝ)) |
10 | lenlt 10154 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ inf(𝐴, ℝ, < ) ∈ ℝ) → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ¬ inf(𝐴, ℝ, < ) < 𝐵)) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ¬ inf(𝐴, ℝ, < ) < 𝐵)) |
12 | simplr 807 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑤 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
13 | ssel 3630 | . . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ → (𝑤 ∈ 𝐴 → 𝑤 ∈ ℝ)) | |
14 | 13 | adantr 480 | . . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (𝑤 ∈ 𝐴 → 𝑤 ∈ ℝ)) |
15 | 14 | imp 444 | . . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ) |
16 | 12, 15 | lenltd 10221 | . . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑤 ∈ 𝐴) → (𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵)) |
17 | 16 | ralbidva 3014 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (∀𝑤 ∈ 𝐴 𝐵 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝑤 < 𝐵)) |
18 | 17 | 3ad2antl1 1243 | . . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (∀𝑤 ∈ 𝐴 𝐵 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝑤 < 𝐵)) |
19 | ralnex 3021 | . . . 4 ⊢ (∀𝑤 ∈ 𝐴 ¬ 𝑤 < 𝐵 ↔ ¬ ∃𝑤 ∈ 𝐴 𝑤 < 𝐵) | |
20 | 18, 19 | syl6bb 276 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (∀𝑤 ∈ 𝐴 𝐵 ≤ 𝑤 ↔ ¬ ∃𝑤 ∈ 𝐴 𝑤 < 𝐵)) |
21 | 6, 11, 20 | 3bitr4d 300 | . 2 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑤 ∈ 𝐴 𝐵 ≤ 𝑤)) |
22 | breq2 4689 | . . 3 ⊢ (𝑤 = 𝑧 → (𝐵 ≤ 𝑤 ↔ 𝐵 ≤ 𝑧)) | |
23 | 22 | cbvralv 3201 | . 2 ⊢ (∀𝑤 ∈ 𝐴 𝐵 ≤ 𝑤 ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) |
24 | 21, 23 | syl6bb 276 | 1 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 ∈ wcel 2030 ≠ wne 2823 ∀wral 2941 ∃wrex 2942 ⊆ wss 3607 ∅c0 3948 class class class wbr 4685 Or wor 5063 infcinf 8388 ℝcr 9973 < clt 10112 ≤ cle 10113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-inf 8390 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 |
This theorem is referenced by: infxrre 12204 minveclem2 23243 minveclem3b 23245 minveclem4 23249 minveclem6 23251 pilem2 24251 pilem3 24252 pntlem3 25343 minvecolem2 27859 minvecolem4 27864 minvecolem5 27865 minvecolem6 27866 taupi 33299 infmrgelbi 37759 |
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