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Mirrors > Home > ILE Home > Th. List > infssuzcldc | GIF version |
Description: The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by Jim Kingdon, 20-Jan-2022.) |
Ref | Expression |
---|---|
infssuzledc.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
infssuzledc.s | ⊢ 𝑆 = {𝑛 ∈ (ℤ≥‘𝑀) ∣ 𝜓} |
infssuzledc.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
infssuzledc.dc | ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓) |
Ref | Expression |
---|---|
infssuzcldc | ⊢ (𝜑 → inf(𝑆, ℝ, < ) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infssuzledc.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | infssuzledc.s | . . . 4 ⊢ 𝑆 = {𝑛 ∈ (ℤ≥‘𝑀) ∣ 𝜓} | |
3 | infssuzledc.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
4 | infssuzledc.dc | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓) | |
5 | 1, 2, 3, 4 | infssuzex 11982 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝑆 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑤 ∈ 𝑆 𝑤 < 𝑦))) |
6 | ssrab2 3255 | . . . . . . 7 ⊢ {𝑛 ∈ (ℤ≥‘𝑀) ∣ 𝜓} ⊆ (ℤ≥‘𝑀) | |
7 | 2, 6 | eqsstri 3202 | . . . . . 6 ⊢ 𝑆 ⊆ (ℤ≥‘𝑀) |
8 | uzssz 9577 | . . . . . 6 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
9 | 7, 8 | sstri 3179 | . . . . 5 ⊢ 𝑆 ⊆ ℤ |
10 | zssre 9290 | . . . . 5 ⊢ ℤ ⊆ ℝ | |
11 | 9, 10 | sstri 3179 | . . . 4 ⊢ 𝑆 ⊆ ℝ |
12 | 11 | a1i 9 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℝ) |
13 | 5, 12 | infrenegsupex 9624 | . 2 ⊢ (𝜑 → inf(𝑆, ℝ, < ) = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < )) |
14 | 1, 2, 3, 4 | infssuzex 11982 | . . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝑆 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝑆 𝑧 < 𝑦))) |
15 | 14, 12 | infsupneg 9626 | . . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}𝑦 < 𝑧))) |
16 | negeq 8180 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑢 → -𝑤 = -𝑢) | |
17 | 16 | eleq1d 2258 | . . . . . . . . 9 ⊢ (𝑤 = 𝑢 → (-𝑤 ∈ 𝑆 ↔ -𝑢 ∈ 𝑆)) |
18 | 17 | elrab 2908 | . . . . . . . 8 ⊢ (𝑢 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆} ↔ (𝑢 ∈ ℝ ∧ -𝑢 ∈ 𝑆)) |
19 | 9 | sseli 3166 | . . . . . . . . . 10 ⊢ (-𝑢 ∈ 𝑆 → -𝑢 ∈ ℤ) |
20 | 19 | adantl 277 | . . . . . . . . 9 ⊢ ((𝑢 ∈ ℝ ∧ -𝑢 ∈ 𝑆) → -𝑢 ∈ ℤ) |
21 | simpl 109 | . . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℝ ∧ -𝑢 ∈ 𝑆) → 𝑢 ∈ ℝ) | |
22 | 21 | recnd 8016 | . . . . . . . . . 10 ⊢ ((𝑢 ∈ ℝ ∧ -𝑢 ∈ 𝑆) → 𝑢 ∈ ℂ) |
23 | znegclb 9316 | . . . . . . . . . 10 ⊢ (𝑢 ∈ ℂ → (𝑢 ∈ ℤ ↔ -𝑢 ∈ ℤ)) | |
24 | 22, 23 | syl 14 | . . . . . . . . 9 ⊢ ((𝑢 ∈ ℝ ∧ -𝑢 ∈ 𝑆) → (𝑢 ∈ ℤ ↔ -𝑢 ∈ ℤ)) |
25 | 20, 24 | mpbird 167 | . . . . . . . 8 ⊢ ((𝑢 ∈ ℝ ∧ -𝑢 ∈ 𝑆) → 𝑢 ∈ ℤ) |
26 | 18, 25 | sylbi 121 | . . . . . . 7 ⊢ (𝑢 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆} → 𝑢 ∈ ℤ) |
27 | 26 | ssriv 3174 | . . . . . 6 ⊢ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆} ⊆ ℤ |
28 | 27 | a1i 9 | . . . . 5 ⊢ (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆} ⊆ ℤ) |
29 | 15, 28 | suprzclex 9381 | . . . 4 ⊢ (𝜑 → sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < ) ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}) |
30 | nfrab1 2670 | . . . . . 6 ⊢ Ⅎ𝑤{𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆} | |
31 | nfcv 2332 | . . . . . 6 ⊢ Ⅎ𝑤ℝ | |
32 | nfcv 2332 | . . . . . 6 ⊢ Ⅎ𝑤 < | |
33 | 30, 31, 32 | nfsup 7021 | . . . . 5 ⊢ Ⅎ𝑤sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < ) |
34 | 33 | nfneg 8184 | . . . . . 6 ⊢ Ⅎ𝑤-sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < ) |
35 | 34 | nfel1 2343 | . . . . 5 ⊢ Ⅎ𝑤-sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < ) ∈ 𝑆 |
36 | negeq 8180 | . . . . . 6 ⊢ (𝑤 = sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < ) → -𝑤 = -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < )) | |
37 | 36 | eleq1d 2258 | . . . . 5 ⊢ (𝑤 = sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < ) → (-𝑤 ∈ 𝑆 ↔ -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < ) ∈ 𝑆)) |
38 | 33, 31, 35, 37 | elrabf 2906 | . . . 4 ⊢ (sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < ) ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆} ↔ (sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < ) ∈ ℝ ∧ -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < ) ∈ 𝑆)) |
39 | 29, 38 | sylib 122 | . . 3 ⊢ (𝜑 → (sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < ) ∈ ℝ ∧ -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < ) ∈ 𝑆)) |
40 | 39 | simprd 114 | . 2 ⊢ (𝜑 → -sup({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝑆}, ℝ, < ) ∈ 𝑆) |
41 | 13, 40 | eqeltrd 2266 | 1 ⊢ (𝜑 → inf(𝑆, ℝ, < ) ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 835 = wceq 1364 ∈ wcel 2160 {crab 2472 ⊆ wss 3144 ‘cfv 5235 (class class class)co 5896 supcsup 7011 infcinf 7012 ℂcc 7839 ℝcr 7840 < clt 8022 -cneg 8159 ℤcz 9283 ℤ≥cuz 9558 ...cfz 10038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-sup 7013 df-inf 7014 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-inn 8950 df-n0 9207 df-z 9284 df-uz 9559 df-fz 10039 df-fzo 10173 |
This theorem is referenced by: zsupssdc 11987 nnmindc 12067 lcmval 12095 lcmcllem 12099 odzcllem 12274 4sqlem13m 12435 4sqlem14 12436 4sqlem17 12439 4sqlem18 12440 |
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