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Mirrors > Home > ILE Home > Th. List > infssuzledc | GIF version |
Description: The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. (Contributed by Jim Kingdon, 13-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 |
---|---|
infssuzledc | ⊢ (𝜑 → inf(𝑆, ℝ, < ) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lttri3 7767 | . . . 4 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 = 𝑏 ↔ (¬ 𝑎 < 𝑏 ∧ ¬ 𝑏 < 𝑎))) | |
2 | 1 | adantl 273 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ)) → (𝑎 = 𝑏 ↔ (¬ 𝑎 < 𝑏 ∧ ¬ 𝑏 < 𝑎))) |
3 | infssuzledc.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
4 | infssuzledc.s | . . . 4 ⊢ 𝑆 = {𝑛 ∈ (ℤ≥‘𝑀) ∣ 𝜓} | |
5 | infssuzledc.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
6 | infssuzledc.dc | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝐴)) → DECID 𝜓) | |
7 | 3, 4, 5, 6 | infssuzex 11490 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝑆 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝑆 𝑧 < 𝑦))) |
8 | 2, 7 | infclti 6862 | . 2 ⊢ (𝜑 → inf(𝑆, ℝ, < ) ∈ ℝ) |
9 | elrabi 2806 | . . . 4 ⊢ (𝐴 ∈ {𝑛 ∈ (ℤ≥‘𝑀) ∣ 𝜓} → 𝐴 ∈ (ℤ≥‘𝑀)) | |
10 | 9, 4 | eleq2s 2209 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ (ℤ≥‘𝑀)) |
11 | eluzelre 9238 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘𝑀) → 𝐴 ∈ ℝ) | |
12 | 5, 10, 11 | 3syl 17 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
13 | 2, 7 | inflbti 6863 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑆 → ¬ 𝐴 < inf(𝑆, ℝ, < ))) |
14 | 5, 13 | mpd 13 | . 2 ⊢ (𝜑 → ¬ 𝐴 < inf(𝑆, ℝ, < )) |
15 | 8, 12, 14 | nltled 7806 | 1 ⊢ (𝜑 → inf(𝑆, ℝ, < ) ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 DECID wdc 802 = wceq 1314 ∈ wcel 1463 {crab 2394 class class class wbr 3895 ‘cfv 5081 (class class class)co 5728 infcinf 6822 ℝcr 7546 < clt 7724 ≤ cle 7725 ℤcz 8958 ℤ≥cuz 9228 ...cfz 9683 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-addcom 7645 ax-addass 7647 ax-distr 7649 ax-i2m1 7650 ax-0lt1 7651 ax-0id 7653 ax-rnegex 7654 ax-cnre 7656 ax-pre-ltirr 7657 ax-pre-ltwlin 7658 ax-pre-lttrn 7659 ax-pre-apti 7660 ax-pre-ltadd 7661 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-reu 2397 df-rmo 2398 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-1st 5992 df-2nd 5993 df-sup 6823 df-inf 6824 df-pnf 7726 df-mnf 7727 df-xr 7728 df-ltxr 7729 df-le 7730 df-sub 7858 df-neg 7859 df-inn 8631 df-n0 8882 df-z 8959 df-uz 9229 df-fz 9684 df-fzo 9813 |
This theorem is referenced by: lcmledvds 11597 |
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