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Mirrors > Home > ILE Home > Th. List > nnminle | GIF version |
Description: The infimum of a decidable subset of the natural numbers is less than an element of the set. The infimum is also a minimum as shown at nnmindc 11989. (Contributed by Jim Kingdon, 26-Sep-2024.) |
Ref | Expression |
---|---|
nnminle | ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss5 3332 | . . . . . 6 ⊢ (𝐴 ⊆ ℕ ↔ 𝐴 = (ℕ ∩ 𝐴)) | |
2 | 1 | biimpi 119 | . . . . 5 ⊢ (𝐴 ⊆ ℕ → 𝐴 = (ℕ ∩ 𝐴)) |
3 | nnuz 9522 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
4 | 3 | ineq1i 3324 | . . . . . 6 ⊢ (ℕ ∩ 𝐴) = ((ℤ≥‘1) ∩ 𝐴) |
5 | dfin5 3128 | . . . . . 6 ⊢ ((ℤ≥‘1) ∩ 𝐴) = {𝑛 ∈ (ℤ≥‘1) ∣ 𝑛 ∈ 𝐴} | |
6 | 4, 5 | eqtri 2191 | . . . . 5 ⊢ (ℕ ∩ 𝐴) = {𝑛 ∈ (ℤ≥‘1) ∣ 𝑛 ∈ 𝐴} |
7 | 2, 6 | eqtrdi 2219 | . . . 4 ⊢ (𝐴 ⊆ ℕ → 𝐴 = {𝑛 ∈ (ℤ≥‘1) ∣ 𝑛 ∈ 𝐴}) |
8 | 7 | 3ad2ant1 1013 | . . 3 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 = {𝑛 ∈ (ℤ≥‘1) ∣ 𝑛 ∈ 𝐴}) |
9 | 8 | infeq1d 6989 | . 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → inf(𝐴, ℝ, < ) = inf({𝑛 ∈ (ℤ≥‘1) ∣ 𝑛 ∈ 𝐴}, ℝ, < )) |
10 | 1zzd 9239 | . . 3 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → 1 ∈ ℤ) | |
11 | eqid 2170 | . . 3 ⊢ {𝑛 ∈ (ℤ≥‘1) ∣ 𝑛 ∈ 𝐴} = {𝑛 ∈ (ℤ≥‘1) ∣ 𝑛 ∈ 𝐴} | |
12 | simp3 994 | . . . 4 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) | |
13 | 12, 8 | eleqtrd 2249 | . . 3 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ {𝑛 ∈ (ℤ≥‘1) ∣ 𝑛 ∈ 𝐴}) |
14 | eleq1w 2231 | . . . . 5 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ 𝐴 ↔ 𝑛 ∈ 𝐴)) | |
15 | 14 | dcbid 833 | . . . 4 ⊢ (𝑥 = 𝑛 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑛 ∈ 𝐴)) |
16 | simpl2 996 | . . . 4 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑛 ∈ (1...𝐵)) → ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴) | |
17 | elfznn 10010 | . . . . 5 ⊢ (𝑛 ∈ (1...𝐵) → 𝑛 ∈ ℕ) | |
18 | 17 | adantl 275 | . . . 4 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑛 ∈ (1...𝐵)) → 𝑛 ∈ ℕ) |
19 | 15, 16, 18 | rspcdva 2839 | . . 3 ⊢ (((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑛 ∈ (1...𝐵)) → DECID 𝑛 ∈ 𝐴) |
20 | 10, 11, 13, 19 | infssuzledc 11905 | . 2 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝑛 ∈ (ℤ≥‘1) ∣ 𝑛 ∈ 𝐴}, ℝ, < ) ≤ 𝐵) |
21 | 9, 20 | eqbrtrd 4011 | 1 ⊢ ((𝐴 ⊆ ℕ ∧ ∀𝑥 ∈ ℕ DECID 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 DECID wdc 829 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ∀wral 2448 {crab 2452 ∩ cin 3120 ⊆ wss 3121 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 infcinf 6960 ℝcr 7773 1c1 7775 < clt 7954 ≤ cle 7955 ℕcn 8878 ℤ≥cuz 9487 ...cfz 9965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 df-fzo 10099 |
This theorem is referenced by: nnwodc 11991 |
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