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Mirrors > Home > ILE Home > Th. List > elfzo1 | GIF version |
Description: Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.) |
Ref | Expression |
---|---|
elfzo1 | ⊢ (𝑁 ∈ (1..^𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzossnn 10114 | . . . 4 ⊢ (1..^𝑀) ⊆ ℕ | |
2 | 1 | sseli 3133 | . . 3 ⊢ (𝑁 ∈ (1..^𝑀) → 𝑁 ∈ ℕ) |
3 | elfzouz2 10086 | . . . 4 ⊢ (𝑁 ∈ (1..^𝑀) → 𝑀 ∈ (ℤ≥‘𝑁)) | |
4 | eluznn 9529 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ) | |
5 | 2, 3, 4 | syl2anc 409 | . . 3 ⊢ (𝑁 ∈ (1..^𝑀) → 𝑀 ∈ ℕ) |
6 | elfzolt2 10081 | . . 3 ⊢ (𝑁 ∈ (1..^𝑀) → 𝑁 < 𝑀) | |
7 | 2, 5, 6 | 3jca 1166 | . 2 ⊢ (𝑁 ∈ (1..^𝑀) → (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀)) |
8 | nnuz 9492 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
9 | 8 | eqimssi 3193 | . . . . 5 ⊢ ℕ ⊆ (ℤ≥‘1) |
10 | 9 | sseli 3133 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1)) |
11 | nnz 9201 | . . . 4 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
12 | id 19 | . . . 4 ⊢ (𝑁 < 𝑀 → 𝑁 < 𝑀) | |
13 | 10, 11, 12 | 3anim123i 1173 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀) → (𝑁 ∈ (ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑁 < 𝑀)) |
14 | elfzo2 10075 | . . 3 ⊢ (𝑁 ∈ (1..^𝑀) ↔ (𝑁 ∈ (ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑁 < 𝑀)) | |
15 | 13, 14 | sylibr 133 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀) → 𝑁 ∈ (1..^𝑀)) |
16 | 7, 15 | impbii 125 | 1 ⊢ (𝑁 ∈ (1..^𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∧ w3a 967 ∈ wcel 2135 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 1c1 7745 < clt 7924 ℕcn 8848 ℤcz 9182 ℤ≥cuz 9457 ..^cfzo 10067 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 df-fzo 10068 |
This theorem is referenced by: modfzo0difsn 10320 modsumfzodifsn 10321 eulerthlema 12141 modprm0 12165 nconstwlpolemgt0 13783 |
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