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Mirrors > Home > MPE Home > Th. List > elfzo1 | Structured version Visualization version 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 12812 | . . . 4 ⊢ (1..^𝑀) ⊆ ℕ | |
2 | 1 | sseli 3823 | . . 3 ⊢ (𝑁 ∈ (1..^𝑀) → 𝑁 ∈ ℕ) |
3 | elfzouz2 12779 | . . . 4 ⊢ (𝑁 ∈ (1..^𝑀) → 𝑀 ∈ (ℤ≥‘𝑁)) | |
4 | eluznn 12041 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ) | |
5 | 2, 3, 4 | syl2anc 579 | . . 3 ⊢ (𝑁 ∈ (1..^𝑀) → 𝑀 ∈ ℕ) |
6 | elfzolt2 12774 | . . 3 ⊢ (𝑁 ∈ (1..^𝑀) → 𝑁 < 𝑀) | |
7 | 2, 5, 6 | 3jca 1162 | . 2 ⊢ (𝑁 ∈ (1..^𝑀) → (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀)) |
8 | nnuz 12005 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
9 | 8 | eqimssi 3884 | . . . . 5 ⊢ ℕ ⊆ (ℤ≥‘1) |
10 | 9 | sseli 3823 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1)) |
11 | nnz 11727 | . . . 4 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
12 | id 22 | . . . 4 ⊢ (𝑁 < 𝑀 → 𝑁 < 𝑀) | |
13 | 10, 11, 12 | 3anim123i 1194 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀) → (𝑁 ∈ (ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑁 < 𝑀)) |
14 | elfzo2 12768 | . . 3 ⊢ (𝑁 ∈ (1..^𝑀) ↔ (𝑁 ∈ (ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑁 < 𝑀)) | |
15 | 13, 14 | sylibr 226 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀) → 𝑁 ∈ (1..^𝑀)) |
16 | 7, 15 | impbii 201 | 1 ⊢ (𝑁 ∈ (1..^𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ w3a 1111 ∈ wcel 2164 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 1c1 10253 < clt 10391 ℕcn 11350 ℤcz 11704 ℤ≥cuz 11968 ..^cfzo 12760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 |
This theorem is referenced by: modfzo0difsn 13037 modsumfzodifsn 13038 cshwshashlem1 16168 cshwshashlem2 16169 pthdivtx 27031 pthdlem2lem 27069 crctcshwlkn0lem3 27111 crctcshwlkn0lem4 27112 crctcshwlkn0lem5 27113 crctcshwlkn0lem6 27114 crctcshwlkn0lem7 27115 clwwisshclwwslem 27352 fiunelros 30771 iccpartlt 42241 bgoldbtbndlem4 42519 |
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