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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 13684 | . . . 4 ⊢ (1..^𝑀) ⊆ ℕ | |
2 | 1 | sseli 3973 | . . 3 ⊢ (𝑁 ∈ (1..^𝑀) → 𝑁 ∈ ℕ) |
3 | elfzouz2 13650 | . . . 4 ⊢ (𝑁 ∈ (1..^𝑀) → 𝑀 ∈ (ℤ≥‘𝑁)) | |
4 | eluznn 12903 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ) | |
5 | 2, 3, 4 | syl2anc 583 | . . 3 ⊢ (𝑁 ∈ (1..^𝑀) → 𝑀 ∈ ℕ) |
6 | elfzolt2 13644 | . . 3 ⊢ (𝑁 ∈ (1..^𝑀) → 𝑁 < 𝑀) | |
7 | 2, 5, 6 | 3jca 1125 | . 2 ⊢ (𝑁 ∈ (1..^𝑀) → (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀)) |
8 | nnuz 12866 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
9 | 8 | eqimssi 4037 | . . . . 5 ⊢ ℕ ⊆ (ℤ≥‘1) |
10 | 9 | sseli 3973 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1)) |
11 | nnz 12580 | . . . 4 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
12 | id 22 | . . . 4 ⊢ (𝑁 < 𝑀 → 𝑁 < 𝑀) | |
13 | 10, 11, 12 | 3anim123i 1148 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀) → (𝑁 ∈ (ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑁 < 𝑀)) |
14 | elfzo2 13638 | . . 3 ⊢ (𝑁 ∈ (1..^𝑀) ↔ (𝑁 ∈ (ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑁 < 𝑀)) | |
15 | 13, 14 | sylibr 233 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀) → 𝑁 ∈ (1..^𝑀)) |
16 | 7, 15 | impbii 208 | 1 ⊢ (𝑁 ∈ (1..^𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5141 ‘cfv 6536 (class class class)co 7404 1c1 11110 < clt 11249 ℕcn 12213 ℤcz 12559 ℤ≥cuz 12823 ..^cfzo 13630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 |
This theorem is referenced by: modfzo0difsn 13911 modsumfzodifsn 13912 cshwshashlem1 17036 cshwshashlem2 17037 pthdivtx 29491 pthdlem2lem 29529 crctcshwlkn0lem3 29571 crctcshwlkn0lem4 29572 crctcshwlkn0lem5 29573 crctcshwlkn0lem6 29574 crctcshwlkn0lem7 29575 clwwisshclwwslem 29772 fiunelros 33702 iccpartlt 46645 bgoldbtbndlem4 47029 |
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