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Mirrors > Home > MPE Home > Th. List > fznnfl | Structured version Visualization version GIF version |
Description: Finite set of sequential integers starting at 1 and ending at a real number. (Contributed by Mario Carneiro, 3-May-2016.) |
Ref | Expression |
---|---|
fznnfl | ⊢ (𝑁 ∈ ℝ → (𝐾 ∈ (1...(⌊‘𝑁)) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flcl 12979 | . . 3 ⊢ (𝑁 ∈ ℝ → (⌊‘𝑁) ∈ ℤ) | |
2 | fznn 12790 | . . 3 ⊢ ((⌊‘𝑁) ∈ ℤ → (𝐾 ∈ (1...(⌊‘𝑁)) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ (⌊‘𝑁)))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑁 ∈ ℝ → (𝐾 ∈ (1...(⌊‘𝑁)) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ (⌊‘𝑁)))) |
4 | nnz 11816 | . . . 4 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℤ) | |
5 | flge 12989 | . . . 4 ⊢ ((𝑁 ∈ ℝ ∧ 𝐾 ∈ ℤ) → (𝐾 ≤ 𝑁 ↔ 𝐾 ≤ (⌊‘𝑁))) | |
6 | 4, 5 | sylan2 584 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ 𝐾 ∈ ℕ) → (𝐾 ≤ 𝑁 ↔ 𝐾 ≤ (⌊‘𝑁))) |
7 | 6 | pm5.32da 571 | . 2 ⊢ (𝑁 ∈ ℝ → ((𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ (⌊‘𝑁)))) |
8 | 3, 7 | bitr4d 274 | 1 ⊢ (𝑁 ∈ ℝ → (𝐾 ∈ (1...(⌊‘𝑁)) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∈ wcel 2051 class class class wbr 4926 ‘cfv 6186 (class class class)co 6975 ℝcr 10333 1c1 10335 ≤ cle 10474 ℕcn 11438 ℤcz 11792 ...cfz 12707 ⌊cfl 12974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-sup 8700 df-inf 8701 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-n0 11707 df-z 11793 df-uz 12058 df-fz 12708 df-fl 12976 |
This theorem is referenced by: o1fsum 15027 fsumharmonic 25307 dvdsflf1o 25482 dvdsflsumcom 25483 fsumfldivdiaglem 25484 dchrisum0lem1a 25780 dchrmusum2 25788 dchrvmasumlem1 25789 dchrvmasum2lem 25790 dchrvmasumlem2 25792 dchrisum0fno1 25805 dchrisum0lem1b 25809 dchrisum0lem1 25810 mulog2sumlem2 25829 vmalogdivsum2 25832 2vmadivsumlem 25834 selberglem2 25840 selberg3lem1 25851 selberg4lem1 25854 |
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