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Mirrors > Home > ILE Home > Th. List > elfzelz | GIF version |
Description: A member of a finite set of sequential integer is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfzelz | ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz 9802 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) | |
2 | eluzelz 9335 | . 2 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ ℤ) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ‘cfv 5123 (class class class)co 5774 ℤcz 9054 ℤ≥cuz 9326 ...cfz 9790 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-neg 7936 df-z 9055 df-uz 9327 df-fz 9791 |
This theorem is referenced by: elfz1eq 9815 fzsplit2 9830 fzdisj 9832 elfznn 9834 fznatpl1 9856 fzdifsuc 9861 fzrev2i 9866 fzrev3i 9868 elfzp12 9879 fznuz 9882 fzrevral 9885 fzshftral 9888 fznn0sub2 9905 elfzmlbm 9908 difelfznle 9912 fzosplit 9954 ser3mono 10251 iseqf1olemkle 10257 iseqf1olemklt 10258 iseqf1olemqcl 10259 iseqf1olemnab 10261 iseqf1olemab 10262 iseqf1olemmo 10265 iseqf1olemqk 10267 seq3f1olemqsumkj 10271 seq3f1olemqsumk 10272 seq3f1olemqsum 10273 seq3f1olemstep 10274 bcval2 10496 bcval4 10498 bccmpl 10500 bcp1nk 10508 bcpasc 10512 bccl2 10514 zfz1isolemiso 10582 seq3coll 10585 seq3shft 10610 sumrbdclem 11146 summodclem2a 11150 fsum0diaglem 11209 fisum0diag 11210 mptfzshft 11211 fsumrev 11212 fsumshft 11213 fsumshftm 11214 fisum0diag2 11216 binomlem 11252 binom11 11255 bcxmas 11258 arisum 11267 geo2sum 11283 cvgratnnlemabsle 11296 cvgratnnlemrate 11299 mertenslemub 11303 mertenslemi1 11304 prodfap0 11314 prodrbdclem 11340 prodmodclem2a 11345 fzm1ndvds 11554 lcmval 11744 lcmcllem 11748 lcmledvds 11751 prmdvdsfz 11819 phivalfi 11888 hashdvds 11897 phiprmpw 11898 trilpolemlt1 13234 supfz 13237 inffz 13238 |
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