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Mirrors > Home > ILE Home > Th. List > fsumzcl | GIF version |
Description: Closure of a finite sum of integers. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.) |
Ref | Expression |
---|---|
fsumcl.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumzcl.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) |
Ref | Expression |
---|---|
fsumzcl | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zsscn 9232 | . . 3 ⊢ ℤ ⊆ ℂ | |
2 | 1 | a1i 9 | . 2 ⊢ (𝜑 → ℤ ⊆ ℂ) |
3 | zaddcl 9264 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) ∈ ℤ) | |
4 | 3 | adantl 277 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑥 + 𝑦) ∈ ℤ) |
5 | fsumcl.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
6 | fsumzcl.2 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) | |
7 | 0zd 9236 | . 2 ⊢ (𝜑 → 0 ∈ ℤ) | |
8 | 2, 4, 5, 6, 7 | fsumcllem 11374 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2146 ⊆ wss 3127 (class class class)co 5865 Fincfn 6730 ℂcc 7784 + caddc 7789 ℤcz 9224 Σcsu 11328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-isom 5217 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-irdg 6361 df-frec 6382 df-1o 6407 df-oadd 6411 df-er 6525 df-en 6731 df-dom 6732 df-fin 6733 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8602 df-inn 8891 df-2 8949 df-3 8950 df-4 8951 df-n0 9148 df-z 9225 df-uz 9500 df-q 9591 df-rp 9623 df-fz 9978 df-fzo 10111 df-seqfrec 10414 df-exp 10488 df-ihash 10722 df-cj 10818 df-re 10819 df-im 10820 df-rsqrt 10974 df-abs 10975 df-clim 11254 df-sumdc 11329 |
This theorem is referenced by: fsumzcl2 11380 eirraplem 11751 |
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