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Mirrors > Home > MPE Home > Th. List > fsumzcl | Structured version Visualization version 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 11712 | . . 3 ⊢ ℤ ⊆ ℂ | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ℤ ⊆ ℂ) |
3 | zaddcl 11745 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) ∈ ℤ) | |
4 | 3 | adantl 475 | . 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 11716 | . 2 ⊢ (𝜑 → 0 ∈ ℤ) | |
8 | 2, 4, 5, 6, 7 | fsumcllem 14840 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 ⊆ wss 3798 (class class class)co 6905 Fincfn 8222 ℂcc 10250 + caddc 10255 ℤcz 11704 Σcsu 14793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 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 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 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-rmo 3125 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-int 4698 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-se 5302 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-isom 6132 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-1o 7826 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-fz 12620 df-fzo 12761 df-seq 13096 df-exp 13155 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-clim 14596 df-sum 14794 |
This theorem is referenced by: fsumzcl2 14846 fsummsnunz 14860 eirrlem 15306 fsumdvds 15407 3dvds 15429 sumeven 15484 sumodd 15485 aalioulem1 24486 aaliou3lem6 24502 sgmnncl 25286 mersenne 25365 lgseisenlem4 25516 lgseisen 25517 lgsquadlem1 25518 vtxdgoddnumeven 26851 jm2.22 38405 jm2.23 38406 etransclem27 41272 etransclem36 41281 etransclem44 41289 etransclem45 41290 fsummmodsnunz 42233 2pwp1prm 42333 lighneallem3 42354 lighneallem4b 42356 lighneallem4 42357 |
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