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Mirrors > Home > MPE Home > Th. List > fsumnn0cl | Structured version Visualization version GIF version |
Description: Closure of a finite sum of nonnegative integers. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
fsumcl.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumnn0cl.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℕ0) |
Ref | Expression |
---|---|
fsumnn0cl | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0sscn 11739 | . . 3 ⊢ ℕ0 ⊆ ℂ | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ℕ0 ⊆ ℂ) |
3 | nn0addcl 11769 | . . 3 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 + 𝑦) ∈ ℕ0) | |
4 | 3 | adantl 482 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)) → (𝑥 + 𝑦) ∈ ℕ0) |
5 | fsumcl.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
6 | fsumnn0cl.2 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℕ0) | |
7 | 0nn0 11749 | . . 3 ⊢ 0 ∈ ℕ0 | |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ ℕ0) |
9 | 2, 4, 5, 6, 8 | fsumcllem 14910 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2079 ⊆ wss 3854 (class class class)co 7007 Fincfn 8347 ℂcc 10370 0cc0 10372 + caddc 10375 ℕ0cn0 11734 Σcsu 14864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-inf2 8939 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-fal 1533 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-sup 8742 df-oi 8810 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-n0 11735 df-z 11819 df-uz 12083 df-rp 12229 df-fz 12732 df-fzo 12873 df-seq 13208 df-exp 13268 df-hash 13529 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-clim 14667 df-sum 14865 |
This theorem is referenced by: bitsinv2 15613 bitsf1ocnv 15614 ramcl 16182 lgsquadlem2 25627 eulerpartlemsf 31190 eulerpartlemb 31199 signsvvf 31422 fsumnncl 41348 mccllem 41374 etransclem35 42050 |
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