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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsummmodsnunz | Structured version Visualization version GIF version |
Description: A finite sum of summands modulo a positive number with an additional summand is an integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.) |
Ref | Expression |
---|---|
fsummmodsnunz | ⊢ ((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑧})(𝐵 mod 𝑁) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2977 | . . 3 ⊢ Ⅎ𝑥(𝐵 mod 𝑁) | |
2 | nfcsb1v 3906 | . . 3 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌(𝐵 mod 𝑁) | |
3 | csbeq1a 3896 | . . 3 ⊢ (𝑘 = 𝑥 → (𝐵 mod 𝑁) = ⦋𝑥 / 𝑘⦌(𝐵 mod 𝑁)) | |
4 | 1, 2, 3 | cbvsumi 15053 | . 2 ⊢ Σ𝑘 ∈ (𝐴 ∪ {𝑧})(𝐵 mod 𝑁) = Σ𝑥 ∈ (𝐴 ∪ {𝑧})⦋𝑥 / 𝑘⦌(𝐵 mod 𝑁) |
5 | snfi 8593 | . . . . 5 ⊢ {𝑧} ∈ Fin | |
6 | unfi 8784 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin) | |
7 | 5, 6 | mpan2 689 | . . . 4 ⊢ (𝐴 ∈ Fin → (𝐴 ∪ {𝑧}) ∈ Fin) |
8 | 7 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → (𝐴 ∪ {𝑧}) ∈ Fin) |
9 | rspcsbela 4386 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ∪ {𝑧}) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → ⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ) | |
10 | 9 | expcom 416 | . . . . . 6 ⊢ (∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ → (𝑥 ∈ (𝐴 ∪ {𝑧}) → ⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ)) |
11 | 10 | 3ad2ant3 1131 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → (𝑥 ∈ (𝐴 ∪ {𝑧}) → ⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ)) |
12 | 11 | imp 409 | . . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) ∧ 𝑥 ∈ (𝐴 ∪ {𝑧})) → ⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ) |
13 | vex 3497 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
14 | csbov1g 7200 | . . . . . . . . 9 ⊢ (𝑥 ∈ V → ⦋𝑥 / 𝑘⦌(𝐵 mod 𝑁) = (⦋𝑥 / 𝑘⦌𝐵 mod 𝑁)) | |
15 | 13, 14 | ax-mp 5 | . . . . . . . 8 ⊢ ⦋𝑥 / 𝑘⦌(𝐵 mod 𝑁) = (⦋𝑥 / 𝑘⦌𝐵 mod 𝑁) |
16 | simpr 487 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ ⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ) → ⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ) | |
17 | simpl 485 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ ⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ) → 𝑁 ∈ ℕ) | |
18 | 16, 17 | zmodcld 13259 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ) → (⦋𝑥 / 𝑘⦌𝐵 mod 𝑁) ∈ ℕ0) |
19 | 18 | nn0zd 12084 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ ⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ) → (⦋𝑥 / 𝑘⦌𝐵 mod 𝑁) ∈ ℤ) |
20 | 15, 19 | eqeltrid 2917 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ ⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ) → ⦋𝑥 / 𝑘⦌(𝐵 mod 𝑁) ∈ ℤ) |
21 | 20 | ex 415 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ → ⦋𝑥 / 𝑘⦌(𝐵 mod 𝑁) ∈ ℤ)) |
22 | 21 | 3ad2ant2 1130 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → (⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ → ⦋𝑥 / 𝑘⦌(𝐵 mod 𝑁) ∈ ℤ)) |
23 | 22 | adantr 483 | . . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) ∧ 𝑥 ∈ (𝐴 ∪ {𝑧})) → (⦋𝑥 / 𝑘⦌𝐵 ∈ ℤ → ⦋𝑥 / 𝑘⦌(𝐵 mod 𝑁) ∈ ℤ)) |
24 | 12, 23 | mpd 15 | . . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) ∧ 𝑥 ∈ (𝐴 ∪ {𝑧})) → ⦋𝑥 / 𝑘⦌(𝐵 mod 𝑁) ∈ ℤ) |
25 | 8, 24 | fsumzcl 15091 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → Σ𝑥 ∈ (𝐴 ∪ {𝑧})⦋𝑥 / 𝑘⦌(𝐵 mod 𝑁) ∈ ℤ) |
26 | 4, 25 | eqeltrid 2917 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝑁 ∈ ℕ ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑧})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑧})(𝐵 mod 𝑁) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⦋csb 3882 ∪ cun 3933 {csn 4566 (class class class)co 7155 Fincfn 8508 ℕcn 11637 ℤcz 11980 mod cmo 13236 Σcsu 15041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-sum 15042 |
This theorem is referenced by: (None) |
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