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Mirrors > Home > MPE Home > Th. List > fsumdvds | Structured version Visualization version GIF version |
Description: If every term in a sum is divisible by 𝑁, then so is the sum. (Contributed by Mario Carneiro, 17-Jan-2015.) |
Ref | Expression |
---|---|
fsumdvds.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumdvds.2 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
fsumdvds.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) |
fsumdvds.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑁 ∥ 𝐵) |
Ref | Expression |
---|---|
fsumdvds | ⊢ (𝜑 → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 11980 | . . . 4 ⊢ 0 ∈ ℤ | |
2 | dvds0 15613 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∥ 0) | |
3 | 1, 2 | mp1i 13 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → 0 ∥ 0) |
4 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 = 0) | |
5 | simplr 765 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 = 0) | |
6 | fsumdvds.4 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑁 ∥ 𝐵) | |
7 | 6 | adantlr 711 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ∥ 𝐵) |
8 | 5, 7 | eqbrtrrd 5081 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 0 ∥ 𝐵) |
9 | fsumdvds.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) | |
10 | 9 | adantlr 711 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) |
11 | 0dvds 15618 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → (0 ∥ 𝐵 ↔ 𝐵 = 0)) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → (0 ∥ 𝐵 ↔ 𝐵 = 0)) |
13 | 8, 12 | mpbid 233 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 = 0) |
14 | 13 | sumeq2dv 15048 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 = 0) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 0) |
15 | fsumdvds.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
16 | 15 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐴 ∈ Fin) |
17 | 16 | olcd 870 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐴 ⊆ (ℤ≥‘0) ∨ 𝐴 ∈ Fin)) |
18 | sumz 15067 | . . . . 5 ⊢ ((𝐴 ⊆ (ℤ≥‘0) ∨ 𝐴 ∈ Fin) → Σ𝑘 ∈ 𝐴 0 = 0) | |
19 | 17, 18 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 = 0) → Σ𝑘 ∈ 𝐴 0 = 0) |
20 | 14, 19 | eqtrd 2853 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → Σ𝑘 ∈ 𝐴 𝐵 = 0) |
21 | 3, 4, 20 | 3brtr4d 5089 | . 2 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
22 | 15 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝐴 ∈ Fin) |
23 | fsumdvds.2 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
24 | 23 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℤ) |
25 | 24 | zcnd 12076 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℂ) |
26 | 9 | adantlr 711 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) |
27 | 26 | zcnd 12076 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
28 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ≠ 0) | |
29 | 22, 25, 27, 28 | fsumdivc 15129 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) = Σ𝑘 ∈ 𝐴 (𝐵 / 𝑁)) |
30 | 6 | adantlr 711 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ∥ 𝐵) |
31 | 24 | adantr 481 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ∈ ℤ) |
32 | simplr 765 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ≠ 0) | |
33 | dvdsval2 15598 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ 𝐵 ∈ ℤ) → (𝑁 ∥ 𝐵 ↔ (𝐵 / 𝑁) ∈ ℤ)) | |
34 | 31, 32, 26, 33 | syl3anc 1363 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → (𝑁 ∥ 𝐵 ↔ (𝐵 / 𝑁) ∈ ℤ)) |
35 | 30, 34 | mpbid 233 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 / 𝑁) ∈ ℤ) |
36 | 22, 35 | fsumzcl 15080 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → Σ𝑘 ∈ 𝐴 (𝐵 / 𝑁) ∈ ℤ) |
37 | 29, 36 | eqeltrd 2910 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) ∈ ℤ) |
38 | 15, 9 | fsumzcl 15080 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
39 | 38 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
40 | dvdsval2 15598 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → (𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵 ↔ (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) ∈ ℤ)) | |
41 | 24, 28, 39, 40 | syl3anc 1363 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → (𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵 ↔ (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) ∈ ℤ)) |
42 | 37, 41 | mpbird 258 | . 2 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
43 | 21, 42 | pm2.61dane 3101 | 1 ⊢ (𝜑 → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ⊆ wss 3933 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 0cc0 10525 / cdiv 11285 ℤcz 11969 ℤ≥cuz 12231 Σcsu 15030 ∥ cdvds 15595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 df-dvds 15596 |
This theorem is referenced by: 3dvds 15668 sylow1lem3 18654 sylow2alem2 18672 poimirlem26 34799 poimirlem27 34800 etransclem37 42433 etransclem38 42434 etransclem44 42440 |
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