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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 11986 | . . . 4 ⊢ 0 ∈ ℤ | |
2 | dvds0 15619 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∥ 0) | |
3 | 1, 2 | mp1i 13 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → 0 ∥ 0) |
4 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 = 0) | |
5 | simplr 767 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 = 0) | |
6 | fsumdvds.4 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑁 ∥ 𝐵) | |
7 | 6 | adantlr 713 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ∥ 𝐵) |
8 | 5, 7 | eqbrtrrd 5082 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 0 ∥ 𝐵) |
9 | fsumdvds.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) | |
10 | 9 | adantlr 713 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) |
11 | 0dvds 15624 | . . . . . . 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 234 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 = 0) |
14 | 13 | sumeq2dv 15054 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 = 0) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 0) |
15 | fsumdvds.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
16 | 15 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐴 ∈ Fin) |
17 | 16 | olcd 870 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐴 ⊆ (ℤ≥‘0) ∨ 𝐴 ∈ Fin)) |
18 | sumz 15073 | . . . . 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 2856 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → Σ𝑘 ∈ 𝐴 𝐵 = 0) |
21 | 3, 4, 20 | 3brtr4d 5090 | . 2 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
22 | 15 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝐴 ∈ Fin) |
23 | fsumdvds.2 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
24 | 23 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℤ) |
25 | 24 | zcnd 12082 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℂ) |
26 | 9 | adantlr 713 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) |
27 | 26 | zcnd 12082 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
28 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ≠ 0) | |
29 | 22, 25, 27, 28 | fsumdivc 15135 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) = Σ𝑘 ∈ 𝐴 (𝐵 / 𝑁)) |
30 | 6 | adantlr 713 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ∥ 𝐵) |
31 | 24 | adantr 483 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ∈ ℤ) |
32 | simplr 767 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ≠ 0) | |
33 | dvdsval2 15604 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ 𝐵 ∈ ℤ) → (𝑁 ∥ 𝐵 ↔ (𝐵 / 𝑁) ∈ ℤ)) | |
34 | 31, 32, 26, 33 | syl3anc 1367 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → (𝑁 ∥ 𝐵 ↔ (𝐵 / 𝑁) ∈ ℤ)) |
35 | 30, 34 | mpbid 234 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 / 𝑁) ∈ ℤ) |
36 | 22, 35 | fsumzcl 15086 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → Σ𝑘 ∈ 𝐴 (𝐵 / 𝑁) ∈ ℤ) |
37 | 29, 36 | eqeltrd 2913 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) ∈ ℤ) |
38 | 15, 9 | fsumzcl 15086 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
39 | 38 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
40 | dvdsval2 15604 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → (𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵 ↔ (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) ∈ ℤ)) | |
41 | 24, 28, 39, 40 | syl3anc 1367 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → (𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵 ↔ (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) ∈ ℤ)) |
42 | 37, 41 | mpbird 259 | . 2 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
43 | 21, 42 | pm2.61dane 3104 | 1 ⊢ (𝜑 → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3935 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 0cc0 10531 / cdiv 11291 ℤcz 11975 ℤ≥cuz 12237 Σcsu 15036 ∥ cdvds 15601 |
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 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-dvds 15602 |
This theorem is referenced by: 3dvds 15674 sylow1lem3 18719 sylow2alem2 18737 poimirlem26 34912 poimirlem27 34913 etransclem37 42550 etransclem38 42551 etransclem44 42557 |
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