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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 12444 | . . . 4 ⊢ 0 ∈ ℤ | |
2 | dvds0 16089 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∥ 0) | |
3 | 1, 2 | mp1i 13 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → 0 ∥ 0) |
4 | simpr 486 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 = 0) | |
5 | simplr 768 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 = 0) | |
6 | fsumdvds.4 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑁 ∥ 𝐵) | |
7 | 6 | adantlr 714 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ∥ 𝐵) |
8 | 5, 7 | eqbrtrrd 5128 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 0 ∥ 𝐵) |
9 | fsumdvds.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) | |
10 | 9 | adantlr 714 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) |
11 | 0dvds 16094 | . . . . . . 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 231 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 = 0) |
14 | 13 | sumeq2dv 15523 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 = 0) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 0) |
15 | fsumdvds.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
16 | 15 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐴 ∈ Fin) |
17 | 16 | olcd 873 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐴 ⊆ (ℤ≥‘0) ∨ 𝐴 ∈ Fin)) |
18 | sumz 15542 | . . . . 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 2778 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → Σ𝑘 ∈ 𝐴 𝐵 = 0) |
21 | 3, 4, 20 | 3brtr4d 5136 | . 2 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
22 | 15 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝐴 ∈ Fin) |
23 | fsumdvds.2 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
24 | 23 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℤ) |
25 | 24 | zcnd 12541 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℂ) |
26 | 9 | adantlr 714 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) |
27 | 26 | zcnd 12541 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
28 | simpr 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ≠ 0) | |
29 | 22, 25, 27, 28 | fsumdivc 15606 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) = Σ𝑘 ∈ 𝐴 (𝐵 / 𝑁)) |
30 | 6 | adantlr 714 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ∥ 𝐵) |
31 | 24 | adantr 482 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ∈ ℤ) |
32 | simplr 768 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ≠ 0) | |
33 | dvdsval2 16074 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ 𝐵 ∈ ℤ) → (𝑁 ∥ 𝐵 ↔ (𝐵 / 𝑁) ∈ ℤ)) | |
34 | 31, 32, 26, 33 | syl3anc 1372 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → (𝑁 ∥ 𝐵 ↔ (𝐵 / 𝑁) ∈ ℤ)) |
35 | 30, 34 | mpbid 231 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → (𝐵 / 𝑁) ∈ ℤ) |
36 | 22, 35 | fsumzcl 15555 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → Σ𝑘 ∈ 𝐴 (𝐵 / 𝑁) ∈ ℤ) |
37 | 29, 36 | eqeltrd 2839 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) ∈ ℤ) |
38 | 15, 9 | fsumzcl 15555 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
39 | 38 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
40 | dvdsval2 16074 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → (𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵 ↔ (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) ∈ ℤ)) | |
41 | 24, 28, 39, 40 | syl3anc 1372 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → (𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵 ↔ (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) ∈ ℤ)) |
42 | 37, 41 | mpbird 257 | . 2 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
43 | 21, 42 | pm2.61dane 3031 | 1 ⊢ (𝜑 → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ⊆ wss 3909 class class class wbr 5104 ‘cfv 6492 (class class class)co 7350 Fincfn 8817 0cc0 10985 / cdiv 11746 ℤcz 12433 ℤ≥cuz 12696 Σcsu 15505 ∥ cdvds 16071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-inf2 9511 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-sup 9312 df-oi 9380 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-n0 12348 df-z 12434 df-uz 12697 df-rp 12845 df-fz 13354 df-fzo 13497 df-seq 13836 df-exp 13897 df-hash 14159 df-cj 14918 df-re 14919 df-im 14920 df-sqrt 15054 df-abs 15055 df-clim 15305 df-sum 15506 df-dvds 16072 |
This theorem is referenced by: 3dvds 16148 sylow1lem3 19311 sylow2alem2 19329 poimirlem26 35990 poimirlem27 35991 etransclem37 44234 etransclem38 44235 etransclem44 44241 |
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