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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 12573 | . . . 4 ⊢ 0 ∈ ℤ | |
2 | dvds0 16219 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∥ 0) | |
3 | 1, 2 | mp1i 13 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → 0 ∥ 0) |
4 | simpr 483 | . . 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 5171 | . . . . . 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 16224 | . . . . . . 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 15653 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 = 0) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 0) |
15 | fsumdvds.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
16 | 15 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐴 ∈ Fin) |
17 | 16 | olcd 870 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐴 ⊆ (ℤ≥‘0) ∨ 𝐴 ∈ Fin)) |
18 | sumz 15672 | . . . . 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 2770 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → Σ𝑘 ∈ 𝐴 𝐵 = 0) |
21 | 3, 4, 20 | 3brtr4d 5179 | . 2 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
22 | 15 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝐴 ∈ Fin) |
23 | fsumdvds.2 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
24 | 23 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℤ) |
25 | 24 | zcnd 12671 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℂ) |
26 | 9 | adantlr 711 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) |
27 | 26 | zcnd 12671 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
28 | simpr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ≠ 0) | |
29 | 22, 25, 27, 28 | fsumdivc 15736 | . . . 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 479 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ∈ ℤ) |
32 | simplr 765 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ≠ 0) | |
33 | dvdsval2 16204 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ 𝐵 ∈ ℤ) → (𝑁 ∥ 𝐵 ↔ (𝐵 / 𝑁) ∈ ℤ)) | |
34 | 31, 32, 26, 33 | syl3anc 1369 | . . . . . 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 15685 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → Σ𝑘 ∈ 𝐴 (𝐵 / 𝑁) ∈ ℤ) |
37 | 29, 36 | eqeltrd 2831 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) ∈ ℤ) |
38 | 15, 9 | fsumzcl 15685 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
39 | 38 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
40 | dvdsval2 16204 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → (𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵 ↔ (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) ∈ ℤ)) | |
41 | 24, 28, 39, 40 | syl3anc 1369 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → (𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵 ↔ (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) ∈ ℤ)) |
42 | 37, 41 | mpbird 256 | . 2 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
43 | 21, 42 | pm2.61dane 3027 | 1 ⊢ (𝜑 → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 843 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ⊆ wss 3947 class class class wbr 5147 ‘cfv 6542 (class class class)co 7411 Fincfn 8941 0cc0 11112 / cdiv 11875 ℤcz 12562 ℤ≥cuz 12826 Σcsu 15636 ∥ cdvds 16201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-sum 15637 df-dvds 16202 |
This theorem is referenced by: 3dvds 16278 sylow1lem3 19509 sylow2alem2 19527 poimirlem26 36817 poimirlem27 36818 etransclem37 45285 etransclem38 45286 etransclem44 45292 |
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