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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 12574 | . . . 4 ⊢ 0 ∈ ℤ | |
2 | dvds0 16220 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∥ 0) | |
3 | 1, 2 | mp1i 13 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → 0 ∥ 0) |
4 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 = 0) | |
5 | simplr 766 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 = 0) | |
6 | fsumdvds.4 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑁 ∥ 𝐵) | |
7 | 6 | adantlr 712 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ∥ 𝐵) |
8 | 5, 7 | eqbrtrrd 5172 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 0 ∥ 𝐵) |
9 | fsumdvds.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) | |
10 | 9 | adantlr 712 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 = 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) |
11 | 0dvds 16225 | . . . . . . 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 15654 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 = 0) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 0) |
15 | fsumdvds.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
16 | 15 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐴 ∈ Fin) |
17 | 16 | olcd 871 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝐴 ⊆ (ℤ≥‘0) ∨ 𝐴 ∈ Fin)) |
18 | sumz 15673 | . . . . 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 2771 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → Σ𝑘 ∈ 𝐴 𝐵 = 0) |
21 | 3, 4, 20 | 3brtr4d 5180 | . 2 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
22 | 15 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝐴 ∈ Fin) |
23 | fsumdvds.2 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
24 | 23 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℤ) |
25 | 24 | zcnd 12672 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℂ) |
26 | 9 | adantlr 712 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℤ) |
27 | 26 | zcnd 12672 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
28 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → 𝑁 ≠ 0) | |
29 | 22, 25, 27, 28 | fsumdivc 15737 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) = Σ𝑘 ∈ 𝐴 (𝐵 / 𝑁)) |
30 | 6 | adantlr 712 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ∥ 𝐵) |
31 | 24 | adantr 480 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ∈ ℤ) |
32 | simplr 766 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 ≠ 0) ∧ 𝑘 ∈ 𝐴) → 𝑁 ≠ 0) | |
33 | dvdsval2 16205 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ 𝐵 ∈ ℤ) → (𝑁 ∥ 𝐵 ↔ (𝐵 / 𝑁) ∈ ℤ)) | |
34 | 31, 32, 26, 33 | syl3anc 1370 | . . . . . 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 15686 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → Σ𝑘 ∈ 𝐴 (𝐵 / 𝑁) ∈ ℤ) |
37 | 29, 36 | eqeltrd 2832 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) ∈ ℤ) |
38 | 15, 9 | fsumzcl 15686 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
39 | 38 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ≠ 0) → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) |
40 | dvdsval2 16205 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0 ∧ Σ𝑘 ∈ 𝐴 𝐵 ∈ ℤ) → (𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵 ↔ (Σ𝑘 ∈ 𝐴 𝐵 / 𝑁) ∈ ℤ)) | |
41 | 24, 28, 39, 40 | syl3anc 1370 | . . 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 3028 | 1 ⊢ (𝜑 → 𝑁 ∥ Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ⊆ wss 3948 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 Fincfn 8943 0cc0 11114 / cdiv 11876 ℤcz 12563 ℤ≥cuz 12827 Σcsu 15637 ∥ cdvds 16202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 df-dvds 16203 |
This theorem is referenced by: 3dvds 16279 sylow1lem3 19510 sylow2alem2 19528 poimirlem26 36818 poimirlem27 36819 etransclem37 45286 etransclem38 45287 etransclem44 45293 |
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