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Mirrors > Home > MPE Home > Th. List > mulsucdiv2z | Structured version Visualization version GIF version |
Description: An integer multiplied with its successor divided by 2 yields an integer, i.e. an integer multiplied with its successor is even. (Contributed by AV, 19-Jul-2021.) |
Ref | Expression |
---|---|
mulsucdiv2z | ⊢ (𝑁 ∈ ℤ → ((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zeo 12681 | . 2 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) | |
2 | peano2z 12636 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
3 | zmulcl 12644 | . . . . . 6 ⊢ (((𝑁 / 2) ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) → ((𝑁 / 2) · (𝑁 + 1)) ∈ ℤ) | |
4 | 2, 3 | sylan2 591 | . . . . 5 ⊢ (((𝑁 / 2) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 / 2) · (𝑁 + 1)) ∈ ℤ) |
5 | zcn 12596 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
6 | 2 | zcnd 12700 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℂ) |
7 | 2cnne0 12455 | . . . . . . . . 9 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
8 | 7 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
9 | div23 11924 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((𝑁 · (𝑁 + 1)) / 2) = ((𝑁 / 2) · (𝑁 + 1))) | |
10 | 5, 6, 8, 9 | syl3anc 1368 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → ((𝑁 · (𝑁 + 1)) / 2) = ((𝑁 / 2) · (𝑁 + 1))) |
11 | 10 | eleq1d 2810 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ ↔ ((𝑁 / 2) · (𝑁 + 1)) ∈ ℤ)) |
12 | 11 | adantl 480 | . . . . 5 ⊢ (((𝑁 / 2) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ ↔ ((𝑁 / 2) · (𝑁 + 1)) ∈ ℤ)) |
13 | 4, 12 | mpbird 256 | . . . 4 ⊢ (((𝑁 / 2) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ) |
14 | 13 | ex 411 | . . 3 ⊢ ((𝑁 / 2) ∈ ℤ → (𝑁 ∈ ℤ → ((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ)) |
15 | zmulcl 12644 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (𝑁 · ((𝑁 + 1) / 2)) ∈ ℤ) | |
16 | 15 | ancoms 457 | . . . . 5 ⊢ ((((𝑁 + 1) / 2) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 · ((𝑁 + 1) / 2)) ∈ ℤ) |
17 | divass 11923 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ (𝑁 + 1) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((𝑁 · (𝑁 + 1)) / 2) = (𝑁 · ((𝑁 + 1) / 2))) | |
18 | 5, 6, 8, 17 | syl3anc 1368 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → ((𝑁 · (𝑁 + 1)) / 2) = (𝑁 · ((𝑁 + 1) / 2))) |
19 | 18 | eleq1d 2810 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ ↔ (𝑁 · ((𝑁 + 1) / 2)) ∈ ℤ)) |
20 | 19 | adantl 480 | . . . . 5 ⊢ ((((𝑁 + 1) / 2) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ ↔ (𝑁 · ((𝑁 + 1) / 2)) ∈ ℤ)) |
21 | 16, 20 | mpbird 256 | . . . 4 ⊢ ((((𝑁 + 1) / 2) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ) |
22 | 21 | ex 411 | . . 3 ⊢ (((𝑁 + 1) / 2) ∈ ℤ → (𝑁 ∈ ℤ → ((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ)) |
23 | 14, 22 | jaoi 855 | . 2 ⊢ (((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ) → (𝑁 ∈ ℤ → ((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ)) |
24 | 1, 23 | mpcom 38 | 1 ⊢ (𝑁 ∈ ℤ → ((𝑁 · (𝑁 + 1)) / 2) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 (class class class)co 7419 ℂcc 11138 0cc0 11140 1c1 11141 + caddc 11143 · cmul 11145 / cdiv 11903 2c2 12300 ℤcz 12591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-n0 12506 df-z 12592 |
This theorem is referenced by: sqoddm1div8z 16334 |
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