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Mirrors > Home > MPE Home > Th. List > even2n | Structured version Visualization version GIF version |
Description: An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.) |
Ref | Expression |
---|---|
even2n | ⊢ (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evenelz 16333 | . 2 ⊢ (2 ∥ 𝑁 → 𝑁 ∈ ℤ) | |
2 | 2z 12641 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → 2 ∈ ℤ) |
4 | id 22 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℤ) | |
5 | 3, 4 | zmulcld 12719 | . . . . 5 ⊢ (𝑛 ∈ ℤ → (2 · 𝑛) ∈ ℤ) |
6 | 5 | adantr 479 | . . . 4 ⊢ ((𝑛 ∈ ℤ ∧ (2 · 𝑛) = 𝑁) → (2 · 𝑛) ∈ ℤ) |
7 | eleq1 2813 | . . . . 5 ⊢ ((2 · 𝑛) = 𝑁 → ((2 · 𝑛) ∈ ℤ ↔ 𝑁 ∈ ℤ)) | |
8 | 7 | adantl 480 | . . . 4 ⊢ ((𝑛 ∈ ℤ ∧ (2 · 𝑛) = 𝑁) → ((2 · 𝑛) ∈ ℤ ↔ 𝑁 ∈ ℤ)) |
9 | 6, 8 | mpbid 231 | . . 3 ⊢ ((𝑛 ∈ ℤ ∧ (2 · 𝑛) = 𝑁) → 𝑁 ∈ ℤ) |
10 | 9 | rexlimiva 3136 | . 2 ⊢ (∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁 → 𝑁 ∈ ℤ) |
11 | divides 16253 | . . . 4 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 2) = 𝑁)) | |
12 | zcn 12610 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ) | |
13 | 2cnd 12337 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → 2 ∈ ℂ) | |
14 | 12, 13 | mulcomd 11281 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → (𝑛 · 2) = (2 · 𝑛)) |
15 | 14 | eqeq1d 2727 | . . . . 5 ⊢ (𝑛 ∈ ℤ → ((𝑛 · 2) = 𝑁 ↔ (2 · 𝑛) = 𝑁)) |
16 | 15 | rexbiia 3081 | . . . 4 ⊢ (∃𝑛 ∈ ℤ (𝑛 · 2) = 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁) |
17 | 11, 16 | bitrdi 286 | . . 3 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁)) |
18 | 2, 17 | mpan 688 | . 2 ⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁)) |
19 | 1, 10, 18 | pm5.21nii 377 | 1 ⊢ (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 class class class wbr 5152 (class class class)co 7423 · cmul 11159 2c2 12314 ℤcz 12605 ∥ cdvds 16251 |
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 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 |
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-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-ltxr 11299 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-n0 12520 df-z 12606 df-dvds 16252 |
This theorem is referenced by: evennn02n 16347 evennn2n 16348 m1expe 16371 |
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